I am done

rl/Lib/site-packages/sympy/integrals/__init__.py

@@ -0,0 +1,45 @@
+"""Integration functions that integrate a SymPy expression.
+
+    Examples
+    ========
+
+    >>> from sympy import integrate, sin
+    >>> from sympy.abc import x
+    >>> integrate(1/x,x)
+    log(x)
+    >>> integrate(sin(x),x)
+    -cos(x)
+"""
+from .integrals import integrate, Integral, line_integrate
+from .transforms import (mellin_transform, inverse_mellin_transform,
+                        MellinTransform, InverseMellinTransform,
+                        laplace_transform, inverse_laplace_transform,
+                        laplace_correspondence, laplace_initial_conds,
+                        LaplaceTransform, InverseLaplaceTransform,
+                        fourier_transform, inverse_fourier_transform,
+                        FourierTransform, InverseFourierTransform,
+                        sine_transform, inverse_sine_transform,
+                        SineTransform, InverseSineTransform,
+                        cosine_transform, inverse_cosine_transform,
+                        CosineTransform, InverseCosineTransform,
+                        hankel_transform, inverse_hankel_transform,
+                        HankelTransform, InverseHankelTransform)
+from .singularityfunctions import singularityintegrate
+
+__all__ = [
+    'integrate', 'Integral', 'line_integrate',
+
+    'mellin_transform', 'inverse_mellin_transform', 'MellinTransform',
+    'InverseMellinTransform', 'laplace_transform',
+    'inverse_laplace_transform', 'LaplaceTransform',
+    'laplace_correspondence', 'laplace_initial_conds',
+    'InverseLaplaceTransform', 'fourier_transform',
+    'inverse_fourier_transform', 'FourierTransform',
+    'InverseFourierTransform', 'sine_transform', 'inverse_sine_transform',
+    'SineTransform', 'InverseSineTransform', 'cosine_transform',
+    'inverse_cosine_transform', 'CosineTransform', 'InverseCosineTransform',
+    'hankel_transform', 'inverse_hankel_transform', 'HankelTransform',
+    'InverseHankelTransform',
+
+    'singularityintegrate',
+]

rl/Lib/site-packages/sympy/integrals/benchmarks/bench_integrate.py

@@ -0,0 +1,21 @@
+from sympy.core.symbol import Symbol
+from sympy.functions.elementary.trigonometric import sin
+from sympy.integrals.integrals import integrate
+
+x = Symbol('x')
+
+
+def bench_integrate_sin():
+    integrate(sin(x), x)
+
+
+def bench_integrate_x1sin():
+    integrate(x**1*sin(x), x)
+
+
+def bench_integrate_x2sin():
+    integrate(x**2*sin(x), x)
+
+
+def bench_integrate_x3sin():
+    integrate(x**3*sin(x), x)

rl/Lib/site-packages/sympy/integrals/benchmarks/bench_trigintegrate.py

@@ -0,0 +1,13 @@
+from sympy.core.symbol import Symbol
+from sympy.functions.elementary.trigonometric import sin
+from sympy.integrals.trigonometry import trigintegrate
+
+x = Symbol('x')
+
+
+def timeit_trigintegrate_sin3x():
+    trigintegrate(sin(x)**3, x)
+
+
+def timeit_trigintegrate_x2():
+    trigintegrate(x**2, x)  # -> None

rl/Lib/site-packages/sympy/integrals/deltafunctions.py

@@ -0,0 +1,201 @@
+from sympy.core.mul import Mul
+from sympy.core.singleton import S
+from sympy.core.sorting import default_sort_key
+from sympy.functions import DiracDelta, Heaviside
+from .integrals import Integral, integrate
+
+
+def change_mul(node, x):
+    """change_mul(node, x)
+
+       Rearranges the operands of a product, bringing to front any simple
+       DiracDelta expression.
+
+       Explanation
+       ===========
+
+       If no simple DiracDelta expression was found, then all the DiracDelta
+       expressions are simplified (using DiracDelta.expand(diracdelta=True, wrt=x)).
+
+       Return: (dirac, new node)
+       Where:
+         o dirac is either a simple DiracDelta expression or None (if no simple
+           expression was found);
+         o new node is either a simplified DiracDelta expressions or None (if it
+           could not be simplified).
+
+       Examples
+       ========
+
+       >>> from sympy import DiracDelta, cos
+       >>> from sympy.integrals.deltafunctions import change_mul
+       >>> from sympy.abc import x, y
+       >>> change_mul(x*y*DiracDelta(x)*cos(x), x)
+       (DiracDelta(x), x*y*cos(x))
+       >>> change_mul(x*y*DiracDelta(x**2 - 1)*cos(x), x)
+       (None, x*y*cos(x)*DiracDelta(x - 1)/2 + x*y*cos(x)*DiracDelta(x + 1)/2)
+       >>> change_mul(x*y*DiracDelta(cos(x))*cos(x), x)
+       (None, None)
+
+       See Also
+       ========
+
+       sympy.functions.special.delta_functions.DiracDelta
+       deltaintegrate
+    """
+
+    new_args = []
+    dirac = None
+
+    #Sorting is needed so that we consistently collapse the same delta;
+    #However, we must preserve the ordering of non-commutative terms
+    c, nc = node.args_cnc()
+    sorted_args = sorted(c, key=default_sort_key)
+    sorted_args.extend(nc)
+
+    for arg in sorted_args:
+        if arg.is_Pow and isinstance(arg.base, DiracDelta):
+            new_args.append(arg.func(arg.base, arg.exp - 1))
+            arg = arg.base
+        if dirac is None and (isinstance(arg, DiracDelta) and arg.is_simple(x)):
+            dirac = arg
+        else:
+            new_args.append(arg)
+    if not dirac:  # there was no simple dirac
+        new_args = []
+        for arg in sorted_args:
+            if isinstance(arg, DiracDelta):
+                new_args.append(arg.expand(diracdelta=True, wrt=x))
+            elif arg.is_Pow and isinstance(arg.base, DiracDelta):
+                new_args.append(arg.func(arg.base.expand(diracdelta=True, wrt=x), arg.exp))
+            else:
+                new_args.append(arg)
+        if new_args != sorted_args:
+            nnode = Mul(*new_args).expand()
+        else:  # if the node didn't change there is nothing to do
+            nnode = None
+        return (None, nnode)
+    return (dirac, Mul(*new_args))
+
+
+def deltaintegrate(f, x):
+    """
+    deltaintegrate(f, x)
+
+    Explanation
+    ===========
+
+    The idea for integration is the following:
+
+    - If we are dealing with a DiracDelta expression, i.e. DiracDelta(g(x)),
+      we try to simplify it.
+
+      If we could simplify it, then we integrate the resulting expression.
+      We already know we can integrate a simplified expression, because only
+      simple DiracDelta expressions are involved.
+
+      If we couldn't simplify it, there are two cases:
+
+      1) The expression is a simple expression: we return the integral,
+         taking care if we are dealing with a Derivative or with a proper
+         DiracDelta.
+
+      2) The expression is not simple (i.e. DiracDelta(cos(x))): we can do
+         nothing at all.
+
+    - If the node is a multiplication node having a DiracDelta term:
+
+      First we expand it.
+
+      If the expansion did work, then we try to integrate the expansion.
+
+      If not, we try to extract a simple DiracDelta term, then we have two
+      cases:
+
+      1) We have a simple DiracDelta term, so we return the integral.
+
+      2) We didn't have a simple term, but we do have an expression with
+         simplified DiracDelta terms, so we integrate this expression.
+
+    Examples
+    ========
+
+        >>> from sympy.abc import x, y, z
+        >>> from sympy.integrals.deltafunctions import deltaintegrate
+        >>> from sympy import sin, cos, DiracDelta
+        >>> deltaintegrate(x*sin(x)*cos(x)*DiracDelta(x - 1), x)
+        sin(1)*cos(1)*Heaviside(x - 1)
+        >>> deltaintegrate(y**2*DiracDelta(x - z)*DiracDelta(y - z), y)
+        z**2*DiracDelta(x - z)*Heaviside(y - z)
+
+    See Also
+    ========
+
+    sympy.functions.special.delta_functions.DiracDelta
+    sympy.integrals.integrals.Integral
+    """
+    if not f.has(DiracDelta):
+        return None
+
+    # g(x) = DiracDelta(h(x))
+    if f.func == DiracDelta:
+        h = f.expand(diracdelta=True, wrt=x)
+        if h == f:  # can't simplify the expression
+            #FIXME: the second term tells whether is DeltaDirac or Derivative
+            #For integrating derivatives of DiracDelta we need the chain rule
+            if f.is_simple(x):
+                if (len(f.args) <= 1 or f.args[1] == 0):
+                    return Heaviside(f.args[0])
+                else:
+                    return (DiracDelta(f.args[0], f.args[1] - 1) /
+                        f.args[0].as_poly().LC())
+        else:  # let's try to integrate the simplified expression
+            fh = integrate(h, x)
+            return fh
+    elif f.is_Mul or f.is_Pow:  # g(x) = a*b*c*f(DiracDelta(h(x)))*d*e
+        g = f.expand()
+        if f != g:  # the expansion worked
+            fh = integrate(g, x)
+            if fh is not None and not isinstance(fh, Integral):
+                return fh
+        else:
+            # no expansion performed, try to extract a simple DiracDelta term
+            deltaterm, rest_mult = change_mul(f, x)
+
+            if not deltaterm:
+                if rest_mult:
+                    fh = integrate(rest_mult, x)
+                    return fh
+            else:
+                from sympy.solvers import solve
+                deltaterm = deltaterm.expand(diracdelta=True, wrt=x)
+                if deltaterm.is_Mul:  # Take out any extracted factors
+                    deltaterm, rest_mult_2 = change_mul(deltaterm, x)
+                    rest_mult = rest_mult*rest_mult_2
+                point = solve(deltaterm.args[0], x)[0]
+
+                # Return the largest hyperreal term left after
+                # repeated integration by parts.  For example,
+                #
+                #   integrate(y*DiracDelta(x, 1),x) == y*DiracDelta(x,0),  not 0
+                #
+                # This is so Integral(y*DiracDelta(x).diff(x),x).doit()
+                # will return y*DiracDelta(x) instead of 0 or DiracDelta(x),
+                # both of which are correct everywhere the value is defined
+                # but give wrong answers for nested integration.
+                n = (0 if len(deltaterm.args)==1 else deltaterm.args[1])
+                m = 0
+                while n >= 0:
+                    r = S.NegativeOne**n*rest_mult.diff(x, n).subs(x, point)
+                    if r.is_zero:
+                        n -= 1
+                        m += 1
+                    else:
+                        if m == 0:
+                            return r*Heaviside(x - point)
+                        else:
+                            return r*DiracDelta(x,m-1)
+                # In some very weak sense, x=0 is still a singularity,
+                # but we hope will not be of any practical consequence.
+                return S.Zero
+    return None

rl/Lib/site-packages/sympy/integrals/heurisch.py

@@ -0,0 +1,780 @@
+from __future__ import annotations
+
+from collections import defaultdict
+from functools import reduce
+from itertools import permutations
+
+from sympy.core.add import Add
+from sympy.core.basic import Basic
+from sympy.core.mul import Mul
+from sympy.core.symbol import Wild, Dummy, Symbol
+from sympy.core.basic import sympify
+from sympy.core.numbers import Rational, pi, I
+from sympy.core.relational import Eq, Ne
+from sympy.core.singleton import S
+from sympy.core.sorting import ordered
+from sympy.core.traversal import iterfreeargs
+
+from sympy.functions import exp, sin, cos, tan, cot, asin, atan
+from sympy.functions import log, sinh, cosh, tanh, coth, asinh
+from sympy.functions import sqrt, erf, erfi, li, Ei
+from sympy.functions import besselj, bessely, besseli, besselk
+from sympy.functions import hankel1, hankel2, jn, yn
+from sympy.functions.elementary.complexes import Abs, re, im, sign, arg
+from sympy.functions.elementary.exponential import LambertW
+from sympy.functions.elementary.integers import floor, ceiling
+from sympy.functions.elementary.piecewise import Piecewise
+from sympy.functions.special.delta_functions import Heaviside, DiracDelta
+
+from sympy.simplify.radsimp import collect
+
+from sympy.logic.boolalg import And, Or
+from sympy.utilities.iterables import uniq
+
+from sympy.polys import quo, gcd, lcm, factor_list, cancel, PolynomialError
+from sympy.polys.monomials import itermonomials
+from sympy.polys.polyroots import root_factors
+
+from sympy.polys.rings import PolyRing
+from sympy.polys.solvers import solve_lin_sys
+from sympy.polys.constructor import construct_domain
+
+from sympy.integrals.integrals import integrate
+
+
+def components(f, x):
+    """
+    Returns a set of all functional components of the given expression
+    which includes symbols, function applications and compositions and
+    non-integer powers. Fractional powers are collected with
+    minimal, positive exponents.
+
+    Examples
+    ========
+
+    >>> from sympy import cos, sin
+    >>> from sympy.abc import x
+    >>> from sympy.integrals.heurisch import components
+
+    >>> components(sin(x)*cos(x)**2, x)
+    {x, sin(x), cos(x)}
+
+    See Also
+    ========
+
+    heurisch
+    """
+    result = set()
+
+    if f.has_free(x):
+        if f.is_symbol and f.is_commutative:
+            result.add(f)
+        elif f.is_Function or f.is_Derivative:
+            for g in f.args:
+                result |= components(g, x)
+
+            result.add(f)
+        elif f.is_Pow:
+            result |= components(f.base, x)
+
+            if not f.exp.is_Integer:
+                if f.exp.is_Rational:
+                    result.add(f.base**Rational(1, f.exp.q))
+                else:
+                    result |= components(f.exp, x) | {f}
+        else:
+            for g in f.args:
+                result |= components(g, x)
+
+    return result
+
+# name -> [] of symbols
+_symbols_cache: dict[str, list[Dummy]] = {}
+
+
+# NB @cacheit is not convenient here
+def _symbols(name, n):
+    """get vector of symbols local to this module"""
+    try:
+        lsyms = _symbols_cache[name]
+    except KeyError:
+        lsyms = []
+        _symbols_cache[name] = lsyms
+
+    while len(lsyms) < n:
+        lsyms.append( Dummy('%s%i' % (name, len(lsyms))) )
+
+    return lsyms[:n]
+
+
+def heurisch_wrapper(f, x, rewrite=False, hints=None, mappings=None, retries=3,
+                     degree_offset=0, unnecessary_permutations=None,
+                     _try_heurisch=None):
+    """
+    A wrapper around the heurisch integration algorithm.
+
+    Explanation
+    ===========
+
+    This method takes the result from heurisch and checks for poles in the
+    denominator. For each of these poles, the integral is reevaluated, and
+    the final integration result is given in terms of a Piecewise.
+
+    Examples
+    ========
+
+    >>> from sympy import cos, symbols
+    >>> from sympy.integrals.heurisch import heurisch, heurisch_wrapper
+    >>> n, x = symbols('n x')
+    >>> heurisch(cos(n*x), x)
+    sin(n*x)/n
+    >>> heurisch_wrapper(cos(n*x), x)
+    Piecewise((sin(n*x)/n, Ne(n, 0)), (x, True))
+
+    See Also
+    ========
+
+    heurisch
+    """
+    from sympy.solvers.solvers import solve, denoms
+    f = sympify(f)
+    if not f.has_free(x):
+        return f*x
+
+    res = heurisch(f, x, rewrite, hints, mappings, retries, degree_offset,
+                   unnecessary_permutations, _try_heurisch)
+    if not isinstance(res, Basic):
+        return res
+
+    # We consider each denominator in the expression, and try to find
+    # cases where one or more symbolic denominator might be zero. The
+    # conditions for these cases are stored in the list slns.
+    #
+    # Since denoms returns a set we use ordered. This is important because the
+    # ordering of slns determines the order of the resulting Piecewise so we
+    # need a deterministic order here to make the output deterministic.
+    slns = []
+    for d in ordered(denoms(res)):
+        try:
+            slns += solve([d], dict=True, exclude=(x,))
+        except NotImplementedError:
+            pass
+    if not slns:
+        return res
+    slns = list(uniq(slns))
+    # Remove the solutions corresponding to poles in the original expression.
+    slns0 = []
+    for d in denoms(f):
+        try:
+            slns0 += solve([d], dict=True, exclude=(x,))
+        except NotImplementedError:
+            pass
+    slns = [s for s in slns if s not in slns0]
+    if not slns:
+        return res
+    if len(slns) > 1:
+        eqs = []
+        for sub_dict in slns:
+            eqs.extend([Eq(key, value) for key, value in sub_dict.items()])
+        slns = solve(eqs, dict=True, exclude=(x,)) + slns
+    # For each case listed in the list slns, we reevaluate the integral.
+    pairs = []
+    for sub_dict in slns:
+        expr = heurisch(f.subs(sub_dict), x, rewrite, hints, mappings, retries,
+                        degree_offset, unnecessary_permutations,
+                        _try_heurisch)
+        cond = And(*[Eq(key, value) for key, value in sub_dict.items()])
+        generic = Or(*[Ne(key, value) for key, value in sub_dict.items()])
+        if expr is None:
+            expr = integrate(f.subs(sub_dict),x)
+        pairs.append((expr, cond))
+    # If there is one condition, put the generic case first. Otherwise,
+    # doing so may lead to longer Piecewise formulas
+    if len(pairs) == 1:
+        pairs = [(heurisch(f, x, rewrite, hints, mappings, retries,
+                              degree_offset, unnecessary_permutations,
+                              _try_heurisch),
+                              generic),
+                 (pairs[0][0], True)]
+    else:
+        pairs.append((heurisch(f, x, rewrite, hints, mappings, retries,
+                              degree_offset, unnecessary_permutations,
+                              _try_heurisch),
+                              True))
+    return Piecewise(*pairs)
+
+class BesselTable:
+    """
+    Derivatives of Bessel functions of orders n and n-1
+    in terms of each other.
+
+    See the docstring of DiffCache.
+    """
+
+    def __init__(self):
+        self.table = {}
+        self.n = Dummy('n')
+        self.z = Dummy('z')
+        self._create_table()
+
+    def _create_table(t):
+        table, n, z = t.table, t.n, t.z
+        for f in (besselj, bessely, hankel1, hankel2):
+            table[f] = (f(n-1, z) - n*f(n, z)/z,
+                        (n-1)*f(n-1, z)/z - f(n, z))
+
+        f = besseli
+        table[f] = (f(n-1, z) - n*f(n, z)/z,
+                    (n-1)*f(n-1, z)/z + f(n, z))
+        f = besselk
+        table[f] = (-f(n-1, z) - n*f(n, z)/z,
+                    (n-1)*f(n-1, z)/z - f(n, z))
+
+        for f in (jn, yn):
+            table[f] = (f(n-1, z) - (n+1)*f(n, z)/z,
+                        (n-1)*f(n-1, z)/z - f(n, z))
+
+    def diffs(t, f, n, z):
+        if f in t.table:
+            diff0, diff1 = t.table[f]
+            repl = [(t.n, n), (t.z, z)]
+            return (diff0.subs(repl), diff1.subs(repl))
+
+    def has(t, f):
+        return f in t.table
+
+_bessel_table = None
+
+class DiffCache:
+    """
+    Store for derivatives of expressions.
+
+    Explanation
+    ===========
+
+    The standard form of the derivative of a Bessel function of order n
+    contains two Bessel functions of orders n-1 and n+1, respectively.
+    Such forms cannot be used in parallel Risch algorithm, because
+    there is a linear recurrence relation between the three functions
+    while the algorithm expects that functions and derivatives are
+    represented in terms of algebraically independent transcendentals.
+
+    The solution is to take two of the functions, e.g., those of orders
+    n and n-1, and to express the derivatives in terms of the pair.
+    To guarantee that the proper form is used the two derivatives are
+    cached as soon as one is encountered.
+
+    Derivatives of other functions are also cached at no extra cost.
+    All derivatives are with respect to the same variable `x`.
+    """
+
+    def __init__(self, x):
+        self.cache = {}
+        self.x = x
+
+        global _bessel_table
+        if not _bessel_table:
+            _bessel_table = BesselTable()
+
+    def get_diff(self, f):
+        cache = self.cache
+
+        if f in cache:
+            pass
+        elif (not hasattr(f, 'func') or
+            not _bessel_table.has(f.func)):
+            cache[f] = cancel(f.diff(self.x))
+        else:
+            n, z = f.args
+            d0, d1 = _bessel_table.diffs(f.func, n, z)
+            dz = self.get_diff(z)
+            cache[f] = d0*dz
+            cache[f.func(n-1, z)] = d1*dz
+
+        return cache[f]
+
+def heurisch(f, x, rewrite=False, hints=None, mappings=None, retries=3,
+             degree_offset=0, unnecessary_permutations=None,
+             _try_heurisch=None):
+    """
+    Compute indefinite integral using heuristic Risch algorithm.
+
+    Explanation
+    ===========
+
+    This is a heuristic approach to indefinite integration in finite
+    terms using the extended heuristic (parallel) Risch algorithm, based
+    on Manuel Bronstein's "Poor Man's Integrator".
+
+    The algorithm supports various classes of functions including
+    transcendental elementary or special functions like Airy,
+    Bessel, Whittaker and Lambert.
+
+    Note that this algorithm is not a decision procedure. If it isn't
+    able to compute the antiderivative for a given function, then this is
+    not a proof that such a functions does not exist.  One should use
+    recursive Risch algorithm in such case.  It's an open question if
+    this algorithm can be made a full decision procedure.
+
+    This is an internal integrator procedure. You should use top level
+    'integrate' function in most cases, as this procedure needs some
+    preprocessing steps and otherwise may fail.
+
+    Specification
+    =============
+
+     heurisch(f, x, rewrite=False, hints=None)
+
+       where
+         f : expression
+         x : symbol
+
+         rewrite -> force rewrite 'f' in terms of 'tan' and 'tanh'
+         hints   -> a list of functions that may appear in anti-derivate
+
+          - hints = None          --> no suggestions at all
+          - hints = [ ]           --> try to figure out
+          - hints = [f1, ..., fn] --> we know better
+
+    Examples
+    ========
+
+    >>> from sympy import tan
+    >>> from sympy.integrals.heurisch import heurisch
+    >>> from sympy.abc import x, y
+
+    >>> heurisch(y*tan(x), x)
+    y*log(tan(x)**2 + 1)/2
+
+    See Manuel Bronstein's "Poor Man's Integrator":
+
+    References
+    ==========
+
+    .. [1] https://www-sop.inria.fr/cafe/Manuel.Bronstein/pmint/index.html
+
+    For more information on the implemented algorithm refer to:
+
+    .. [2] K. Geddes, L. Stefanus, On the Risch-Norman Integration
+       Method and its Implementation in Maple, Proceedings of
+       ISSAC'89, ACM Press, 212-217.
+
+    .. [3] J. H. Davenport, On the Parallel Risch Algorithm (I),
+       Proceedings of EUROCAM'82, LNCS 144, Springer, 144-157.
+
+    .. [4] J. H. Davenport, On the Parallel Risch Algorithm (III):
+       Use of Tangents, SIGSAM Bulletin 16 (1982), 3-6.
+
+    .. [5] J. H. Davenport, B. M. Trager, On the Parallel Risch
+       Algorithm (II), ACM Transactions on Mathematical
+       Software 11 (1985), 356-362.
+
+    See Also
+    ========
+
+    sympy.integrals.integrals.Integral.doit
+    sympy.integrals.integrals.Integral
+    sympy.integrals.heurisch.components
+    """
+    f = sympify(f)
+
+    # There are some functions that Heurisch cannot currently handle,
+    # so do not even try.
+    # Set _try_heurisch=True to skip this check
+    if _try_heurisch is not True:
+        if f.has(Abs, re, im, sign, Heaviside, DiracDelta, floor, ceiling, arg):
+            return
+
+    if not f.has_free(x):
+        return f*x
+
+    if not f.is_Add:
+        indep, f = f.as_independent(x)
+    else:
+        indep = S.One
+
+    rewritables = {
+        (sin, cos, cot): tan,
+        (sinh, cosh, coth): tanh,
+    }
+
+    if rewrite:
+        for candidates, rule in rewritables.items():
+            f = f.rewrite(candidates, rule)
+    else:
+        for candidates in rewritables.keys():
+            if f.has(*candidates):
+                break
+        else:
+            rewrite = True
+
+    terms = components(f, x)
+    dcache = DiffCache(x)
+
+    if hints is not None:
+        if not hints:
+            a = Wild('a', exclude=[x])
+            b = Wild('b', exclude=[x])
+            c = Wild('c', exclude=[x])
+
+            for g in set(terms):  # using copy of terms
+                if g.is_Function:
+                    if isinstance(g, li):
+                        M = g.args[0].match(a*x**b)
+
+                        if M is not None:
+                            terms.add( x*(li(M[a]*x**M[b]) - (M[a]*x**M[b])**(-1/M[b])*Ei((M[b]+1)*log(M[a]*x**M[b])/M[b])) )
+                            #terms.add( x*(li(M[a]*x**M[b]) - (x**M[b])**(-1/M[b])*Ei((M[b]+1)*log(M[a]*x**M[b])/M[b])) )
+                            #terms.add( x*(li(M[a]*x**M[b]) - x*Ei((M[b]+1)*log(M[a]*x**M[b])/M[b])) )
+                            #terms.add( li(M[a]*x**M[b]) - Ei((M[b]+1)*log(M[a]*x**M[b])/M[b]) )
+
+                    elif isinstance(g, exp):
+                        M = g.args[0].match(a*x**2)
+
+                        if M is not None:
+                            if M[a].is_positive:
+                                terms.add(erfi(sqrt(M[a])*x))
+                            else: # M[a].is_negative or unknown
+                                terms.add(erf(sqrt(-M[a])*x))
+
+                        M = g.args[0].match(a*x**2 + b*x + c)
+
+                        if M is not None:
+                            if M[a].is_positive:
+                                terms.add(sqrt(pi/4*(-M[a]))*exp(M[c] - M[b]**2/(4*M[a]))*
+                                          erfi(sqrt(M[a])*x + M[b]/(2*sqrt(M[a]))))
+                            elif M[a].is_negative:
+                                terms.add(sqrt(pi/4*(-M[a]))*exp(M[c] - M[b]**2/(4*M[a]))*
+                                          erf(sqrt(-M[a])*x - M[b]/(2*sqrt(-M[a]))))
+
+                        M = g.args[0].match(a*log(x)**2)
+
+                        if M is not None:
+                            if M[a].is_positive:
+                                terms.add(erfi(sqrt(M[a])*log(x) + 1/(2*sqrt(M[a]))))
+                            if M[a].is_negative:
+                                terms.add(erf(sqrt(-M[a])*log(x) - 1/(2*sqrt(-M[a]))))
+
+                elif g.is_Pow:
+                    if g.exp.is_Rational and g.exp.q == 2:
+                        M = g.base.match(a*x**2 + b)
+
+                        if M is not None and M[b].is_positive:
+                            if M[a].is_positive:
+                                terms.add(asinh(sqrt(M[a]/M[b])*x))
+                            elif M[a].is_negative:
+                                terms.add(asin(sqrt(-M[a]/M[b])*x))
+
+                        M = g.base.match(a*x**2 - b)
+
+                        if M is not None and M[b].is_positive:
+                            if M[a].is_positive:
+                                dF = 1/sqrt(M[a]*x**2 - M[b])
+                                F = log(2*sqrt(M[a])*sqrt(M[a]*x**2 - M[b]) + 2*M[a]*x)/sqrt(M[a])
+                                dcache.cache[F] = dF  # hack: F.diff(x) doesn't automatically simplify to f
+                                terms.add(F)
+                            elif M[a].is_negative:
+                                terms.add(-M[b]/2*sqrt(-M[a])*
+                                           atan(sqrt(-M[a])*x/sqrt(M[a]*x**2 - M[b])))
+
+        else:
+            terms |= set(hints)
+
+    for g in set(terms):  # using copy of terms
+        terms |= components(dcache.get_diff(g), x)
+
+    # XXX: The commented line below makes heurisch more deterministic wrt
+    # PYTHONHASHSEED and the iteration order of sets. There are other places
+    # where sets are iterated over but this one is possibly the most important.
+    # Theoretically the order here should not matter but different orderings
+    # can expose potential bugs in the different code paths so potentially it
+    # is better to keep the non-determinism.
+    #
+    # terms = list(ordered(terms))
+
+    # TODO: caching is significant factor for why permutations work at all. Change this.
+    V = _symbols('x', len(terms))
+
+
+    # sort mapping expressions from largest to smallest (last is always x).
+    mapping = list(reversed(list(zip(*ordered(                          #
+        [(a[0].as_independent(x)[1], a) for a in zip(terms, V)])))[1])) #
+    rev_mapping = {v: k for k, v in mapping}                            #
+    if mappings is None:                                                #
+        # optimizing the number of permutations of mapping              #
+        assert mapping[-1][0] == x # if not, find it and correct this comment
+        unnecessary_permutations = [mapping.pop(-1)]
+        # only permute types of objects and let the ordering
+        # of types take care of the order of replacement
+        types = defaultdict(list)
+        for i in mapping:
+            types[type(i)].append(i)
+        mapping = [types[i] for i in types]
+        def _iter_mappings():
+            for i in permutations(mapping):
+                yield [j for i in i for j in i]
+        mappings = _iter_mappings()
+    else:
+        unnecessary_permutations = unnecessary_permutations or []
+
+    def _substitute(expr):
+        return expr.subs(mapping)
+
+    for mapping in mappings:
+        mapping = list(mapping)
+        mapping = mapping + unnecessary_permutations
+        diffs = [ _substitute(dcache.get_diff(g)) for g in terms ]
+        denoms = [ g.as_numer_denom()[1] for g in diffs ]
+        if all(h.is_polynomial(*V) for h in denoms) and _substitute(f).is_rational_function(*V):
+            denom = reduce(lambda p, q: lcm(p, q, *V), denoms)
+            break
+    else:
+        if not rewrite:
+            result = heurisch(f, x, rewrite=True, hints=hints,
+                unnecessary_permutations=unnecessary_permutations)
+
+            if result is not None:
+                return indep*result
+        return None
+
+    numers = [ cancel(denom*g) for g in diffs ]
+    def _derivation(h):
+        return Add(*[ d * h.diff(v) for d, v in zip(numers, V) ])
+
+    def _deflation(p):
+        for y in V:
+            if not p.has(y):
+                continue
+
+            if _derivation(p) is not S.Zero:
+                c, q = p.as_poly(y).primitive()
+                return _deflation(c)*gcd(q, q.diff(y)).as_expr()
+
+        return p
+
+    def _splitter(p):
+        for y in V:
+            if not p.has(y):
+                continue
+
+            if _derivation(y) is not S.Zero:
+                c, q = p.as_poly(y).primitive()
+
+                q = q.as_expr()
+
+                h = gcd(q, _derivation(q), y)
+                s = quo(h, gcd(q, q.diff(y), y), y)
+
+                c_split = _splitter(c)
+
+                if s.as_poly(y).degree() == 0:
+                    return (c_split[0], q * c_split[1])
+
+                q_split = _splitter(cancel(q / s))
+
+                return (c_split[0]*q_split[0]*s, c_split[1]*q_split[1])
+
+        return (S.One, p)
+
+    special = {}
+
+    for term in terms:
+        if term.is_Function:
+            if isinstance(term, tan):
+                special[1 + _substitute(term)**2] = False
+            elif isinstance(term, tanh):
+                special[1 + _substitute(term)] = False
+                special[1 - _substitute(term)] = False
+            elif isinstance(term, LambertW):
+                special[_substitute(term)] = True
+
+    F = _substitute(f)
+
+    P, Q = F.as_numer_denom()
+
+    u_split = _splitter(denom)
+    v_split = _splitter(Q)
+
+    polys = set(list(v_split) + [ u_split[0] ] + list(special.keys()))
+
+    s = u_split[0] * Mul(*[ k for k, v in special.items() if v ])
+    polified = [ p.as_poly(*V) for p in [s, P, Q] ]
+
+    if None in polified:
+        return None
+
+    #--- definitions for _integrate
+    a, b, c = [ p.total_degree() for p in polified ]
+
+    poly_denom = (s * v_split[0] * _deflation(v_split[1])).as_expr()
+
+    def _exponent(g):
+        if g.is_Pow:
+            if g.exp.is_Rational and g.exp.q != 1:
+                if g.exp.p > 0:
+                    return g.exp.p + g.exp.q - 1
+                else:
+                    return abs(g.exp.p + g.exp.q)
+            else:
+                return 1
+        elif not g.is_Atom and g.args:
+            return max(_exponent(h) for h in g.args)
+        else:
+            return 1
+
+    A, B = _exponent(f), a + max(b, c)
+
+    if A > 1 and B > 1:
+        monoms = tuple(ordered(itermonomials(V, A + B - 1 + degree_offset)))
+    else:
+        monoms = tuple(ordered(itermonomials(V, A + B + degree_offset)))
+
+    poly_coeffs = _symbols('A', len(monoms))
+
+    poly_part = Add(*[ poly_coeffs[i]*monomial
+        for i, monomial in enumerate(monoms) ])
+
+    reducibles = set()
+
+    for poly in ordered(polys):
+        coeff, factors = factor_list(poly, *V)
+        reducibles.add(coeff)
+        reducibles.update(fact for fact, mul in factors)
+
+    def _integrate(field=None):
+        atans = set()
+        pairs = set()
+
+        if field == 'Q':
+            irreducibles = set(reducibles)
+        else:
+            setV = set(V)
+            irreducibles = set()
+            for poly in ordered(reducibles):
+                zV = setV & set(iterfreeargs(poly))
+                for z in ordered(zV):
+                    s = set(root_factors(poly, z, filter=field))
+                    irreducibles |= s
+                    break
+
+        log_part, atan_part = [], []
+
+        for poly in ordered(irreducibles):
+            m = collect(poly, I, evaluate=False)
+            y = m.get(I, S.Zero)
+            if y:
+                x = m.get(S.One, S.Zero)
+                if x.has(I) or y.has(I):
+                    continue  # nontrivial x + I*y
+                pairs.add((x, y))
+                irreducibles.remove(poly)
+
+        while pairs:
+            x, y = pairs.pop()
+            if (x, -y) in pairs:
+                pairs.remove((x, -y))
+                # Choosing b with no minus sign
+                if y.could_extract_minus_sign():
+                    y = -y
+                irreducibles.add(x*x + y*y)
+                atans.add(atan(x/y))
+            else:
+                irreducibles.add(x + I*y)
+
+
+        B = _symbols('B', len(irreducibles))
+        C = _symbols('C', len(atans))
+
+        # Note: the ordering matters here
+        for poly, b in reversed(list(zip(ordered(irreducibles), B))):
+            if poly.has(*V):
+                poly_coeffs.append(b)
+                log_part.append(b * log(poly))
+
+        for poly, c in reversed(list(zip(ordered(atans), C))):
+            if poly.has(*V):
+                poly_coeffs.append(c)
+                atan_part.append(c * poly)
+
+        # TODO: Currently it's better to use symbolic expressions here instead
+        # of rational functions, because it's simpler and FracElement doesn't
+        # give big speed improvement yet. This is because cancellation is slow
+        # due to slow polynomial GCD algorithms. If this gets improved then
+        # revise this code.
+        candidate = poly_part/poly_denom + Add(*log_part) + Add(*atan_part)
+        h = F - _derivation(candidate) / denom
+        raw_numer = h.as_numer_denom()[0]
+
+        # Rewrite raw_numer as a polynomial in K[coeffs][V] where K is a field
+        # that we have to determine. We can't use simply atoms() because log(3),
+        # sqrt(y) and similar expressions can appear, leading to non-trivial
+        # domains.
+        syms = set(poly_coeffs) | set(V)
+        non_syms = set()
+
+        def find_non_syms(expr):
+            if expr.is_Integer or expr.is_Rational:
+                pass # ignore trivial numbers
+            elif expr in syms:
+                pass # ignore variables
+            elif not expr.has_free(*syms):
+                non_syms.add(expr)
+            elif expr.is_Add or expr.is_Mul or expr.is_Pow:
+                list(map(find_non_syms, expr.args))
+            else:
+                # TODO: Non-polynomial expression. This should have been
+                # filtered out at an earlier stage.
+                raise PolynomialError
+
+        try:
+            find_non_syms(raw_numer)
+        except PolynomialError:
+            return None
+        else:
+            ground, _ = construct_domain(non_syms, field=True)
+
+        coeff_ring = PolyRing(poly_coeffs, ground)
+        ring = PolyRing(V, coeff_ring)
+        try:
+            numer = ring.from_expr(raw_numer)
+        except ValueError:
+            raise PolynomialError
+        solution = solve_lin_sys(numer.coeffs(), coeff_ring, _raw=False)
+
+        if solution is None:
+            return None
+        else:
+            return candidate.xreplace(solution).xreplace(
+                dict(zip(poly_coeffs, [S.Zero]*len(poly_coeffs))))
+
+    if all(isinstance(_, Symbol) for _ in V):
+        more_free = F.free_symbols - set(V)
+    else:
+        Fd = F.as_dummy()
+        more_free = Fd.xreplace(dict(zip(V, (Dummy() for _ in V)))
+            ).free_symbols & Fd.free_symbols
+    if not more_free:
+        # all free generators are identified in V
+        solution = _integrate('Q')
+
+        if solution is None:
+            solution = _integrate()
+    else:
+        solution = _integrate()
+
+    if solution is not None:
+        antideriv = solution.subs(rev_mapping)
+        antideriv = cancel(antideriv).expand()
+
+        if antideriv.is_Add:
+            antideriv = antideriv.as_independent(x)[1]
+
+        return indep*antideriv
+    else:
+        if retries >= 0:
+            result = heurisch(f, x, mappings=mappings, rewrite=rewrite, hints=hints, retries=retries - 1, unnecessary_permutations=unnecessary_permutations)
+
+            if result is not None:
+                return indep*result
+
+        return None

rl/Lib/site-packages/sympy/integrals/meijerint_doc.py

@@ -0,0 +1,37 @@
+""" This module cooks up a docstring when imported. Its only purpose is to
+    be displayed in the sphinx documentation. """
+
+from __future__ import annotations
+from typing import Any
+
+from sympy.integrals.meijerint import _create_lookup_table
+from sympy.core.add import Add
+from sympy.core.basic import Basic
+from sympy.core.relational import Eq
+from sympy.core.symbol import Symbol
+from sympy.printing.latex import latex
+
+t: dict[tuple[type[Basic], ...], list[Any]] = {}
+_create_lookup_table(t)
+
+
+doc = ""
+for about, category in t.items():
+    if about == ():
+        doc += 'Elementary functions:\n\n'
+    else:
+        doc += 'Functions involving ' + ', '.join('`%s`' % latex(
+            list(category[0][0].atoms(func))[0]) for func in about) + ':\n\n'
+    for formula, gs, cond, hint in category:
+        if not isinstance(gs, list):
+            g = Symbol('\\text{generated}')
+        else:
+            g = Add(*[fac*f for (fac, f) in gs])
+        obj = Eq(formula, g)
+        if cond is True:
+            cond = ""
+        else:
+            cond = ',\\text{ if } %s' % latex(cond)
+        doc += ".. math::\n  %s%s\n\n" % (latex(obj), cond)
+
+__doc__ = doc

rl/Lib/site-packages/sympy/integrals/quadrature.py

@@ -0,0 +1,617 @@
+from sympy.core import S, Dummy, pi
+from sympy.functions.combinatorial.factorials import factorial
+from sympy.functions.elementary.trigonometric import sin, cos
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.special.gamma_functions import gamma
+from sympy.polys.orthopolys import (legendre_poly, laguerre_poly,
+                                    hermite_poly, jacobi_poly)
+from sympy.polys.rootoftools import RootOf
+
+
+def gauss_legendre(n, n_digits):
+    r"""
+    Computes the Gauss-Legendre quadrature [1]_ points and weights.
+
+    Explanation
+    ===========
+
+    The Gauss-Legendre quadrature approximates the integral:
+
+    .. math::
+        \int_{-1}^1 f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i)
+
+    The nodes `x_i` of an order `n` quadrature rule are the roots of `P_n`
+    and the weights `w_i` are given by:
+
+    .. math::
+        w_i = \frac{2}{\left(1-x_i^2\right) \left(P'_n(x_i)\right)^2}
+
+    Parameters
+    ==========
+
+    n :
+        The order of quadrature.
+    n_digits :
+        Number of significant digits of the points and weights to return.
+
+    Returns
+    =======
+
+    (x, w) : the ``x`` and ``w`` are lists of points and weights as Floats.
+             The points `x_i` and weights `w_i` are returned as ``(x, w)``
+             tuple of lists.
+
+    Examples
+    ========
+
+    >>> from sympy.integrals.quadrature import gauss_legendre
+    >>> x, w = gauss_legendre(3, 5)
+    >>> x
+    [-0.7746, 0, 0.7746]
+    >>> w
+    [0.55556, 0.88889, 0.55556]
+    >>> x, w = gauss_legendre(4, 5)
+    >>> x
+    [-0.86114, -0.33998, 0.33998, 0.86114]
+    >>> w
+    [0.34785, 0.65215, 0.65215, 0.34785]
+
+    See Also
+    ========
+
+    gauss_laguerre, gauss_gen_laguerre, gauss_hermite, gauss_chebyshev_t, gauss_chebyshev_u, gauss_jacobi, gauss_lobatto
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Gaussian_quadrature
+    .. [2] https://people.sc.fsu.edu/~jburkardt/cpp_src/legendre_rule/legendre_rule.html
+    """
+    x = Dummy("x")
+    p = legendre_poly(n, x, polys=True)
+    pd = p.diff(x)
+    xi = []
+    w = []
+    for r in p.real_roots():
+        if isinstance(r, RootOf):
+            r = r.eval_rational(S.One/10**(n_digits+2))
+        xi.append(r.n(n_digits))
+        w.append((2/((1-r**2) * pd.subs(x, r)**2)).n(n_digits))
+    return xi, w
+
+
+def gauss_laguerre(n, n_digits):
+    r"""
+    Computes the Gauss-Laguerre quadrature [1]_ points and weights.
+
+    Explanation
+    ===========
+
+    The Gauss-Laguerre quadrature approximates the integral:
+
+    .. math::
+        \int_0^{\infty} e^{-x} f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i)
+
+
+    The nodes `x_i` of an order `n` quadrature rule are the roots of `L_n`
+    and the weights `w_i` are given by:
+
+    .. math::
+        w_i = \frac{x_i}{(n+1)^2 \left(L_{n+1}(x_i)\right)^2}
+
+    Parameters
+    ==========
+
+    n :
+        The order of quadrature.
+    n_digits :
+        Number of significant digits of the points and weights to return.
+
+    Returns
+    =======
+
+    (x, w) : The ``x`` and ``w`` are lists of points and weights as Floats.
+             The points `x_i` and weights `w_i` are returned as ``(x, w)``
+             tuple of lists.
+
+    Examples
+    ========
+
+    >>> from sympy.integrals.quadrature import gauss_laguerre
+    >>> x, w = gauss_laguerre(3, 5)
+    >>> x
+    [0.41577, 2.2943, 6.2899]
+    >>> w
+    [0.71109, 0.27852, 0.010389]
+    >>> x, w = gauss_laguerre(6, 5)
+    >>> x
+    [0.22285, 1.1889, 2.9927, 5.7751, 9.8375, 15.983]
+    >>> w
+    [0.45896, 0.417, 0.11337, 0.010399, 0.00026102, 8.9855e-7]
+
+    See Also
+    ========
+
+    gauss_legendre, gauss_gen_laguerre, gauss_hermite, gauss_chebyshev_t, gauss_chebyshev_u, gauss_jacobi, gauss_lobatto
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Gauss%E2%80%93Laguerre_quadrature
+    .. [2] https://people.sc.fsu.edu/~jburkardt/cpp_src/laguerre_rule/laguerre_rule.html
+    """
+    x = Dummy("x")
+    p = laguerre_poly(n, x, polys=True)
+    p1 = laguerre_poly(n+1, x, polys=True)
+    xi = []
+    w = []
+    for r in p.real_roots():
+        if isinstance(r, RootOf):
+            r = r.eval_rational(S.One/10**(n_digits+2))
+        xi.append(r.n(n_digits))
+        w.append((r/((n+1)**2 * p1.subs(x, r)**2)).n(n_digits))
+    return xi, w
+
+
+def gauss_hermite(n, n_digits):
+    r"""
+    Computes the Gauss-Hermite quadrature [1]_ points and weights.
+
+    Explanation
+    ===========
+
+    The Gauss-Hermite quadrature approximates the integral:
+
+    .. math::
+        \int_{-\infty}^{\infty} e^{-x^2} f(x)\,dx \approx
+            \sum_{i=1}^n w_i f(x_i)
+
+    The nodes `x_i` of an order `n` quadrature rule are the roots of `H_n`
+    and the weights `w_i` are given by:
+
+    .. math::
+        w_i = \frac{2^{n-1} n! \sqrt{\pi}}{n^2 \left(H_{n-1}(x_i)\right)^2}
+
+    Parameters
+    ==========
+
+    n :
+        The order of quadrature.
+    n_digits :
+        Number of significant digits of the points and weights to return.
+
+    Returns
+    =======
+
+    (x, w) : The ``x`` and ``w`` are lists of points and weights as Floats.
+             The points `x_i` and weights `w_i` are returned as ``(x, w)``
+             tuple of lists.
+
+    Examples
+    ========
+
+    >>> from sympy.integrals.quadrature import gauss_hermite
+    >>> x, w = gauss_hermite(3, 5)
+    >>> x
+    [-1.2247, 0, 1.2247]
+    >>> w
+    [0.29541, 1.1816, 0.29541]
+
+    >>> x, w = gauss_hermite(6, 5)
+    >>> x
+    [-2.3506, -1.3358, -0.43608, 0.43608, 1.3358, 2.3506]
+    >>> w
+    [0.00453, 0.15707, 0.72463, 0.72463, 0.15707, 0.00453]
+
+    See Also
+    ========
+
+    gauss_legendre, gauss_laguerre, gauss_gen_laguerre, gauss_chebyshev_t, gauss_chebyshev_u, gauss_jacobi, gauss_lobatto
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Gauss-Hermite_Quadrature
+    .. [2] https://people.sc.fsu.edu/~jburkardt/cpp_src/hermite_rule/hermite_rule.html
+    .. [3] https://people.sc.fsu.edu/~jburkardt/cpp_src/gen_hermite_rule/gen_hermite_rule.html
+    """
+    x = Dummy("x")
+    p = hermite_poly(n, x, polys=True)
+    p1 = hermite_poly(n-1, x, polys=True)
+    xi = []
+    w = []
+    for r in p.real_roots():
+        if isinstance(r, RootOf):
+            r = r.eval_rational(S.One/10**(n_digits+2))
+        xi.append(r.n(n_digits))
+        w.append(((2**(n-1) * factorial(n) * sqrt(pi)) /
+                 (n**2 * p1.subs(x, r)**2)).n(n_digits))
+    return xi, w
+
+
+def gauss_gen_laguerre(n, alpha, n_digits):
+    r"""
+    Computes the generalized Gauss-Laguerre quadrature [1]_ points and weights.
+
+    Explanation
+    ===========
+
+    The generalized Gauss-Laguerre quadrature approximates the integral:
+
+    .. math::
+        \int_{0}^\infty x^{\alpha} e^{-x} f(x)\,dx \approx
+            \sum_{i=1}^n w_i f(x_i)
+
+    The nodes `x_i` of an order `n` quadrature rule are the roots of
+    `L^{\alpha}_n` and the weights `w_i` are given by:
+
+    .. math::
+        w_i = \frac{\Gamma(\alpha+n)}
+                {n \Gamma(n) L^{\alpha}_{n-1}(x_i) L^{\alpha+1}_{n-1}(x_i)}
+
+    Parameters
+    ==========
+
+    n :
+        The order of quadrature.
+
+    alpha :
+        The exponent of the singularity, `\alpha > -1`.
+
+    n_digits :
+        Number of significant digits of the points and weights to return.
+
+    Returns
+    =======
+
+    (x, w) : the ``x`` and ``w`` are lists of points and weights as Floats.
+             The points `x_i` and weights `w_i` are returned as ``(x, w)``
+             tuple of lists.
+
+    Examples
+    ========
+
+    >>> from sympy import S
+    >>> from sympy.integrals.quadrature import gauss_gen_laguerre
+    >>> x, w = gauss_gen_laguerre(3, -S.Half, 5)
+    >>> x
+    [0.19016, 1.7845, 5.5253]
+    >>> w
+    [1.4493, 0.31413, 0.00906]
+
+    >>> x, w = gauss_gen_laguerre(4, 3*S.Half, 5)
+    >>> x
+    [0.97851, 2.9904, 6.3193, 11.712]
+    >>> w
+    [0.53087, 0.67721, 0.11895, 0.0023152]
+
+    See Also
+    ========
+
+    gauss_legendre, gauss_laguerre, gauss_hermite, gauss_chebyshev_t, gauss_chebyshev_u, gauss_jacobi, gauss_lobatto
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Gauss%E2%80%93Laguerre_quadrature
+    .. [2] https://people.sc.fsu.edu/~jburkardt/cpp_src/gen_laguerre_rule/gen_laguerre_rule.html
+    """
+    x = Dummy("x")
+    p = laguerre_poly(n, x, alpha=alpha, polys=True)
+    p1 = laguerre_poly(n-1, x, alpha=alpha, polys=True)
+    p2 = laguerre_poly(n-1, x, alpha=alpha+1, polys=True)
+    xi = []
+    w = []
+    for r in p.real_roots():
+        if isinstance(r, RootOf):
+            r = r.eval_rational(S.One/10**(n_digits+2))
+        xi.append(r.n(n_digits))
+        w.append((gamma(alpha+n) /
+                 (n*gamma(n)*p1.subs(x, r)*p2.subs(x, r))).n(n_digits))
+    return xi, w
+
+
+def gauss_chebyshev_t(n, n_digits):
+    r"""
+    Computes the Gauss-Chebyshev quadrature [1]_ points and weights of
+    the first kind.
+
+    Explanation
+    ===========
+
+    The Gauss-Chebyshev quadrature of the first kind approximates the integral:
+
+    .. math::
+        \int_{-1}^{1} \frac{1}{\sqrt{1-x^2}} f(x)\,dx \approx
+            \sum_{i=1}^n w_i f(x_i)
+
+    The nodes `x_i` of an order `n` quadrature rule are the roots of `T_n`
+    and the weights `w_i` are given by:
+
+    .. math::
+        w_i = \frac{\pi}{n}
+
+    Parameters
+    ==========
+
+    n :
+        The order of quadrature.
+
+    n_digits :
+        Number of significant digits of the points and weights to return.
+
+    Returns
+    =======
+
+    (x, w) : the ``x`` and ``w`` are lists of points and weights as Floats.
+             The points `x_i` and weights `w_i` are returned as ``(x, w)``
+             tuple of lists.
+
+    Examples
+    ========
+
+    >>> from sympy.integrals.quadrature import gauss_chebyshev_t
+    >>> x, w = gauss_chebyshev_t(3, 5)
+    >>> x
+    [0.86602, 0, -0.86602]
+    >>> w
+    [1.0472, 1.0472, 1.0472]
+
+    >>> x, w = gauss_chebyshev_t(6, 5)
+    >>> x
+    [0.96593, 0.70711, 0.25882, -0.25882, -0.70711, -0.96593]
+    >>> w
+    [0.5236, 0.5236, 0.5236, 0.5236, 0.5236, 0.5236]
+
+    See Also
+    ========
+
+    gauss_legendre, gauss_laguerre, gauss_hermite, gauss_gen_laguerre, gauss_chebyshev_u, gauss_jacobi, gauss_lobatto
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Chebyshev%E2%80%93Gauss_quadrature
+    .. [2] https://people.sc.fsu.edu/~jburkardt/cpp_src/chebyshev1_rule/chebyshev1_rule.html
+    """
+    xi = []
+    w = []
+    for i in range(1, n+1):
+        xi.append((cos((2*i-S.One)/(2*n)*S.Pi)).n(n_digits))
+        w.append((S.Pi/n).n(n_digits))
+    return xi, w
+
+
+def gauss_chebyshev_u(n, n_digits):
+    r"""
+    Computes the Gauss-Chebyshev quadrature [1]_ points and weights of
+    the second kind.
+
+    Explanation
+    ===========
+
+    The Gauss-Chebyshev quadrature of the second kind approximates the
+    integral:
+
+    .. math::
+        \int_{-1}^{1} \sqrt{1-x^2} f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i)
+
+    The nodes `x_i` of an order `n` quadrature rule are the roots of `U_n`
+    and the weights `w_i` are given by:
+
+    .. math::
+        w_i = \frac{\pi}{n+1} \sin^2 \left(\frac{i}{n+1}\pi\right)
+
+    Parameters
+    ==========
+
+    n : the order of quadrature
+
+    n_digits : number of significant digits of the points and weights to return
+
+    Returns
+    =======
+
+    (x, w) : the ``x`` and ``w`` are lists of points and weights as Floats.
+             The points `x_i` and weights `w_i` are returned as ``(x, w)``
+             tuple of lists.
+
+    Examples
+    ========
+
+    >>> from sympy.integrals.quadrature import gauss_chebyshev_u
+    >>> x, w = gauss_chebyshev_u(3, 5)
+    >>> x
+    [0.70711, 0, -0.70711]
+    >>> w
+    [0.3927, 0.7854, 0.3927]
+
+    >>> x, w = gauss_chebyshev_u(6, 5)
+    >>> x
+    [0.90097, 0.62349, 0.22252, -0.22252, -0.62349, -0.90097]
+    >>> w
+    [0.084489, 0.27433, 0.42658, 0.42658, 0.27433, 0.084489]
+
+    See Also
+    ========
+
+    gauss_legendre, gauss_laguerre, gauss_hermite, gauss_gen_laguerre, gauss_chebyshev_t, gauss_jacobi, gauss_lobatto
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Chebyshev%E2%80%93Gauss_quadrature
+    .. [2] https://people.sc.fsu.edu/~jburkardt/cpp_src/chebyshev2_rule/chebyshev2_rule.html
+    """
+    xi = []
+    w = []
+    for i in range(1, n+1):
+        xi.append((cos(i/(n+S.One)*S.Pi)).n(n_digits))
+        w.append((S.Pi/(n+S.One)*sin(i*S.Pi/(n+S.One))**2).n(n_digits))
+    return xi, w
+
+
+def gauss_jacobi(n, alpha, beta, n_digits):
+    r"""
+    Computes the Gauss-Jacobi quadrature [1]_ points and weights.
+
+    Explanation
+    ===========
+
+    The Gauss-Jacobi quadrature of the first kind approximates the integral:
+
+    .. math::
+        \int_{-1}^1 (1-x)^\alpha (1+x)^\beta f(x)\,dx \approx
+            \sum_{i=1}^n w_i f(x_i)
+
+    The nodes `x_i` of an order `n` quadrature rule are the roots of
+    `P^{(\alpha,\beta)}_n` and the weights `w_i` are given by:
+
+    .. math::
+        w_i = -\frac{2n+\alpha+\beta+2}{n+\alpha+\beta+1}
+              \frac{\Gamma(n+\alpha+1)\Gamma(n+\beta+1)}
+              {\Gamma(n+\alpha+\beta+1)(n+1)!}
+              \frac{2^{\alpha+\beta}}{P'_n(x_i)
+              P^{(\alpha,\beta)}_{n+1}(x_i)}
+
+    Parameters
+    ==========
+
+    n : the order of quadrature
+
+    alpha : the first parameter of the Jacobi Polynomial, `\alpha > -1`
+
+    beta : the second parameter of the Jacobi Polynomial, `\beta > -1`
+
+    n_digits : number of significant digits of the points and weights to return
+
+    Returns
+    =======
+
+    (x, w) : the ``x`` and ``w`` are lists of points and weights as Floats.
+             The points `x_i` and weights `w_i` are returned as ``(x, w)``
+             tuple of lists.
+
+    Examples
+    ========
+
+    >>> from sympy import S
+    >>> from sympy.integrals.quadrature import gauss_jacobi
+    >>> x, w = gauss_jacobi(3, S.Half, -S.Half, 5)
+    >>> x
+    [-0.90097, -0.22252, 0.62349]
+    >>> w
+    [1.7063, 1.0973, 0.33795]
+
+    >>> x, w = gauss_jacobi(6, 1, 1, 5)
+    >>> x
+    [-0.87174, -0.5917, -0.2093, 0.2093, 0.5917, 0.87174]
+    >>> w
+    [0.050584, 0.22169, 0.39439, 0.39439, 0.22169, 0.050584]
+
+    See Also
+    ========
+
+    gauss_legendre, gauss_laguerre, gauss_hermite, gauss_gen_laguerre,
+    gauss_chebyshev_t, gauss_chebyshev_u, gauss_lobatto
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Gauss%E2%80%93Jacobi_quadrature
+    .. [2] https://people.sc.fsu.edu/~jburkardt/cpp_src/jacobi_rule/jacobi_rule.html
+    .. [3] https://people.sc.fsu.edu/~jburkardt/cpp_src/gegenbauer_rule/gegenbauer_rule.html
+    """
+    x = Dummy("x")
+    p = jacobi_poly(n, alpha, beta, x, polys=True)
+    pd = p.diff(x)
+    pn = jacobi_poly(n+1, alpha, beta, x, polys=True)
+    xi = []
+    w = []
+    for r in p.real_roots():
+        if isinstance(r, RootOf):
+            r = r.eval_rational(S.One/10**(n_digits+2))
+        xi.append(r.n(n_digits))
+        w.append((
+            - (2*n+alpha+beta+2) / (n+alpha+beta+S.One) *
+            (gamma(n+alpha+1)*gamma(n+beta+1)) /
+            (gamma(n+alpha+beta+S.One)*gamma(n+2)) *
+            2**(alpha+beta) / (pd.subs(x, r) * pn.subs(x, r))).n(n_digits))
+    return xi, w
+
+
+def gauss_lobatto(n, n_digits):
+    r"""
+    Computes the Gauss-Lobatto quadrature [1]_ points and weights.
+
+    Explanation
+    ===========
+
+    The Gauss-Lobatto quadrature approximates the integral:
+
+    .. math::
+        \int_{-1}^1 f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i)
+
+    The nodes `x_i` of an order `n` quadrature rule are the roots of `P'_(n-1)`
+    and the weights `w_i` are given by:
+
+    .. math::
+        &w_i = \frac{2}{n(n-1) \left[P_{n-1}(x_i)\right]^2},\quad x\neq\pm 1\\
+        &w_i = \frac{2}{n(n-1)},\quad x=\pm 1
+
+    Parameters
+    ==========
+
+    n : the order of quadrature
+
+    n_digits : number of significant digits of the points and weights to return
+
+    Returns
+    =======
+
+    (x, w) : the ``x`` and ``w`` are lists of points and weights as Floats.
+             The points `x_i` and weights `w_i` are returned as ``(x, w)``
+             tuple of lists.
+
+    Examples
+    ========
+
+    >>> from sympy.integrals.quadrature import gauss_lobatto
+    >>> x, w = gauss_lobatto(3, 5)
+    >>> x
+    [-1, 0, 1]
+    >>> w
+    [0.33333, 1.3333, 0.33333]
+    >>> x, w = gauss_lobatto(4, 5)
+    >>> x
+    [-1, -0.44721, 0.44721, 1]
+    >>> w
+    [0.16667, 0.83333, 0.83333, 0.16667]
+
+    See Also
+    ========
+
+    gauss_legendre,gauss_laguerre, gauss_gen_laguerre, gauss_hermite, gauss_chebyshev_t, gauss_chebyshev_u, gauss_jacobi
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Gaussian_quadrature#Gauss.E2.80.93Lobatto_rules
+    .. [2] https://web.archive.org/web/20200118141346/http://people.math.sfu.ca/~cbm/aands/page_888.htm
+    """
+    x = Dummy("x")
+    p = legendre_poly(n-1, x, polys=True)
+    pd = p.diff(x)
+    xi = []
+    w = []
+    for r in pd.real_roots():
+        if isinstance(r, RootOf):
+            r = r.eval_rational(S.One/10**(n_digits+2))
+        xi.append(r.n(n_digits))
+        w.append((2/(n*(n-1) * p.subs(x, r)**2)).n(n_digits))
+
+    xi.insert(0, -1)
+    xi.append(1)
+    w.insert(0, (S(2)/(n*(n-1))).n(n_digits))
+    w.append((S(2)/(n*(n-1))).n(n_digits))
+    return xi, w

rl/Lib/site-packages/sympy/integrals/rationaltools.py

@@ -0,0 +1,429 @@
+"""This module implements tools for integrating rational functions. """
+
+from sympy.core.function import Lambda
+from sympy.core.numbers import I
+from sympy.core.singleton import S
+from sympy.core.symbol import (Dummy, Symbol, symbols)
+from sympy.functions.elementary.exponential import log
+from sympy.functions.elementary.trigonometric import atan
+from sympy.polys.polyroots import roots
+from sympy.polys.polytools import cancel
+from sympy.polys.rootoftools import RootSum
+from sympy.polys import Poly, resultant, ZZ
+
+
+def ratint(f, x, **flags):
+    """
+    Performs indefinite integration of rational functions.
+
+    Explanation
+    ===========
+
+    Given a field :math:`K` and a rational function :math:`f = p/q`,
+    where :math:`p` and :math:`q` are polynomials in :math:`K[x]`,
+    returns a function :math:`g` such that :math:`f = g'`.
+
+    Examples
+    ========
+
+    >>> from sympy.integrals.rationaltools import ratint
+    >>> from sympy.abc import x
+
+    >>> ratint(36/(x**5 - 2*x**4 - 2*x**3 + 4*x**2 + x - 2), x)
+    (12*x + 6)/(x**2 - 1) + 4*log(x - 2) - 4*log(x + 1)
+
+    References
+    ==========
+
+    .. [1] M. Bronstein, Symbolic Integration I: Transcendental
+       Functions, Second Edition, Springer-Verlag, 2005, pp. 35-70
+
+    See Also
+    ========
+
+    sympy.integrals.integrals.Integral.doit
+    sympy.integrals.rationaltools.ratint_logpart
+    sympy.integrals.rationaltools.ratint_ratpart
+
+    """
+    if isinstance(f, tuple):
+        p, q = f
+    else:
+        p, q = f.as_numer_denom()
+
+    p, q = Poly(p, x, composite=False, field=True), Poly(q, x, composite=False, field=True)
+
+    coeff, p, q = p.cancel(q)
+    poly, p = p.div(q)
+
+    result = poly.integrate(x).as_expr()
+
+    if p.is_zero:
+        return coeff*result
+
+    g, h = ratint_ratpart(p, q, x)
+
+    P, Q = h.as_numer_denom()
+
+    P = Poly(P, x)
+    Q = Poly(Q, x)
+
+    q, r = P.div(Q)
+
+    result += g + q.integrate(x).as_expr()
+
+    if not r.is_zero:
+        symbol = flags.get('symbol', 't')
+
+        if not isinstance(symbol, Symbol):
+            t = Dummy(symbol)
+        else:
+            t = symbol.as_dummy()
+
+        L = ratint_logpart(r, Q, x, t)
+
+        real = flags.get('real')
+
+        if real is None:
+            if isinstance(f, tuple):
+                p, q = f
+                atoms = p.atoms() | q.atoms()
+            else:
+                atoms = f.atoms()
+
+            for elt in atoms - {x}:
+                if not elt.is_extended_real:
+                    real = False
+                    break
+            else:
+                real = True
+
+        eps = S.Zero
+
+        if not real:
+            for h, q in L:
+                _, h = h.primitive()
+                eps += RootSum(
+                    q, Lambda(t, t*log(h.as_expr())), quadratic=True)
+        else:
+            for h, q in L:
+                _, h = h.primitive()
+                R = log_to_real(h, q, x, t)
+
+                if R is not None:
+                    eps += R
+                else:
+                    eps += RootSum(
+                        q, Lambda(t, t*log(h.as_expr())), quadratic=True)
+
+        result += eps
+
+    return coeff*result
+
+
+def ratint_ratpart(f, g, x):
+    """
+    Horowitz-Ostrogradsky algorithm.
+
+    Explanation
+    ===========
+
+    Given a field K and polynomials f and g in K[x], such that f and g
+    are coprime and deg(f) < deg(g), returns fractions A and B in K(x),
+    such that f/g = A' + B and B has square-free denominator.
+
+    Examples
+    ========
+
+        >>> from sympy.integrals.rationaltools import ratint_ratpart
+        >>> from sympy.abc import x, y
+        >>> from sympy import Poly
+        >>> ratint_ratpart(Poly(1, x, domain='ZZ'),
+        ... Poly(x + 1, x, domain='ZZ'), x)
+        (0, 1/(x + 1))
+        >>> ratint_ratpart(Poly(1, x, domain='EX'),
+        ... Poly(x**2 + y**2, x, domain='EX'), x)
+        (0, 1/(x**2 + y**2))
+        >>> ratint_ratpart(Poly(36, x, domain='ZZ'),
+        ... Poly(x**5 - 2*x**4 - 2*x**3 + 4*x**2 + x - 2, x, domain='ZZ'), x)
+        ((12*x + 6)/(x**2 - 1), 12/(x**2 - x - 2))
+
+    See Also
+    ========
+
+    ratint, ratint_logpart
+    """
+    from sympy.solvers.solvers import solve
+
+    f = Poly(f, x)
+    g = Poly(g, x)
+
+    u, v, _ = g.cofactors(g.diff())
+
+    n = u.degree()
+    m = v.degree()
+
+    A_coeffs = [ Dummy('a' + str(n - i)) for i in range(0, n) ]
+    B_coeffs = [ Dummy('b' + str(m - i)) for i in range(0, m) ]
+
+    C_coeffs = A_coeffs + B_coeffs
+
+    A = Poly(A_coeffs, x, domain=ZZ[C_coeffs])
+    B = Poly(B_coeffs, x, domain=ZZ[C_coeffs])
+
+    H = f - A.diff()*v + A*(u.diff()*v).quo(u) - B*u
+
+    result = solve(H.coeffs(), C_coeffs)
+
+    A = A.as_expr().subs(result)
+    B = B.as_expr().subs(result)
+
+    rat_part = cancel(A/u.as_expr(), x)
+    log_part = cancel(B/v.as_expr(), x)
+
+    return rat_part, log_part
+
+
+def ratint_logpart(f, g, x, t=None):
+    r"""
+    Lazard-Rioboo-Trager algorithm.
+
+    Explanation
+    ===========
+
+    Given a field K and polynomials f and g in K[x], such that f and g
+    are coprime, deg(f) < deg(g) and g is square-free, returns a list
+    of tuples (s_i, q_i) of polynomials, for i = 1..n, such that s_i
+    in K[t, x] and q_i in K[t], and::
+
+                           ___    ___
+                 d  f   d  \  `   \  `
+                 -- - = --  )      )   a log(s_i(a, x))
+                 dx g   dx /__,   /__,
+                          i=1..n a | q_i(a) = 0
+
+    Examples
+    ========
+
+    >>> from sympy.integrals.rationaltools import ratint_logpart
+    >>> from sympy.abc import x
+    >>> from sympy import Poly
+    >>> ratint_logpart(Poly(1, x, domain='ZZ'),
+    ... Poly(x**2 + x + 1, x, domain='ZZ'), x)
+    [(Poly(x + 3*_t/2 + 1/2, x, domain='QQ[_t]'),
+    ...Poly(3*_t**2 + 1, _t, domain='ZZ'))]
+    >>> ratint_logpart(Poly(12, x, domain='ZZ'),
+    ... Poly(x**2 - x - 2, x, domain='ZZ'), x)
+    [(Poly(x - 3*_t/8 - 1/2, x, domain='QQ[_t]'),
+    ...Poly(-_t**2 + 16, _t, domain='ZZ'))]
+
+    See Also
+    ========
+
+    ratint, ratint_ratpart
+    """
+    f, g = Poly(f, x), Poly(g, x)
+
+    t = t or Dummy('t')
+    a, b = g, f - g.diff()*Poly(t, x)
+
+    res, R = resultant(a, b, includePRS=True)
+    res = Poly(res, t, composite=False)
+
+    assert res, "BUG: resultant(%s, %s) cannot be zero" % (a, b)
+
+    R_map, H = {}, []
+
+    for r in R:
+        R_map[r.degree()] = r
+
+    def _include_sign(c, sqf):
+        if c.is_extended_real and (c < 0) == True:
+            h, k = sqf[0]
+            c_poly = c.as_poly(h.gens)
+            sqf[0] = h*c_poly, k
+
+    C, res_sqf = res.sqf_list()
+    _include_sign(C, res_sqf)
+
+    for q, i in res_sqf:
+        _, q = q.primitive()
+
+        if g.degree() == i:
+            H.append((g, q))
+        else:
+            h = R_map[i]
+            h_lc = Poly(h.LC(), t, field=True)
+
+            c, h_lc_sqf = h_lc.sqf_list(all=True)
+            _include_sign(c, h_lc_sqf)
+
+            for a, j in h_lc_sqf:
+                h = h.quo(Poly(a.gcd(q)**j, x))
+
+            inv, coeffs = h_lc.invert(q), [S.One]
+
+            for coeff in h.coeffs()[1:]:
+                coeff = coeff.as_poly(inv.gens)
+                T = (inv*coeff).rem(q)
+                coeffs.append(T.as_expr())
+
+            h = Poly(dict(list(zip(h.monoms(), coeffs))), x)
+
+            H.append((h, q))
+
+    return H
+
+
+def log_to_atan(f, g):
+    """
+    Convert complex logarithms to real arctangents.
+
+    Explanation
+    ===========
+
+    Given a real field K and polynomials f and g in K[x], with g != 0,
+    returns a sum h of arctangents of polynomials in K[x], such that:
+
+                   dh   d         f + I g
+                   -- = -- I log( ------- )
+                   dx   dx        f - I g
+
+    Examples
+    ========
+
+        >>> from sympy.integrals.rationaltools import log_to_atan
+        >>> from sympy.abc import x
+        >>> from sympy import Poly, sqrt, S
+        >>> log_to_atan(Poly(x, x, domain='ZZ'), Poly(1, x, domain='ZZ'))
+        2*atan(x)
+        >>> log_to_atan(Poly(x + S(1)/2, x, domain='QQ'),
+        ... Poly(sqrt(3)/2, x, domain='EX'))
+        2*atan(2*sqrt(3)*x/3 + sqrt(3)/3)
+
+    See Also
+    ========
+
+    log_to_real
+    """
+    if f.degree() < g.degree():
+        f, g = -g, f
+
+    f = f.to_field()
+    g = g.to_field()
+
+    p, q = f.div(g)
+
+    if q.is_zero:
+        return 2*atan(p.as_expr())
+    else:
+        s, t, h = g.gcdex(-f)
+        u = (f*s + g*t).quo(h)
+        A = 2*atan(u.as_expr())
+
+        return A + log_to_atan(s, t)
+
+
+def log_to_real(h, q, x, t):
+    r"""
+    Convert complex logarithms to real functions.
+
+    Explanation
+    ===========
+
+    Given real field K and polynomials h in K[t,x] and q in K[t],
+    returns real function f such that:
+                          ___
+                  df   d  \  `
+                  -- = --  )  a log(h(a, x))
+                  dx   dx /__,
+                         a | q(a) = 0
+
+    Examples
+    ========
+
+        >>> from sympy.integrals.rationaltools import log_to_real
+        >>> from sympy.abc import x, y
+        >>> from sympy import Poly, S
+        >>> log_to_real(Poly(x + 3*y/2 + S(1)/2, x, domain='QQ[y]'),
+        ... Poly(3*y**2 + 1, y, domain='ZZ'), x, y)
+        2*sqrt(3)*atan(2*sqrt(3)*x/3 + sqrt(3)/3)/3
+        >>> log_to_real(Poly(x**2 - 1, x, domain='ZZ'),
+        ... Poly(-2*y + 1, y, domain='ZZ'), x, y)
+        log(x**2 - 1)/2
+
+    See Also
+    ========
+
+    log_to_atan
+    """
+    from sympy.simplify.radsimp import collect
+    u, v = symbols('u,v', cls=Dummy)
+
+    H = h.as_expr().xreplace({t: u + I*v}).expand()
+    Q = q.as_expr().xreplace({t: u + I*v}).expand()
+
+    H_map = collect(H, I, evaluate=False)
+    Q_map = collect(Q, I, evaluate=False)
+
+    a, b = H_map.get(S.One, S.Zero), H_map.get(I, S.Zero)
+    c, d = Q_map.get(S.One, S.Zero), Q_map.get(I, S.Zero)
+
+    R = Poly(resultant(c, d, v), u)
+
+    R_u = roots(R, filter='R')
+
+    if len(R_u) != R.count_roots():
+        return None
+
+    result = S.Zero
+
+    for r_u in R_u.keys():
+        C = Poly(c.xreplace({u: r_u}), v)
+        if not C:
+            # t was split into real and imaginary parts
+            # and denom Q(u, v) = c + I*d. We just found
+            # that c(r_u) is 0 so the roots are in d
+            C = Poly(d.xreplace({u: r_u}), v)
+            # we were going to reject roots from C that
+            # did not set d to zero, but since we are now
+            # using C = d and c is already 0, there is
+            # nothing to check
+            d = S.Zero
+
+        R_v = roots(C, filter='R')
+
+        if len(R_v) != C.count_roots():
+            return None
+
+        R_v_paired = [] # take one from each pair of conjugate roots
+        for r_v in R_v:
+            if r_v not in R_v_paired and -r_v not in R_v_paired:
+                if r_v.is_negative or r_v.could_extract_minus_sign():
+                    R_v_paired.append(-r_v)
+                elif not r_v.is_zero:
+                    R_v_paired.append(r_v)
+
+        for r_v in R_v_paired:
+
+            D = d.xreplace({u: r_u, v: r_v})
+
+            if D.evalf(chop=True) != 0:
+                continue
+
+            A = Poly(a.xreplace({u: r_u, v: r_v}), x)
+            B = Poly(b.xreplace({u: r_u, v: r_v}), x)
+
+            AB = (A**2 + B**2).as_expr()
+
+            result += r_u*log(AB) + r_v*log_to_atan(A, B)
+
+    R_q = roots(q, filter='R')
+
+    if len(R_q) != q.count_roots():
+        return None
+
+    for r in R_q.keys():
+        result += r*log(h.as_expr().subs(t, r))
+
+    return result

rl/Lib/site-packages/sympy/integrals/rde.py

@@ -0,0 +1,800 @@
+"""
+Algorithms for solving the Risch differential equation.
+
+Given a differential field K of characteristic 0 that is a simple
+monomial extension of a base field k and f, g in K, the Risch
+Differential Equation problem is to decide if there exist y in K such
+that Dy + f*y == g and to find one if there are some.  If t is a
+monomial over k and the coefficients of f and g are in k(t), then y is
+in k(t), and the outline of the algorithm here is given as:
+
+1. Compute the normal part n of the denominator of y.  The problem is
+then reduced to finding y' in k<t>, where y == y'/n.
+2. Compute the special part s of the denominator of y.   The problem is
+then reduced to finding y'' in k[t], where y == y''/(n*s)
+3. Bound the degree of y''.
+4. Reduce the equation Dy + f*y == g to a similar equation with f, g in
+k[t].
+5. Find the solutions in k[t] of bounded degree of the reduced equation.
+
+See Chapter 6 of "Symbolic Integration I: Transcendental Functions" by
+Manuel Bronstein.  See also the docstring of risch.py.
+"""
+
+from operator import mul
+from functools import reduce
+
+from sympy.core import oo
+from sympy.core.symbol import Dummy
+
+from sympy.polys import Poly, gcd, ZZ, cancel
+
+from sympy.functions.elementary.complexes import (im, re)
+from sympy.functions.elementary.miscellaneous import sqrt
+
+from sympy.integrals.risch import (gcdex_diophantine, frac_in, derivation,
+    splitfactor, NonElementaryIntegralException, DecrementLevel, recognize_log_derivative)
+
+# TODO: Add messages to NonElementaryIntegralException errors
+
+
+def order_at(a, p, t):
+    """
+    Computes the order of a at p, with respect to t.
+
+    Explanation
+    ===========
+
+    For a, p in k[t], the order of a at p is defined as nu_p(a) = max({n
+    in Z+ such that p**n|a}), where a != 0.  If a == 0, nu_p(a) = +oo.
+
+    To compute the order at a rational function, a/b, use the fact that
+    nu_p(a/b) == nu_p(a) - nu_p(b).
+    """
+    if a.is_zero:
+        return oo
+    if p == Poly(t, t):
+        return a.as_poly(t).ET()[0][0]
+
+    # Uses binary search for calculating the power. power_list collects the tuples
+    # (p^k,k) where each k is some power of 2. After deciding the largest k
+    # such that k is power of 2 and p^k|a the loop iteratively calculates
+    # the actual power.
+    power_list = []
+    p1 = p
+    r = a.rem(p1)
+    tracks_power = 1
+    while r.is_zero:
+        power_list.append((p1,tracks_power))
+        p1 = p1*p1
+        tracks_power *= 2
+        r = a.rem(p1)
+    n = 0
+    product = Poly(1, t)
+    while len(power_list) != 0:
+        final = power_list.pop()
+        productf = product*final[0]
+        r = a.rem(productf)
+        if r.is_zero:
+            n += final[1]
+            product = productf
+    return n
+
+
+def order_at_oo(a, d, t):
+    """
+    Computes the order of a/d at oo (infinity), with respect to t.
+
+    For f in k(t), the order or f at oo is defined as deg(d) - deg(a), where
+    f == a/d.
+    """
+    if a.is_zero:
+        return oo
+    return d.degree(t) - a.degree(t)
+
+
+def weak_normalizer(a, d, DE, z=None):
+    """
+    Weak normalization.
+
+    Explanation
+    ===========
+
+    Given a derivation D on k[t] and f == a/d in k(t), return q in k[t]
+    such that f - Dq/q is weakly normalized with respect to t.
+
+    f in k(t) is said to be "weakly normalized" with respect to t if
+    residue_p(f) is not a positive integer for any normal irreducible p
+    in k[t] such that f is in R_p (Definition 6.1.1).  If f has an
+    elementary integral, this is equivalent to no logarithm of
+    integral(f) whose argument depends on t has a positive integer
+    coefficient, where the arguments of the logarithms not in k(t) are
+    in k[t].
+
+    Returns (q, f - Dq/q)
+    """
+    z = z or Dummy('z')
+    dn, ds = splitfactor(d, DE)
+
+    # Compute d1, where dn == d1*d2**2*...*dn**n is a square-free
+    # factorization of d.
+    g = gcd(dn, dn.diff(DE.t))
+    d_sqf_part = dn.quo(g)
+    d1 = d_sqf_part.quo(gcd(d_sqf_part, g))
+
+    a1, b = gcdex_diophantine(d.quo(d1).as_poly(DE.t), d1.as_poly(DE.t),
+        a.as_poly(DE.t))
+    r = (a - Poly(z, DE.t)*derivation(d1, DE)).as_poly(DE.t).resultant(
+        d1.as_poly(DE.t))
+    r = Poly(r, z)
+
+    if not r.expr.has(z):
+        return (Poly(1, DE.t), (a, d))
+
+    N = [i for i in r.real_roots() if i in ZZ and i > 0]
+
+    q = reduce(mul, [gcd(a - Poly(n, DE.t)*derivation(d1, DE), d1) for n in N],
+        Poly(1, DE.t))
+
+    dq = derivation(q, DE)
+    sn = q*a - d*dq
+    sd = q*d
+    sn, sd = sn.cancel(sd, include=True)
+
+    return (q, (sn, sd))
+
+
+def normal_denom(fa, fd, ga, gd, DE):
+    """
+    Normal part of the denominator.
+
+    Explanation
+    ===========
+
+    Given a derivation D on k[t] and f, g in k(t) with f weakly
+    normalized with respect to t, either raise NonElementaryIntegralException,
+    in which case the equation Dy + f*y == g has no solution in k(t), or the
+    quadruplet (a, b, c, h) such that a, h in k[t], b, c in k<t>, and for any
+    solution y in k(t) of Dy + f*y == g, q = y*h in k<t> satisfies
+    a*Dq + b*q == c.
+
+    This constitutes step 1 in the outline given in the rde.py docstring.
+    """
+    dn, ds = splitfactor(fd, DE)
+    en, es = splitfactor(gd, DE)
+
+    p = dn.gcd(en)
+    h = en.gcd(en.diff(DE.t)).quo(p.gcd(p.diff(DE.t)))
+
+    a = dn*h
+    c = a*h
+    if c.div(en)[1]:
+        # en does not divide dn*h**2
+        raise NonElementaryIntegralException
+    ca = c*ga
+    ca, cd = ca.cancel(gd, include=True)
+
+    ba = a*fa - dn*derivation(h, DE)*fd
+    ba, bd = ba.cancel(fd, include=True)
+
+    # (dn*h, dn*h*f - dn*Dh, dn*h**2*g, h)
+    return (a, (ba, bd), (ca, cd), h)
+
+
+def special_denom(a, ba, bd, ca, cd, DE, case='auto'):
+    """
+    Special part of the denominator.
+
+    Explanation
+    ===========
+
+    case is one of {'exp', 'tan', 'primitive'} for the hyperexponential,
+    hypertangent, and primitive cases, respectively.  For the
+    hyperexponential (resp. hypertangent) case, given a derivation D on
+    k[t] and a in k[t], b, c, in k<t> with Dt/t in k (resp. Dt/(t**2 + 1) in
+    k, sqrt(-1) not in k), a != 0, and gcd(a, t) == 1 (resp.
+    gcd(a, t**2 + 1) == 1), return the quadruplet (A, B, C, 1/h) such that
+    A, B, C, h in k[t] and for any solution q in k<t> of a*Dq + b*q == c,
+    r = qh in k[t] satisfies A*Dr + B*r == C.
+
+    For ``case == 'primitive'``, k<t> == k[t], so it returns (a, b, c, 1) in
+    this case.
+
+    This constitutes step 2 of the outline given in the rde.py docstring.
+    """
+    # TODO: finish writing this and write tests
+
+    if case == 'auto':
+        case = DE.case
+
+    if case == 'exp':
+        p = Poly(DE.t, DE.t)
+    elif case == 'tan':
+        p = Poly(DE.t**2 + 1, DE.t)
+    elif case in ('primitive', 'base'):
+        B = ba.to_field().quo(bd)
+        C = ca.to_field().quo(cd)
+        return (a, B, C, Poly(1, DE.t))
+    else:
+        raise ValueError("case must be one of {'exp', 'tan', 'primitive', "
+            "'base'}, not %s." % case)
+
+    nb = order_at(ba, p, DE.t) - order_at(bd, p, DE.t)
+    nc = order_at(ca, p, DE.t) - order_at(cd, p, DE.t)
+
+    n = min(0, nc - min(0, nb))
+    if not nb:
+        # Possible cancellation.
+        from .prde import parametric_log_deriv
+        if case == 'exp':
+            dcoeff = DE.d.quo(Poly(DE.t, DE.t))
+            with DecrementLevel(DE):  # We are guaranteed to not have problems,
+                                      # because case != 'base'.
+                alphaa, alphad = frac_in(-ba.eval(0)/bd.eval(0)/a.eval(0), DE.t)
+                etaa, etad = frac_in(dcoeff, DE.t)
+                A = parametric_log_deriv(alphaa, alphad, etaa, etad, DE)
+                if A is not None:
+                    Q, m, z = A
+                    if Q == 1:
+                        n = min(n, m)
+
+        elif case == 'tan':
+            dcoeff = DE.d.quo(Poly(DE.t**2+1, DE.t))
+            with DecrementLevel(DE):  # We are guaranteed to not have problems,
+                                      # because case != 'base'.
+                alphaa, alphad = frac_in(im(-ba.eval(sqrt(-1))/bd.eval(sqrt(-1))/a.eval(sqrt(-1))), DE.t)
+                betaa, betad = frac_in(re(-ba.eval(sqrt(-1))/bd.eval(sqrt(-1))/a.eval(sqrt(-1))), DE.t)
+                etaa, etad = frac_in(dcoeff, DE.t)
+
+                if recognize_log_derivative(Poly(2, DE.t)*betaa, betad, DE):
+                    A = parametric_log_deriv(alphaa*Poly(sqrt(-1), DE.t)*betad+alphad*betaa, alphad*betad, etaa, etad, DE)
+                    if A is not None:
+                       Q, m, z = A
+                       if Q == 1:
+                           n = min(n, m)
+    N = max(0, -nb, n - nc)
+    pN = p**N
+    pn = p**-n
+
+    A = a*pN
+    B = ba*pN.quo(bd) + Poly(n, DE.t)*a*derivation(p, DE).quo(p)*pN
+    C = (ca*pN*pn).quo(cd)
+    h = pn
+
+    # (a*p**N, (b + n*a*Dp/p)*p**N, c*p**(N - n), p**-n)
+    return (A, B, C, h)
+
+
+def bound_degree(a, b, cQ, DE, case='auto', parametric=False):
+    """
+    Bound on polynomial solutions.
+
+    Explanation
+    ===========
+
+    Given a derivation D on k[t] and ``a``, ``b``, ``c`` in k[t] with ``a != 0``, return
+    n in ZZ such that deg(q) <= n for any solution q in k[t] of
+    a*Dq + b*q == c, when parametric=False, or deg(q) <= n for any solution
+    c1, ..., cm in Const(k) and q in k[t] of a*Dq + b*q == Sum(ci*gi, (i, 1, m))
+    when parametric=True.
+
+    For ``parametric=False``, ``cQ`` is ``c``, a ``Poly``; for ``parametric=True``, ``cQ`` is Q ==
+    [q1, ..., qm], a list of Polys.
+
+    This constitutes step 3 of the outline given in the rde.py docstring.
+    """
+    # TODO: finish writing this and write tests
+
+    if case == 'auto':
+        case = DE.case
+
+    da = a.degree(DE.t)
+    db = b.degree(DE.t)
+
+    # The parametric and regular cases are identical, except for this part
+    if parametric:
+        dc = max(i.degree(DE.t) for i in cQ)
+    else:
+        dc = cQ.degree(DE.t)
+
+    alpha = cancel(-b.as_poly(DE.t).LC().as_expr()/
+        a.as_poly(DE.t).LC().as_expr())
+
+    if case == 'base':
+        n = max(0, dc - max(db, da - 1))
+        if db == da - 1 and alpha.is_Integer:
+            n = max(0, alpha, dc - db)
+
+    elif case == 'primitive':
+        if db > da:
+            n = max(0, dc - db)
+        else:
+            n = max(0, dc - da + 1)
+
+        etaa, etad = frac_in(DE.d, DE.T[DE.level - 1])
+
+        t1 = DE.t
+        with DecrementLevel(DE):
+            alphaa, alphad = frac_in(alpha, DE.t)
+            if db == da - 1:
+                from .prde import limited_integrate
+                # if alpha == m*Dt + Dz for z in k and m in ZZ:
+                try:
+                    (za, zd), m = limited_integrate(alphaa, alphad, [(etaa, etad)],
+                        DE)
+                except NonElementaryIntegralException:
+                    pass
+                else:
+                    if len(m) != 1:
+                        raise ValueError("Length of m should be 1")
+                    n = max(n, m[0])
+
+            elif db == da:
+                # if alpha == Dz/z for z in k*:
+                    # beta = -lc(a*Dz + b*z)/(z*lc(a))
+                    # if beta == m*Dt + Dw for w in k and m in ZZ:
+                        # n = max(n, m)
+                from .prde import is_log_deriv_k_t_radical_in_field
+                A = is_log_deriv_k_t_radical_in_field(alphaa, alphad, DE)
+                if A is not None:
+                    aa, z = A
+                    if aa == 1:
+                        beta = -(a*derivation(z, DE).as_poly(t1) +
+                            b*z.as_poly(t1)).LC()/(z.as_expr()*a.LC())
+                        betaa, betad = frac_in(beta, DE.t)
+                        from .prde import limited_integrate
+                        try:
+                            (za, zd), m = limited_integrate(betaa, betad,
+                                [(etaa, etad)], DE)
+                        except NonElementaryIntegralException:
+                            pass
+                        else:
+                            if len(m) != 1:
+                                raise ValueError("Length of m should be 1")
+                            n = max(n, m[0].as_expr())
+
+    elif case == 'exp':
+        from .prde import parametric_log_deriv
+
+        n = max(0, dc - max(db, da))
+        if da == db:
+            etaa, etad = frac_in(DE.d.quo(Poly(DE.t, DE.t)), DE.T[DE.level - 1])
+            with DecrementLevel(DE):
+                alphaa, alphad = frac_in(alpha, DE.t)
+                A = parametric_log_deriv(alphaa, alphad, etaa, etad, DE)
+                if A is not None:
+                    # if alpha == m*Dt/t + Dz/z for z in k* and m in ZZ:
+                        # n = max(n, m)
+                    a, m, z = A
+                    if a == 1:
+                        n = max(n, m)
+
+    elif case in ('tan', 'other_nonlinear'):
+        delta = DE.d.degree(DE.t)
+        lam = DE.d.LC()
+        alpha = cancel(alpha/lam)
+        n = max(0, dc - max(da + delta - 1, db))
+        if db == da + delta - 1 and alpha.is_Integer:
+            n = max(0, alpha, dc - db)
+
+    else:
+        raise ValueError("case must be one of {'exp', 'tan', 'primitive', "
+            "'other_nonlinear', 'base'}, not %s." % case)
+
+    return n
+
+
+def spde(a, b, c, n, DE):
+    """
+    Rothstein's Special Polynomial Differential Equation algorithm.
+
+    Explanation
+    ===========
+
+    Given a derivation D on k[t], an integer n and ``a``,``b``,``c`` in k[t] with
+    ``a != 0``, either raise NonElementaryIntegralException, in which case the
+    equation a*Dq + b*q == c has no solution of degree at most ``n`` in
+    k[t], or return the tuple (B, C, m, alpha, beta) such that B, C,
+    alpha, beta in k[t], m in ZZ, and any solution q in k[t] of degree
+    at most n of a*Dq + b*q == c must be of the form
+    q == alpha*h + beta, where h in k[t], deg(h) <= m, and Dh + B*h == C.
+
+    This constitutes step 4 of the outline given in the rde.py docstring.
+    """
+    zero = Poly(0, DE.t)
+
+    alpha = Poly(1, DE.t)
+    beta = Poly(0, DE.t)
+
+    while True:
+        if c.is_zero:
+            return (zero, zero, 0, zero, beta)  # -1 is more to the point
+        if (n < 0) is True:
+            raise NonElementaryIntegralException
+
+        g = a.gcd(b)
+        if not c.rem(g).is_zero:  # g does not divide c
+            raise NonElementaryIntegralException
+
+        a, b, c = a.quo(g), b.quo(g), c.quo(g)
+
+        if a.degree(DE.t) == 0:
+            b = b.to_field().quo(a)
+            c = c.to_field().quo(a)
+            return (b, c, n, alpha, beta)
+
+        r, z = gcdex_diophantine(b, a, c)
+        b += derivation(a, DE)
+        c = z - derivation(r, DE)
+        n -= a.degree(DE.t)
+
+        beta += alpha * r
+        alpha *= a
+
+def no_cancel_b_large(b, c, n, DE):
+    """
+    Poly Risch Differential Equation - No cancellation: deg(b) large enough.
+
+    Explanation
+    ===========
+
+    Given a derivation D on k[t], ``n`` either an integer or +oo, and ``b``,``c``
+    in k[t] with ``b != 0`` and either D == d/dt or
+    deg(b) > max(0, deg(D) - 1), either raise NonElementaryIntegralException, in
+    which case the equation ``Dq + b*q == c`` has no solution of degree at
+    most n in k[t], or a solution q in k[t] of this equation with
+    ``deg(q) < n``.
+    """
+    q = Poly(0, DE.t)
+
+    while not c.is_zero:
+        m = c.degree(DE.t) - b.degree(DE.t)
+        if not 0 <= m <= n:  # n < 0 or m < 0 or m > n
+            raise NonElementaryIntegralException
+
+        p = Poly(c.as_poly(DE.t).LC()/b.as_poly(DE.t).LC()*DE.t**m, DE.t,
+            expand=False)
+        q = q + p
+        n = m - 1
+        c = c - derivation(p, DE) - b*p
+
+    return q
+
+
+def no_cancel_b_small(b, c, n, DE):
+    """
+    Poly Risch Differential Equation - No cancellation: deg(b) small enough.
+
+    Explanation
+    ===========
+
+    Given a derivation D on k[t], ``n`` either an integer or +oo, and ``b``,``c``
+    in k[t] with deg(b) < deg(D) - 1 and either D == d/dt or
+    deg(D) >= 2, either raise NonElementaryIntegralException, in which case the
+    equation Dq + b*q == c has no solution of degree at most n in k[t],
+    or a solution q in k[t] of this equation with deg(q) <= n, or the
+    tuple (h, b0, c0) such that h in k[t], b0, c0, in k, and for any
+    solution q in k[t] of degree at most n of Dq + bq == c, y == q - h
+    is a solution in k of Dy + b0*y == c0.
+    """
+    q = Poly(0, DE.t)
+
+    while not c.is_zero:
+        if n == 0:
+            m = 0
+        else:
+            m = c.degree(DE.t) - DE.d.degree(DE.t) + 1
+
+        if not 0 <= m <= n:  # n < 0 or m < 0 or m > n
+            raise NonElementaryIntegralException
+
+        if m > 0:
+            p = Poly(c.as_poly(DE.t).LC()/(m*DE.d.as_poly(DE.t).LC())*DE.t**m,
+                DE.t, expand=False)
+        else:
+            if b.degree(DE.t) != c.degree(DE.t):
+                raise NonElementaryIntegralException
+            if b.degree(DE.t) == 0:
+                return (q, b.as_poly(DE.T[DE.level - 1]),
+                    c.as_poly(DE.T[DE.level - 1]))
+            p = Poly(c.as_poly(DE.t).LC()/b.as_poly(DE.t).LC(), DE.t,
+                expand=False)
+
+        q = q + p
+        n = m - 1
+        c = c - derivation(p, DE) - b*p
+
+    return q
+
+
+# TODO: better name for this function
+def no_cancel_equal(b, c, n, DE):
+    """
+    Poly Risch Differential Equation - No cancellation: deg(b) == deg(D) - 1
+
+    Explanation
+    ===========
+
+    Given a derivation D on k[t] with deg(D) >= 2, n either an integer
+    or +oo, and b, c in k[t] with deg(b) == deg(D) - 1, either raise
+    NonElementaryIntegralException, in which case the equation Dq + b*q == c has
+    no solution of degree at most n in k[t], or a solution q in k[t] of
+    this equation with deg(q) <= n, or the tuple (h, m, C) such that h
+    in k[t], m in ZZ, and C in k[t], and for any solution q in k[t] of
+    degree at most n of Dq + b*q == c, y == q - h is a solution in k[t]
+    of degree at most m of Dy + b*y == C.
+    """
+    q = Poly(0, DE.t)
+    lc = cancel(-b.as_poly(DE.t).LC()/DE.d.as_poly(DE.t).LC())
+    if lc.is_Integer and lc.is_positive:
+        M = lc
+    else:
+        M = -1
+
+    while not c.is_zero:
+        m = max(M, c.degree(DE.t) - DE.d.degree(DE.t) + 1)
+
+        if not 0 <= m <= n:  # n < 0 or m < 0 or m > n
+            raise NonElementaryIntegralException
+
+        u = cancel(m*DE.d.as_poly(DE.t).LC() + b.as_poly(DE.t).LC())
+        if u.is_zero:
+            return (q, m, c)
+        if m > 0:
+            p = Poly(c.as_poly(DE.t).LC()/u*DE.t**m, DE.t, expand=False)
+        else:
+            if c.degree(DE.t) != DE.d.degree(DE.t) - 1:
+                raise NonElementaryIntegralException
+            else:
+                p = c.as_poly(DE.t).LC()/b.as_poly(DE.t).LC()
+
+        q = q + p
+        n = m - 1
+        c = c - derivation(p, DE) - b*p
+
+    return q
+
+
+def cancel_primitive(b, c, n, DE):
+    """
+    Poly Risch Differential Equation - Cancellation: Primitive case.
+
+    Explanation
+    ===========
+
+    Given a derivation D on k[t], n either an integer or +oo, ``b`` in k, and
+    ``c`` in k[t] with Dt in k and ``b != 0``, either raise
+    NonElementaryIntegralException, in which case the equation Dq + b*q == c
+    has no solution of degree at most n in k[t], or a solution q in k[t] of
+    this equation with deg(q) <= n.
+    """
+    # Delayed imports
+    from .prde import is_log_deriv_k_t_radical_in_field
+    with DecrementLevel(DE):
+        ba, bd = frac_in(b, DE.t)
+        A = is_log_deriv_k_t_radical_in_field(ba, bd, DE)
+        if A is not None:
+            n, z = A
+            if n == 1:  # b == Dz/z
+                raise NotImplementedError("is_deriv_in_field() is required to "
+                    " solve this problem.")
+                # if z*c == Dp for p in k[t] and deg(p) <= n:
+                #     return p/z
+                # else:
+                #     raise NonElementaryIntegralException
+
+    if c.is_zero:
+        return c  # return 0
+
+    if n < c.degree(DE.t):
+        raise NonElementaryIntegralException
+
+    q = Poly(0, DE.t)
+    while not c.is_zero:
+        m = c.degree(DE.t)
+        if n < m:
+            raise NonElementaryIntegralException
+        with DecrementLevel(DE):
+            a2a, a2d = frac_in(c.LC(), DE.t)
+            sa, sd = rischDE(ba, bd, a2a, a2d, DE)
+        stm = Poly(sa.as_expr()/sd.as_expr()*DE.t**m, DE.t, expand=False)
+        q += stm
+        n = m - 1
+        c -= b*stm + derivation(stm, DE)
+
+    return q
+
+
+def cancel_exp(b, c, n, DE):
+    """
+    Poly Risch Differential Equation - Cancellation: Hyperexponential case.
+
+    Explanation
+    ===========
+
+    Given a derivation D on k[t], n either an integer or +oo, ``b`` in k, and
+    ``c`` in k[t] with Dt/t in k and ``b != 0``, either raise
+    NonElementaryIntegralException, in which case the equation Dq + b*q == c
+    has no solution of degree at most n in k[t], or a solution q in k[t] of
+    this equation with deg(q) <= n.
+    """
+    from .prde import parametric_log_deriv
+    eta = DE.d.quo(Poly(DE.t, DE.t)).as_expr()
+
+    with DecrementLevel(DE):
+        etaa, etad = frac_in(eta, DE.t)
+        ba, bd = frac_in(b, DE.t)
+        A = parametric_log_deriv(ba, bd, etaa, etad, DE)
+        if A is not None:
+            a, m, z = A
+            if a == 1:
+                raise NotImplementedError("is_deriv_in_field() is required to "
+                    "solve this problem.")
+                # if c*z*t**m == Dp for p in k<t> and q = p/(z*t**m) in k[t] and
+                # deg(q) <= n:
+                #     return q
+                # else:
+                #     raise NonElementaryIntegralException
+
+    if c.is_zero:
+        return c  # return 0
+
+    if n < c.degree(DE.t):
+        raise NonElementaryIntegralException
+
+    q = Poly(0, DE.t)
+    while not c.is_zero:
+        m = c.degree(DE.t)
+        if n < m:
+            raise NonElementaryIntegralException
+        # a1 = b + m*Dt/t
+        a1 = b.as_expr()
+        with DecrementLevel(DE):
+            # TODO: Write a dummy function that does this idiom
+            a1a, a1d = frac_in(a1, DE.t)
+            a1a = a1a*etad + etaa*a1d*Poly(m, DE.t)
+            a1d = a1d*etad
+
+            a2a, a2d = frac_in(c.LC(), DE.t)
+
+            sa, sd = rischDE(a1a, a1d, a2a, a2d, DE)
+        stm = Poly(sa.as_expr()/sd.as_expr()*DE.t**m, DE.t, expand=False)
+        q += stm
+        n = m - 1
+        c -= b*stm + derivation(stm, DE)  # deg(c) becomes smaller
+    return q
+
+
+def solve_poly_rde(b, cQ, n, DE, parametric=False):
+    """
+    Solve a Polynomial Risch Differential Equation with degree bound ``n``.
+
+    This constitutes step 4 of the outline given in the rde.py docstring.
+
+    For parametric=False, cQ is c, a Poly; for parametric=True, cQ is Q ==
+    [q1, ..., qm], a list of Polys.
+    """
+    # No cancellation
+    if not b.is_zero and (DE.case == 'base' or
+            b.degree(DE.t) > max(0, DE.d.degree(DE.t) - 1)):
+
+        if parametric:
+            # Delayed imports
+            from .prde import prde_no_cancel_b_large
+            return prde_no_cancel_b_large(b, cQ, n, DE)
+        return no_cancel_b_large(b, cQ, n, DE)
+
+    elif (b.is_zero or b.degree(DE.t) < DE.d.degree(DE.t) - 1) and \
+            (DE.case == 'base' or DE.d.degree(DE.t) >= 2):
+
+        if parametric:
+            from .prde import prde_no_cancel_b_small
+            return prde_no_cancel_b_small(b, cQ, n, DE)
+
+        R = no_cancel_b_small(b, cQ, n, DE)
+
+        if isinstance(R, Poly):
+            return R
+        else:
+            # XXX: Might k be a field? (pg. 209)
+            h, b0, c0 = R
+            with DecrementLevel(DE):
+                b0, c0 = b0.as_poly(DE.t), c0.as_poly(DE.t)
+                if b0 is None:  # See above comment
+                    raise ValueError("b0 should be a non-Null value")
+                if c0 is  None:
+                    raise ValueError("c0 should be a non-Null value")
+                y = solve_poly_rde(b0, c0, n, DE).as_poly(DE.t)
+            return h + y
+
+    elif DE.d.degree(DE.t) >= 2 and b.degree(DE.t) == DE.d.degree(DE.t) - 1 and \
+            n > -b.as_poly(DE.t).LC()/DE.d.as_poly(DE.t).LC():
+
+        # TODO: Is this check necessary, and if so, what should it do if it fails?
+        # b comes from the first element returned from spde()
+        if not b.as_poly(DE.t).LC().is_number:
+            raise TypeError("Result should be a number")
+
+        if parametric:
+            raise NotImplementedError("prde_no_cancel_b_equal() is not yet "
+                "implemented.")
+
+        R = no_cancel_equal(b, cQ, n, DE)
+
+        if isinstance(R, Poly):
+            return R
+        else:
+            h, m, C = R
+            # XXX: Or should it be rischDE()?
+            y = solve_poly_rde(b, C, m, DE)
+            return h + y
+
+    else:
+        # Cancellation
+        if b.is_zero:
+            raise NotImplementedError("Remaining cases for Poly (P)RDE are "
+            "not yet implemented (is_deriv_in_field() required).")
+        else:
+            if DE.case == 'exp':
+                if parametric:
+                    raise NotImplementedError("Parametric RDE cancellation "
+                        "hyperexponential case is not yet implemented.")
+                return cancel_exp(b, cQ, n, DE)
+
+            elif DE.case == 'primitive':
+                if parametric:
+                    raise NotImplementedError("Parametric RDE cancellation "
+                        "primitive case is not yet implemented.")
+                return cancel_primitive(b, cQ, n, DE)
+
+            else:
+                raise NotImplementedError("Other Poly (P)RDE cancellation "
+                    "cases are not yet implemented (%s)." % DE.case)
+
+        if parametric:
+            raise NotImplementedError("Remaining cases for Poly PRDE not yet "
+                "implemented.")
+        raise NotImplementedError("Remaining cases for Poly RDE not yet "
+            "implemented.")
+
+
+def rischDE(fa, fd, ga, gd, DE):
+    """
+    Solve a Risch Differential Equation: Dy + f*y == g.
+
+    Explanation
+    ===========
+
+    See the outline in the docstring of rde.py for more information
+    about the procedure used.  Either raise NonElementaryIntegralException, in
+    which case there is no solution y in the given differential field,
+    or return y in k(t) satisfying Dy + f*y == g, or raise
+    NotImplementedError, in which case, the algorithms necessary to
+    solve the given Risch Differential Equation have not yet been
+    implemented.
+    """
+    _, (fa, fd) = weak_normalizer(fa, fd, DE)
+    a, (ba, bd), (ca, cd), hn = normal_denom(fa, fd, ga, gd, DE)
+    A, B, C, hs = special_denom(a, ba, bd, ca, cd, DE)
+    try:
+        # Until this is fully implemented, use oo.  Note that this will almost
+        # certainly cause non-termination in spde() (unless A == 1), and
+        # *might* lead to non-termination in the next step for a nonelementary
+        # integral (I don't know for certain yet).  Fortunately, spde() is
+        # currently written recursively, so this will just give
+        # RuntimeError: maximum recursion depth exceeded.
+        n = bound_degree(A, B, C, DE)
+    except NotImplementedError:
+        # Useful for debugging:
+        # import warnings
+        # warnings.warn("rischDE: Proceeding with n = oo; may cause "
+        #     "non-termination.")
+        n = oo
+
+    B, C, m, alpha, beta = spde(A, B, C, n, DE)
+    if C.is_zero:
+        y = C
+    else:
+        y = solve_poly_rde(B, C, m, DE)
+
+    return (alpha*y + beta, hn*hs)

rl/Lib/site-packages/sympy/integrals/singularityfunctions.py

@@ -0,0 +1,63 @@
+from sympy.functions import SingularityFunction, DiracDelta
+from sympy.integrals import integrate
+
+
+def singularityintegrate(f, x):
+    """
+    This function handles the indefinite integrations of Singularity functions.
+    The ``integrate`` function calls this function internally whenever an
+    instance of SingularityFunction is passed as argument.
+
+    Explanation
+    ===========
+
+    The idea for integration is the following:
+
+    - If we are dealing with a SingularityFunction expression,
+      i.e. ``SingularityFunction(x, a, n)``, we just return
+      ``SingularityFunction(x, a, n + 1)/(n + 1)`` if ``n >= 0`` and
+      ``SingularityFunction(x, a, n + 1)`` if ``n < 0``.
+
+    - If the node is a multiplication or power node having a
+      SingularityFunction term we rewrite the whole expression in terms of
+      Heaviside and DiracDelta and then integrate the output. Lastly, we
+      rewrite the output of integration back in terms of SingularityFunction.
+
+    - If none of the above case arises, we return None.
+
+    Examples
+    ========
+
+    >>> from sympy.integrals.singularityfunctions import singularityintegrate
+    >>> from sympy import SingularityFunction, symbols, Function
+    >>> x, a, n, y = symbols('x a n y')
+    >>> f = Function('f')
+    >>> singularityintegrate(SingularityFunction(x, a, 3), x)
+    SingularityFunction(x, a, 4)/4
+    >>> singularityintegrate(5*SingularityFunction(x, 5, -2), x)
+    5*SingularityFunction(x, 5, -1)
+    >>> singularityintegrate(6*SingularityFunction(x, 5, -1), x)
+    6*SingularityFunction(x, 5, 0)
+    >>> singularityintegrate(x*SingularityFunction(x, 0, -1), x)
+    0
+    >>> singularityintegrate(SingularityFunction(x, 1, -1) * f(x), x)
+    f(1)*SingularityFunction(x, 1, 0)
+
+    """
+
+    if not f.has(SingularityFunction):
+        return None
+
+    if isinstance(f, SingularityFunction):
+        x, a, n = f.args
+        if n.is_positive or n.is_zero:
+            return SingularityFunction(x, a, n + 1)/(n + 1)
+        elif n in (-1, -2, -3, -4):
+            return SingularityFunction(x, a, n + 1)
+
+    if f.is_Mul or f.is_Pow:
+
+        expr = f.rewrite(DiracDelta)
+        expr = integrate(expr, x)
+        return expr.rewrite(SingularityFunction)
+    return None

rl/Lib/site-packages/sympy/integrals/tests/test_deltafunctions.py

@@ -0,0 +1,79 @@
+from sympy.core.function import Function
+from sympy.core.numbers import (Rational, pi)
+from sympy.core.singleton import S
+from sympy.core.symbol import symbols
+from sympy.functions.elementary.trigonometric import (cos, sin)
+from sympy.functions.special.delta_functions import (DiracDelta, Heaviside)
+from sympy.integrals.deltafunctions import change_mul, deltaintegrate
+
+f = Function("f")
+x_1, x_2, x, y, z = symbols("x_1 x_2 x y z")
+
+
+def test_change_mul():
+    assert change_mul(x, x) == (None, None)
+    assert change_mul(x*y, x) == (None, None)
+    assert change_mul(x*y*DiracDelta(x), x) == (DiracDelta(x), x*y)
+    assert change_mul(x*y*DiracDelta(x)*DiracDelta(y), x) == \
+        (DiracDelta(x), x*y*DiracDelta(y))
+    assert change_mul(DiracDelta(x)**2, x) == \
+        (DiracDelta(x), DiracDelta(x))
+    assert change_mul(y*DiracDelta(x)**2, x) == \
+        (DiracDelta(x), y*DiracDelta(x))
+
+
+def test_deltaintegrate():
+    assert deltaintegrate(x, x) is None
+    assert deltaintegrate(x + DiracDelta(x), x) is None
+    assert deltaintegrate(DiracDelta(x, 0), x) == Heaviside(x)
+    for n in range(10):
+        assert deltaintegrate(DiracDelta(x, n + 1), x) == DiracDelta(x, n)
+    assert deltaintegrate(DiracDelta(x), x) == Heaviside(x)
+    assert deltaintegrate(DiracDelta(-x), x) == Heaviside(x)
+    assert deltaintegrate(DiracDelta(x - y), x) == Heaviside(x - y)
+    assert deltaintegrate(DiracDelta(y - x), x) == Heaviside(x - y)
+
+    assert deltaintegrate(x*DiracDelta(x), x) == 0
+    assert deltaintegrate((x - y)*DiracDelta(x - y), x) == 0
+
+    assert deltaintegrate(DiracDelta(x)**2, x) == DiracDelta(0)*Heaviside(x)
+    assert deltaintegrate(y*DiracDelta(x)**2, x) == \
+        y*DiracDelta(0)*Heaviside(x)
+    assert deltaintegrate(DiracDelta(x, 1), x) == DiracDelta(x, 0)
+    assert deltaintegrate(y*DiracDelta(x, 1), x) == y*DiracDelta(x, 0)
+    assert deltaintegrate(DiracDelta(x, 1)**2, x) == -DiracDelta(0, 2)*Heaviside(x)
+    assert deltaintegrate(y*DiracDelta(x, 1)**2, x) == -y*DiracDelta(0, 2)*Heaviside(x)
+
+
+    assert deltaintegrate(DiracDelta(x) * f(x), x) == f(0) * Heaviside(x)
+    assert deltaintegrate(DiracDelta(-x) * f(x), x) == f(0) * Heaviside(x)
+    assert deltaintegrate(DiracDelta(x - 1) * f(x), x) == f(1) * Heaviside(x - 1)
+    assert deltaintegrate(DiracDelta(1 - x) * f(x), x) == f(1) * Heaviside(x - 1)
+    assert deltaintegrate(DiracDelta(x**2 + x - 2), x) == \
+        Heaviside(x - 1)/3 + Heaviside(x + 2)/3
+
+    p = cos(x)*(DiracDelta(x) + DiracDelta(x**2 - 1))*sin(x)*(x - pi)
+    assert deltaintegrate(p, x) - (-pi*(cos(1)*Heaviside(-1 + x)*sin(1)/2 - \
+        cos(1)*Heaviside(1 + x)*sin(1)/2) + \
+        cos(1)*Heaviside(1 + x)*sin(1)/2 + \
+        cos(1)*Heaviside(-1 + x)*sin(1)/2) == 0
+
+    p = x_2*DiracDelta(x - x_2)*DiracDelta(x_2 - x_1)
+    assert deltaintegrate(p, x_2) == x*DiracDelta(x - x_1)*Heaviside(x_2 - x)
+
+    p = x*y**2*z*DiracDelta(y - x)*DiracDelta(y - z)*DiracDelta(x - z)
+    assert deltaintegrate(p, y) == x**3*z*DiracDelta(x - z)**2*Heaviside(y - x)
+    assert deltaintegrate((x + 1)*DiracDelta(2*x), x) == S.Half * Heaviside(x)
+    assert deltaintegrate((x + 1)*DiracDelta(x*Rational(2, 3) + Rational(4, 9)), x) == \
+        S.Half * Heaviside(x + Rational(2, 3))
+
+    a, b, c = symbols('a b c', commutative=False)
+    assert deltaintegrate(DiracDelta(x - y)*f(x - b)*f(x - a), x) == \
+        f(y - b)*f(y - a)*Heaviside(x - y)
+
+    p = f(x - a)*DiracDelta(x - y)*f(x - c)*f(x - b)
+    assert deltaintegrate(p, x) == f(y - a)*f(y - c)*f(y - b)*Heaviside(x - y)
+
+    p = DiracDelta(x - z)*f(x - b)*f(x - a)*DiracDelta(x - y)
+    assert deltaintegrate(p, x) == DiracDelta(y - z)*f(y - b)*f(y - a) * \
+        Heaviside(x - y)

rl/Lib/site-packages/sympy/integrals/tests/test_failing_integrals.py

@@ -0,0 +1,277 @@
+# A collection of failing integrals from the issues.
+
+from sympy.core.numbers import (I, Rational, oo, pi)
+from sympy.core.singleton import S
+from sympy.core.symbol import symbols
+from sympy.functions.elementary.complexes import sign
+from sympy.functions.elementary.exponential import (exp, log)
+from sympy.functions.elementary.hyperbolic import (sech, sinh)
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.piecewise import Piecewise
+from sympy.functions.elementary.trigonometric import (acos, atan, cos, sin, tan)
+from sympy.functions.special.delta_functions import DiracDelta
+from sympy.functions.special.gamma_functions import gamma
+from sympy.integrals.integrals import (Integral, integrate)
+from sympy.simplify.fu import fu
+
+
+from sympy.testing.pytest import XFAIL, slow, tooslow
+
+from sympy.abc import x, k, c, y, b, h, a, m, z, n, t
+
+
+@tooslow
+@XFAIL
+def test_issue_3880():
+    # integrate_hyperexponential(Poly(t*2*(1 - t0**2)*t0*(x**3 + x**2), t), Poly((1 + t0**2)**2*2*(x**2 + x + 1), t), [Poly(1, x), Poly(1 + t0**2, t0), Poly(t, t)], [x, t0, t], [exp, tan])
+    assert not integrate(exp(x)*cos(2*x)*sin(2*x) * (x**3 + x**2)/(2*(x**2 + x + 1)), x).has(Integral)
+
+
+def test_issue_4212_real():
+    xr = symbols('xr', real=True)
+    negabsx = Piecewise((-xr, xr < 0), (xr, True))
+    assert integrate(sign(xr), xr) == negabsx
+
+
+@XFAIL
+def test_issue_4212():
+    # XXX: Maybe this should be expected to fail without real assumptions on x.
+    # As a complex function sign(x) is not analytic and so there is no complex
+    # function whose complex derivative is sign(x). With real assumptions this
+    # works (see test_issue_4212_real above).
+    assert not integrate(sign(x), x).has(Integral)
+
+
+def test_issue_4511():
+    # This works, but gives a slightly over-complicated answer.
+    f = integrate(cos(x)**2 / (1 - sin(x)), x)
+    assert fu(f) == x - cos(x) - 1
+    assert f == ((x*tan(x/2)**2 + x - 2)/(tan(x/2)**2 + 1)).expand()
+
+
+def test_integrate_DiracDelta_no_meijerg():
+    assert integrate(integrate(integrate(
+        DiracDelta(x - y - z), (z, 0, oo)), (y, 0, 1), meijerg=False), (x, 0, 1)) == S.Half
+
+
+@XFAIL
+def test_integrate_DiracDelta_fails():
+    # issue 6427
+    # works without meijerg. See test_integrate_DiracDelta_no_meijerg above.
+    assert integrate(integrate(integrate(
+        DiracDelta(x - y - z), (z, 0, oo)), (y, 0, 1)), (x, 0, 1)) == S.Half
+
+
+@XFAIL
+@slow
+def test_issue_4525():
+    # Warning: takes a long time
+    assert not integrate((x**m * (1 - x)**n * (a + b*x + c*x**2))/(1 + x**2), (x, 0, 1)).has(Integral)
+
+
+@XFAIL
+@tooslow
+def test_issue_4540():
+    # Note, this integral is probably nonelementary
+    assert not integrate(
+        (sin(1/x) - x*exp(x)) /
+        ((-sin(1/x) + x*exp(x))*x + x*sin(1/x)), x).has(Integral)
+
+
+@XFAIL
+@slow
+def test_issue_4891():
+    # Requires the hypergeometric function.
+    assert not integrate(cos(x)**y, x).has(Integral)
+
+
+@XFAIL
+@slow
+def test_issue_1796a():
+    assert not integrate(exp(2*b*x)*exp(-a*x**2), x).has(Integral)
+
+
+@XFAIL
+def test_issue_4895b():
+    assert not integrate(exp(2*b*x)*exp(-a*x**2), (x, -oo, 0)).has(Integral)
+
+
+@XFAIL
+def test_issue_4895c():
+    assert not integrate(exp(2*b*x)*exp(-a*x**2), (x, -oo, oo)).has(Integral)
+
+
+@XFAIL
+def test_issue_4895d():
+    assert not integrate(exp(2*b*x)*exp(-a*x**2), (x, 0, oo)).has(Integral)
+
+
+@XFAIL
+@slow
+def test_issue_4941():
+    assert not integrate(sqrt(1 + sinh(x/20)**2), (x, -25, 25)).has(Integral)
+
+
+@XFAIL
+def test_issue_4992():
+    # Nonelementary integral.  Requires hypergeometric/Meijer-G handling.
+    assert not integrate(log(x) * x**(k - 1) * exp(-x) / gamma(k), (x, 0, oo)).has(Integral)
+
+
+@XFAIL
+def test_issue_16396a():
+    i = integrate(1/(1+sqrt(tan(x))), (x, pi/3, pi/6))
+    assert not i.has(Integral)
+
+
+@XFAIL
+def test_issue_16396b():
+    i = integrate(x*sin(x)/(1+cos(x)**2), (x, 0, pi))
+    assert not i.has(Integral)
+
+
+@XFAIL
+def test_issue_16046():
+    assert integrate(exp(exp(I*x)), [x, 0, 2*pi]) == 2*pi
+
+
+@XFAIL
+def test_issue_15925a():
+    assert not integrate(sqrt((1+sin(x))**2+(cos(x))**2), (x, -pi/2, pi/2)).has(Integral)
+
+
+def test_issue_15925b():
+    f = sqrt((-12*cos(x)**2*sin(x))**2+(12*cos(x)*sin(x)**2)**2)
+    assert integrate(f, (x, 0, pi/6)) == Rational(3, 2)
+
+
+@XFAIL
+def test_issue_15925b_manual():
+    assert not integrate(sqrt((-12*cos(x)**2*sin(x))**2+(12*cos(x)*sin(x)**2)**2),
+                         (x, 0, pi/6), manual=True).has(Integral)
+
+
+@XFAIL
+@tooslow
+def test_issue_15227():
+    i = integrate(log(1-x)*log((1+x)**2)/x, (x, 0, 1))
+    assert not i.has(Integral)
+    # assert i == -5*zeta(3)/4
+
+
+@XFAIL
+@slow
+def test_issue_14716():
+    i = integrate(log(x + 5)*cos(pi*x),(x, S.Half, 1))
+    assert not i.has(Integral)
+    # Mathematica can not solve it either, but
+    # integrate(log(x + 5)*cos(pi*x),(x, S.Half, 1)).transform(x, y - 5).doit()
+    # works
+    # assert i == -log(Rational(11, 2))/pi - Si(pi*Rational(11, 2))/pi + Si(6*pi)/pi
+
+
+@XFAIL
+def test_issue_14709a():
+    i = integrate(x*acos(1 - 2*x/h), (x, 0, h))
+    assert not i.has(Integral)
+    # assert i == 5*h**2*pi/16
+
+
+@slow
+@XFAIL
+def test_issue_14398():
+    assert not integrate(exp(x**2)*cos(x), x).has(Integral)
+
+
+@XFAIL
+def test_issue_14074():
+    i = integrate(log(sin(x)), (x, 0, pi/2))
+    assert not i.has(Integral)
+    # assert i == -pi*log(2)/2
+
+
+@XFAIL
+@slow
+def test_issue_14078b():
+    i = integrate((atan(4*x)-atan(2*x))/x, (x, 0, oo))
+    assert not i.has(Integral)
+    # assert i == pi*log(2)/2
+
+
+@XFAIL
+def test_issue_13792():
+    i =  integrate(log(1/x) / (1 - x), (x, 0, 1))
+    assert not i.has(Integral)
+    # assert i in [polylog(2, -exp_polar(I*pi)), pi**2/6]
+
+
+@XFAIL
+def test_issue_11845a():
+    assert not integrate(exp(y - x**3), (x, 0, 1)).has(Integral)
+
+
+@XFAIL
+def test_issue_11845b():
+    assert not integrate(exp(-y - x**3), (x, 0, 1)).has(Integral)
+
+
+@XFAIL
+def test_issue_11813():
+    assert not integrate((a - x)**Rational(-1, 2)*x, (x, 0, a)).has(Integral)
+
+
+@XFAIL
+def test_issue_11254c():
+    assert not integrate(sech(x)**2, (x, 0, 1)).has(Integral)
+
+
+@XFAIL
+def test_issue_10584():
+    assert not integrate(sqrt(x**2 + 1/x**2), x).has(Integral)
+
+
+@XFAIL
+def test_issue_9101():
+    assert not integrate(log(x + sqrt(x**2 + y**2 + z**2)), z).has(Integral)
+
+
+@XFAIL
+def test_issue_7147():
+    assert not integrate(x/sqrt(a*x**2 + b*x + c)**3, x).has(Integral)
+
+
+@XFAIL
+def test_issue_7109():
+    assert not integrate(sqrt(a**2/(a**2 - x**2)), x).has(Integral)
+
+
+@XFAIL
+def test_integrate_Piecewise_rational_over_reals():
+    f = Piecewise(
+        (0,                                              t - 478.515625*pi <  0),
+        (13.2075145209219*pi/(0.000871222*t + 0.995)**2, t - 478.515625*pi >= 0))
+
+    assert abs((integrate(f, (t, 0, oo)) - 15235.9375*pi).evalf()) <= 1e-7
+
+
+@XFAIL
+def test_issue_4311_slow():
+    # Not slow when bypassing heurish
+    assert not integrate(x*abs(9-x**2), x).has(Integral)
+
+@XFAIL
+def test_issue_20370():
+    a = symbols('a', positive=True)
+    assert integrate((1 + a * cos(x))**-1, (x, 0, 2 * pi)) == (2 * pi / sqrt(1 - a**2))
+
+
+@XFAIL
+def test_polylog():
+    # log(1/x)*log(x+1)-polylog(2, -x)
+    assert not integrate(log(1/x)/(x + 1), x).has(Integral)
+
+
+@XFAIL
+def test_polylog_manual():
+    # Make sure _parts_rule does not go into an infinite loop here
+    assert not integrate(log(1/x)/(x + 1), x, manual=True).has(Integral)

rl/Lib/site-packages/sympy/integrals/tests/test_heurisch.py

@@ -0,0 +1,389 @@
+from sympy.concrete.summations import Sum
+from sympy.core.add import Add
+from sympy.core.function import (Derivative, Function, diff)
+from sympy.core.numbers import (I, Rational, pi)
+from sympy.core.relational import Eq, Ne
+from sympy.core.symbol import (Symbol, symbols)
+from sympy.functions.elementary.exponential import (LambertW, exp, log)
+from sympy.functions.elementary.hyperbolic import (asinh, cosh, sinh, tanh)
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.piecewise import Piecewise
+from sympy.functions.elementary.trigonometric import (acos, asin, atan, cos, sin, tan)
+from sympy.functions.special.bessel import (besselj, besselk, bessely, jn)
+from sympy.functions.special.error_functions import erf
+from sympy.integrals.integrals import Integral
+from sympy.logic.boolalg import And
+from sympy.matrices import Matrix
+from sympy.simplify.ratsimp import ratsimp
+from sympy.simplify.simplify import simplify
+from sympy.integrals.heurisch import components, heurisch, heurisch_wrapper
+from sympy.testing.pytest import XFAIL, slow
+from sympy.integrals.integrals import integrate
+x, y, z, nu = symbols('x,y,z,nu')
+f = Function('f')
+
+
+def test_components():
+    assert components(x*y, x) == {x}
+    assert components(1/(x + y), x) == {x}
+    assert components(sin(x), x) == {sin(x), x}
+    assert components(sin(x)*sqrt(log(x)), x) == \
+        {log(x), sin(x), sqrt(log(x)), x}
+    assert components(x*sin(exp(x)*y), x) == \
+        {sin(y*exp(x)), x, exp(x)}
+    assert components(x**Rational(17, 54)/sqrt(sin(x)), x) == \
+        {sin(x), x**Rational(1, 54), sqrt(sin(x)), x}
+
+    assert components(f(x), x) == \
+        {x, f(x)}
+    assert components(Derivative(f(x), x), x) == \
+        {x, f(x), Derivative(f(x), x)}
+    assert components(f(x)*diff(f(x), x), x) == \
+        {x, f(x), Derivative(f(x), x), Derivative(f(x), x)}
+
+
+def test_issue_10680():
+    assert isinstance(integrate(x**log(x**log(x**log(x))),x), Integral)
+
+
+def test_issue_21166():
+    assert integrate(sin(x/sqrt(abs(x))), (x, -1, 1)) == 0
+
+
+def test_heurisch_polynomials():
+    assert heurisch(1, x) == x
+    assert heurisch(x, x) == x**2/2
+    assert heurisch(x**17, x) == x**18/18
+    # For coverage
+    assert heurisch_wrapper(y, x) == y*x
+
+
+def test_heurisch_fractions():
+    assert heurisch(1/x, x) == log(x)
+    assert heurisch(1/(2 + x), x) == log(x + 2)
+    assert heurisch(1/(x + sin(y)), x) == log(x + sin(y))
+
+    # Up to a constant, where C = pi*I*Rational(5, 12), Mathematica gives identical
+    # result in the first case. The difference is because SymPy changes
+    # signs of expressions without any care.
+    # XXX ^ ^ ^ is this still correct?
+    assert heurisch(5*x**5/(
+        2*x**6 - 5), x) in [5*log(2*x**6 - 5) / 12, 5*log(-2*x**6 + 5) / 12]
+    assert heurisch(5*x**5/(2*x**6 + 5), x) == 5*log(2*x**6 + 5) / 12
+
+    assert heurisch(1/x**2, x) == -1/x
+    assert heurisch(-1/x**5, x) == 1/(4*x**4)
+
+
+def test_heurisch_log():
+    assert heurisch(log(x), x) == x*log(x) - x
+    assert heurisch(log(3*x), x) == -x + x*log(3) + x*log(x)
+    assert heurisch(log(x**2), x) in [x*log(x**2) - 2*x, 2*x*log(x) - 2*x]
+
+
+def test_heurisch_exp():
+    assert heurisch(exp(x), x) == exp(x)
+    assert heurisch(exp(-x), x) == -exp(-x)
+    assert heurisch(exp(17*x), x) == exp(17*x) / 17
+    assert heurisch(x*exp(x), x) == x*exp(x) - exp(x)
+    assert heurisch(x*exp(x**2), x) == exp(x**2) / 2
+
+    assert heurisch(exp(-x**2), x) is None
+
+    assert heurisch(2**x, x) == 2**x/log(2)
+    assert heurisch(x*2**x, x) == x*2**x/log(2) - 2**x*log(2)**(-2)
+
+    assert heurisch(Integral(x**z*y, (y, 1, 2), (z, 2, 3)).function, x) == (x*x**z*y)/(z+1)
+    assert heurisch(Sum(x**z, (z, 1, 2)).function, z) == x**z/log(x)
+
+    # https://github.com/sympy/sympy/issues/23707
+    anti = -exp(z)/(sqrt(x - y)*exp(z*sqrt(x - y)) - exp(z*sqrt(x - y)))
+    assert heurisch(exp(z)*exp(-z*sqrt(x - y)), z) == anti
+
+
+def test_heurisch_trigonometric():
+    assert heurisch(sin(x), x) == -cos(x)
+    assert heurisch(pi*sin(x) + 1, x) == x - pi*cos(x)
+
+    assert heurisch(cos(x), x) == sin(x)
+    assert heurisch(tan(x), x) in [
+        log(1 + tan(x)**2)/2,
+        log(tan(x) + I) + I*x,
+        log(tan(x) - I) - I*x,
+    ]
+
+    assert heurisch(sin(x)*sin(y), x) == -cos(x)*sin(y)
+    assert heurisch(sin(x)*sin(y), y) == -cos(y)*sin(x)
+
+    # gives sin(x) in answer when run via setup.py and cos(x) when run via py.test
+    assert heurisch(sin(x)*cos(x), x) in [sin(x)**2 / 2, -cos(x)**2 / 2]
+    assert heurisch(cos(x)/sin(x), x) == log(sin(x))
+
+    assert heurisch(x*sin(7*x), x) == sin(7*x) / 49 - x*cos(7*x) / 7
+    assert heurisch(1/pi/4 * x**2*cos(x), x) == 1/pi/4*(x**2*sin(x) -
+                    2*sin(x) + 2*x*cos(x))
+
+    assert heurisch(acos(x/4) * asin(x/4), x) == 2*x - (sqrt(16 - x**2))*asin(x/4) \
+        + (sqrt(16 - x**2))*acos(x/4) + x*asin(x/4)*acos(x/4)
+
+    assert heurisch(sin(x)/(cos(x)**2+1), x) == -atan(cos(x)) #fixes issue 13723
+    assert heurisch(1/(cos(x)+2), x) == 2*sqrt(3)*atan(sqrt(3)*tan(x/2)/3)/3
+    assert heurisch(2*sin(x)*cos(x)/(sin(x)**4 + 1), x) == atan(sqrt(2)*sin(x)
+        - 1) - atan(sqrt(2)*sin(x) + 1)
+
+    assert heurisch(1/cosh(x), x) == 2*atan(tanh(x/2))
+
+
+def test_heurisch_hyperbolic():
+    assert heurisch(sinh(x), x) == cosh(x)
+    assert heurisch(cosh(x), x) == sinh(x)
+
+    assert heurisch(x*sinh(x), x) == x*cosh(x) - sinh(x)
+    assert heurisch(x*cosh(x), x) == x*sinh(x) - cosh(x)
+
+    assert heurisch(
+        x*asinh(x/2), x) == x**2*asinh(x/2)/2 + asinh(x/2) - x*sqrt(4 + x**2)/4
+
+
+def test_heurisch_mixed():
+    assert heurisch(sin(x)*exp(x), x) == exp(x)*sin(x)/2 - exp(x)*cos(x)/2
+    assert heurisch(sin(x/sqrt(-x)), x) == 2*x*cos(x/sqrt(-x))/sqrt(-x) - 2*sin(x/sqrt(-x))
+
+
+def test_heurisch_radicals():
+    assert heurisch(1/sqrt(x), x) == 2*sqrt(x)
+    assert heurisch(1/sqrt(x)**3, x) == -2/sqrt(x)
+    assert heurisch(sqrt(x)**3, x) == 2*sqrt(x)**5/5
+
+    assert heurisch(sin(x)*sqrt(cos(x)), x) == -2*sqrt(cos(x))**3/3
+    y = Symbol('y')
+    assert heurisch(sin(y*sqrt(x)), x) == 2/y**2*sin(y*sqrt(x)) - \
+        2*sqrt(x)*cos(y*sqrt(x))/y
+    assert heurisch_wrapper(sin(y*sqrt(x)), x) == Piecewise(
+        (-2*sqrt(x)*cos(sqrt(x)*y)/y + 2*sin(sqrt(x)*y)/y**2, Ne(y, 0)),
+        (0, True))
+    y = Symbol('y', positive=True)
+    assert heurisch_wrapper(sin(y*sqrt(x)), x) == 2/y**2*sin(y*sqrt(x)) - \
+        2*sqrt(x)*cos(y*sqrt(x))/y
+
+
+def test_heurisch_special():
+    assert heurisch(erf(x), x) == x*erf(x) + exp(-x**2)/sqrt(pi)
+    assert heurisch(exp(-x**2)*erf(x), x) == sqrt(pi)*erf(x)**2 / 4
+
+
+def test_heurisch_symbolic_coeffs():
+    assert heurisch(1/(x + y), x) == log(x + y)
+    assert heurisch(1/(x + sqrt(2)), x) == log(x + sqrt(2))
+    assert simplify(diff(heurisch(log(x + y + z), y), y)) == log(x + y + z)
+
+
+def test_heurisch_symbolic_coeffs_1130():
+    y = Symbol('y')
+    assert heurisch_wrapper(1/(x**2 + y), x) == Piecewise(
+    (log(x - sqrt(-y))/(2*sqrt(-y)) - log(x + sqrt(-y))/(2*sqrt(-y)),
+    Ne(y, 0)), (-1/x, True))
+    y = Symbol('y', positive=True)
+    assert heurisch_wrapper(1/(x**2 + y), x) == (atan(x/sqrt(y))/sqrt(y))
+
+
+def test_heurisch_hacking():
+    assert heurisch(sqrt(1 + 7*x**2), x, hints=[]) == \
+        x*sqrt(1 + 7*x**2)/2 + sqrt(7)*asinh(sqrt(7)*x)/14
+    assert heurisch(sqrt(1 - 7*x**2), x, hints=[]) == \
+        x*sqrt(1 - 7*x**2)/2 + sqrt(7)*asin(sqrt(7)*x)/14
+
+    assert heurisch(1/sqrt(1 + 7*x**2), x, hints=[]) == \
+        sqrt(7)*asinh(sqrt(7)*x)/7
+    assert heurisch(1/sqrt(1 - 7*x**2), x, hints=[]) == \
+        sqrt(7)*asin(sqrt(7)*x)/7
+
+    assert heurisch(exp(-7*x**2), x, hints=[]) == \
+        sqrt(7*pi)*erf(sqrt(7)*x)/14
+
+    assert heurisch(1/sqrt(9 - 4*x**2), x, hints=[]) == \
+        asin(x*Rational(2, 3))/2
+
+    assert heurisch(1/sqrt(9 + 4*x**2), x, hints=[]) == \
+        asinh(x*Rational(2, 3))/2
+
+    assert heurisch(1/sqrt(3*x**2-4), x, hints=[]) == \
+           sqrt(3)*log(3*x + sqrt(3)*sqrt(3*x**2 - 4))/3
+
+
+def test_heurisch_function():
+    assert heurisch(f(x), x) is None
+
+@XFAIL
+def test_heurisch_function_derivative():
+    # TODO: it looks like this used to work just by coincindence and
+    # thanks to sloppy implementation. Investigate why this used to
+    # work at all and if support for this can be restored.
+
+    df = diff(f(x), x)
+
+    assert heurisch(f(x)*df, x) == f(x)**2/2
+    assert heurisch(f(x)**2*df, x) == f(x)**3/3
+    assert heurisch(df/f(x), x) == log(f(x))
+
+
+def test_heurisch_wrapper():
+    f = 1/(y + x)
+    assert heurisch_wrapper(f, x) == log(x + y)
+    f = 1/(y - x)
+    assert heurisch_wrapper(f, x) == -log(x - y)
+    f = 1/((y - x)*(y + x))
+    assert heurisch_wrapper(f, x) == Piecewise(
+        (-log(x - y)/(2*y) + log(x + y)/(2*y), Ne(y, 0)), (1/x, True))
+    # issue 6926
+    f = sqrt(x**2/((y - x)*(y + x)))
+    assert heurisch_wrapper(f, x) == x*sqrt(-x**2/(x**2 - y**2)) \
+    - y**2*sqrt(-x**2/(x**2 - y**2))/x
+
+
+def test_issue_3609():
+    assert heurisch(1/(x * (1 + log(x)**2)), x) == atan(log(x))
+
+### These are examples from the Poor Man's Integrator
+### http://www-sop.inria.fr/cafe/Manuel.Bronstein/pmint/examples/
+
+
+def test_pmint_rat():
+    # TODO: heurisch() is off by a constant: -3/4. Possibly different permutation
+    # would give the optimal result?
+
+    def drop_const(expr, x):
+        if expr.is_Add:
+            return Add(*[ arg for arg in expr.args if arg.has(x) ])
+        else:
+            return expr
+
+    f = (x**7 - 24*x**4 - 4*x**2 + 8*x - 8)/(x**8 + 6*x**6 + 12*x**4 + 8*x**2)
+    g = (4 + 8*x**2 + 6*x + 3*x**3)/(x**5 + 4*x**3 + 4*x) + log(x)
+
+    assert drop_const(ratsimp(heurisch(f, x)), x) == g
+
+
+def test_pmint_trig():
+    f = (x - tan(x)) / tan(x)**2 + tan(x)
+    g = -x**2/2 - x/tan(x) + log(tan(x)**2 + 1)/2
+
+    assert heurisch(f, x) == g
+
+
+def test_pmint_logexp():
+    f = (1 + x + x*exp(x))*(x + log(x) + exp(x) - 1)/(x + log(x) + exp(x))**2/x
+    g = log(x + exp(x) + log(x)) + 1/(x + exp(x) + log(x))
+
+    assert ratsimp(heurisch(f, x)) == g
+
+
+def test_pmint_erf():
+    f = exp(-x**2)*erf(x)/(erf(x)**3 - erf(x)**2 - erf(x) + 1)
+    g = sqrt(pi)*log(erf(x) - 1)/8 - sqrt(pi)*log(erf(x) + 1)/8 - sqrt(pi)/(4*erf(x) - 4)
+
+    assert ratsimp(heurisch(f, x)) == g
+
+
+def test_pmint_LambertW():
+    f = LambertW(x)
+    g = x*LambertW(x) - x + x/LambertW(x)
+
+    assert heurisch(f, x) == g
+
+
+def test_pmint_besselj():
+    f = besselj(nu + 1, x)/besselj(nu, x)
+    g = nu*log(x) - log(besselj(nu, x))
+
+    assert heurisch(f, x) == g
+
+    f = (nu*besselj(nu, x) - x*besselj(nu + 1, x))/x
+    g = besselj(nu, x)
+
+    assert heurisch(f, x) == g
+
+    f = jn(nu + 1, x)/jn(nu, x)
+    g = nu*log(x) - log(jn(nu, x))
+
+    assert heurisch(f, x) == g
+
+
+@slow
+def test_pmint_bessel_products():
+    f = x*besselj(nu, x)*bessely(nu, 2*x)
+    g = -2*x*besselj(nu, x)*bessely(nu - 1, 2*x)/3 + x*besselj(nu - 1, x)*bessely(nu, 2*x)/3
+
+    assert heurisch(f, x) == g
+
+    f = x*besselj(nu, x)*besselk(nu, 2*x)
+    g = -2*x*besselj(nu, x)*besselk(nu - 1, 2*x)/5 - x*besselj(nu - 1, x)*besselk(nu, 2*x)/5
+
+    assert heurisch(f, x) == g
+
+
+def test_pmint_WrightOmega():
+    def omega(x):
+        return LambertW(exp(x))
+
+    f = (1 + omega(x) * (2 + cos(omega(x)) * (x + omega(x))))/(1 + omega(x))/(x + omega(x))
+    g = log(x + LambertW(exp(x))) + sin(LambertW(exp(x)))
+
+    assert heurisch(f, x) == g
+
+
+def test_RR():
+    # Make sure the algorithm does the right thing if the ring is RR. See
+    # issue 8685.
+    assert heurisch(sqrt(1 + 0.25*x**2), x, hints=[]) == \
+        0.5*x*sqrt(0.25*x**2 + 1) + 1.0*asinh(0.5*x)
+
+# TODO: convert the rest of PMINT tests:
+# Airy functions
+# f = (x - AiryAi(x)*AiryAi(1, x)) / (x**2 - AiryAi(x)**2)
+# g = Rational(1,2)*ln(x + AiryAi(x)) + Rational(1,2)*ln(x - AiryAi(x))
+# f = x**2 * AiryAi(x)
+# g = -AiryAi(x) + AiryAi(1, x)*x
+# Whittaker functions
+# f = WhittakerW(mu + 1, nu, x) / (WhittakerW(mu, nu, x) * x)
+# g = x/2 - mu*ln(x) - ln(WhittakerW(mu, nu, x))
+
+
+def test_issue_22527():
+    t, R = symbols(r't R')
+    z = Function('z')(t)
+    def f(x):
+      return x/sqrt(R**2 - x**2)
+    Uz = integrate(f(z), z)
+    Ut = integrate(f(t), t)
+    assert Ut == Uz.subs(z, t)
+
+
+def test_heurisch_complex_erf_issue_26338():
+    r = symbols('r', real=True)
+    a = exp(-r**2/(2*(2 - I)**2))
+    assert heurisch(a, r, hints=[]) is None  # None, not a wrong soln
+    a = sqrt(pi)*erf((1 + I)/2)/2
+    assert integrate(exp(-I*r**2/2), (r, 0, 1)) == a - I*a
+
+    a = exp(-x**2/(2*(2 - I)**2))
+    assert heurisch(a, x, hints=[]) is None  # None, not a wrong soln
+    a = sqrt(pi)*erf((1 + I)/2)/2
+    assert integrate(exp(-I*x**2/2), (x, 0, 1)) == a - I*a
+
+
+def test_issue_15498():
+    Z0 = Function('Z0')
+    k01, k10, t, s= symbols('k01 k10 t s', real=True, positive=True)
+    m = Matrix([[exp(-k10*t)]])
+    _83 = Rational(83, 100)  # 0.83 works, too
+    [a, b, c, d, e, f, g] = [100, 0.5, _83, 50, 0.6, 2, 120]
+    AIF_btf = a*(d*e*(1 - exp(-(t - b)/e)) + f*g*(1 - exp(-(t - b)/g)))
+    AIF_atf = a*(d*e*exp(-(t - b)/e)*(exp((c - b)/e) - 1
+        ) + f*g*exp(-(t - b)/g)*(exp((c - b)/g) - 1))
+    AIF_sym = Piecewise((0, t < b), (AIF_btf, And(b <= t, t < c)), (AIF_atf, c <= t))
+    aif_eq = Eq(Z0(t), AIF_sym)
+    f_vec = Matrix([[k01*Z0(t)]])
+    integrand = m*m.subs(t, s)**-1*f_vec.subs(aif_eq.lhs, aif_eq.rhs).subs(t, s)
+    solution = integrate(integrand[0], (s, 0, t))
+    assert solution is not None  # does not hang and takes less than 10 s

rl/Lib/site-packages/sympy/integrals/tests/test_intpoly.py

@@ -0,0 +1,627 @@
+from sympy.functions.elementary.complexes import Abs
+from sympy.functions.elementary.miscellaneous import sqrt
+
+from sympy.core import S, Rational
+
+from sympy.integrals.intpoly import (decompose, best_origin, distance_to_side,
+                                     polytope_integrate, point_sort,
+                                     hyperplane_parameters, main_integrate3d,
+                                     main_integrate, polygon_integrate,
+                                     lineseg_integrate, integration_reduction,
+                                     integration_reduction_dynamic, is_vertex)
+
+from sympy.geometry.line import Segment2D
+from sympy.geometry.polygon import Polygon
+from sympy.geometry.point import Point, Point2D
+from sympy.abc import x, y, z
+
+from sympy.testing.pytest import slow
+
+
+def test_decompose():
+    assert decompose(x) == {1: x}
+    assert decompose(x**2) == {2: x**2}
+    assert decompose(x*y) == {2: x*y}
+    assert decompose(x + y) == {1: x + y}
+    assert decompose(x**2 + y) == {1: y, 2: x**2}
+    assert decompose(8*x**2 + 4*y + 7) == {0: 7, 1: 4*y, 2: 8*x**2}
+    assert decompose(x**2 + 3*y*x) == {2: x**2 + 3*x*y}
+    assert decompose(9*x**2 + y + 4*x + x**3 + y**2*x + 3) ==\
+        {0: 3, 1: 4*x + y, 2: 9*x**2, 3: x**3 + x*y**2}
+
+    assert decompose(x, True) == {x}
+    assert decompose(x ** 2, True) == {x**2}
+    assert decompose(x * y, True) == {x * y}
+    assert decompose(x + y, True) == {x, y}
+    assert decompose(x ** 2 + y, True) == {y, x ** 2}
+    assert decompose(8 * x ** 2 + 4 * y + 7, True) == {7, 4*y, 8*x**2}
+    assert decompose(x ** 2 + 3 * y * x, True) == {x ** 2, 3 * x * y}
+    assert decompose(9 * x ** 2 + y + 4 * x + x ** 3 + y ** 2 * x + 3, True) == \
+           {3, y, 4*x, 9*x**2, x*y**2, x**3}
+
+
+def test_best_origin():
+    expr1 = y ** 2 * x ** 5 + y ** 5 * x ** 7 + 7 * x + x ** 12 + y ** 7 * x
+
+    l1 = Segment2D(Point(0, 3), Point(1, 1))
+    l2 = Segment2D(Point(S(3) / 2, 0), Point(S(3) / 2, 3))
+    l3 = Segment2D(Point(0, S(3) / 2), Point(3, S(3) / 2))
+    l4 = Segment2D(Point(0, 2), Point(2, 0))
+    l5 = Segment2D(Point(0, 2), Point(1, 1))
+    l6 = Segment2D(Point(2, 0), Point(1, 1))
+
+    assert best_origin((2, 1), 3, l1, expr1) == (0, 3)
+    # XXX: Should these return exact Rational output? Maybe best_origin should
+    #      sympify its arguments...
+    assert best_origin((2, 0), 3, l2, x ** 7) == (1.5, 0)
+    assert best_origin((0, 2), 3, l3, x ** 7) == (0, 1.5)
+    assert best_origin((1, 1), 2, l4, x ** 7 * y ** 3) == (0, 2)
+    assert best_origin((1, 1), 2, l4, x ** 3 * y ** 7) == (2, 0)
+    assert best_origin((1, 1), 2, l5, x ** 2 * y ** 9) == (0, 2)
+    assert best_origin((1, 1), 2, l6, x ** 9 * y ** 2) == (2, 0)
+
+
+@slow
+def test_polytope_integrate():
+    #  Convex 2-Polytopes
+    #  Vertex representation
+    assert polytope_integrate(Polygon(Point(0, 0), Point(0, 2),
+                                      Point(4, 0)), 1) == 4
+    assert polytope_integrate(Polygon(Point(0, 0), Point(0, 1),
+                                      Point(1, 1), Point(1, 0)), x * y) ==\
+                                      Rational(1, 4)
+    assert polytope_integrate(Polygon(Point(0, 3), Point(5, 3), Point(1, 1)),
+                              6*x**2 - 40*y) == Rational(-935, 3)
+
+    assert polytope_integrate(Polygon(Point(0, 0), Point(0, sqrt(3)),
+                                      Point(sqrt(3), sqrt(3)),
+                                      Point(sqrt(3), 0)), 1) == 3
+
+    hexagon = Polygon(Point(0, 0), Point(-sqrt(3) / 2, S.Half),
+                      Point(-sqrt(3) / 2, S(3) / 2), Point(0, 2),
+                      Point(sqrt(3) / 2, S(3) / 2), Point(sqrt(3) / 2, S.Half))
+
+    assert polytope_integrate(hexagon, 1) == S(3*sqrt(3)) / 2
+
+    #  Hyperplane representation
+    assert polytope_integrate([((-1, 0), 0), ((1, 2), 4),
+                               ((0, -1), 0)], 1) == 4
+    assert polytope_integrate([((-1, 0), 0), ((0, 1), 1),
+                               ((1, 0), 1), ((0, -1), 0)], x * y) == Rational(1, 4)
+    assert polytope_integrate([((0, 1), 3), ((1, -2), -1),
+                               ((-2, -1), -3)], 6*x**2 - 40*y) == Rational(-935, 3)
+    assert polytope_integrate([((-1, 0), 0), ((0, sqrt(3)), 3),
+                               ((sqrt(3), 0), 3), ((0, -1), 0)], 1) == 3
+
+    hexagon = [((Rational(-1, 2), -sqrt(3) / 2), 0),
+               ((-1, 0), sqrt(3) / 2),
+               ((Rational(-1, 2), sqrt(3) / 2), sqrt(3)),
+               ((S.Half, sqrt(3) / 2), sqrt(3)),
+               ((1, 0), sqrt(3) / 2),
+               ((S.Half, -sqrt(3) / 2), 0)]
+    assert polytope_integrate(hexagon, 1) == S(3*sqrt(3)) / 2
+
+    #  Non-convex polytopes
+    #  Vertex representation
+    assert polytope_integrate(Polygon(Point(-1, -1), Point(-1, 1),
+                                      Point(1, 1), Point(0, 0),
+                                      Point(1, -1)), 1) == 3
+    assert polytope_integrate(Polygon(Point(-1, -1), Point(-1, 1),
+                                      Point(0, 0), Point(1, 1),
+                                      Point(1, -1), Point(0, 0)), 1) == 2
+    #  Hyperplane representation
+    assert polytope_integrate([((-1, 0), 1), ((0, 1), 1), ((1, -1), 0),
+                               ((1, 1), 0), ((0, -1), 1)], 1) == 3
+    assert polytope_integrate([((-1, 0), 1), ((1, 1), 0), ((-1, 1), 0),
+                               ((1, 0), 1), ((-1, -1), 0),
+                               ((1, -1), 0)], 1) == 2
+
+    #  Tests for 2D polytopes mentioned in Chin et al(Page 10):
+    #  http://dilbert.engr.ucdavis.edu/~suku/quadrature/cls-integration.pdf
+    fig1 = Polygon(Point(1.220, -0.827), Point(-1.490, -4.503),
+                   Point(-3.766, -1.622), Point(-4.240, -0.091),
+                   Point(-3.160, 4), Point(-0.981, 4.447),
+                   Point(0.132, 4.027))
+    assert polytope_integrate(fig1, x**2 + x*y + y**2) ==\
+        S(2031627344735367)/(8*10**12)
+
+    fig2 = Polygon(Point(4.561, 2.317), Point(1.491, -1.315),
+                   Point(-3.310, -3.164), Point(-4.845, -3.110),
+                   Point(-4.569, 1.867))
+    assert polytope_integrate(fig2, x**2 + x*y + y**2) ==\
+        S(517091313866043)/(16*10**11)
+
+    fig3 = Polygon(Point(-2.740, -1.888), Point(-3.292, 4.233),
+                   Point(-2.723, -0.697), Point(-0.643, -3.151))
+    assert polytope_integrate(fig3, x**2 + x*y + y**2) ==\
+        S(147449361647041)/(8*10**12)
+
+    fig4 = Polygon(Point(0.211, -4.622), Point(-2.684, 3.851),
+                   Point(0.468, 4.879), Point(4.630, -1.325),
+                   Point(-0.411, -1.044))
+    assert polytope_integrate(fig4, x**2 + x*y + y**2) ==\
+        S(180742845225803)/(10**12)
+
+    #  Tests for many polynomials with maximum degree given(2D case).
+    tri = Polygon(Point(0, 3), Point(5, 3), Point(1, 1))
+    polys = []
+    expr1 = x**9*y + x**7*y**3 + 2*x**2*y**8
+    expr2 = x**6*y**4 + x**5*y**5 + 2*y**10
+    expr3 = x**10 + x**9*y + x**8*y**2 + x**5*y**5
+    polys.extend((expr1, expr2, expr3))
+    result_dict = polytope_integrate(tri, polys, max_degree=10)
+    assert result_dict[expr1] == Rational(615780107, 594)
+    assert result_dict[expr2] == Rational(13062161, 27)
+    assert result_dict[expr3] == Rational(1946257153, 924)
+
+    tri = Polygon(Point(0, 3), Point(5, 3), Point(1, 1))
+    expr1 = x**7*y**1 + 2*x**2*y**6
+    expr2 = x**6*y**4 + x**5*y**5 + 2*y**10
+    expr3 = x**10 + x**9*y + x**8*y**2 + x**5*y**5
+    polys.extend((expr1, expr2, expr3))
+    assert polytope_integrate(tri, polys, max_degree=9) == \
+        {x**7*y + 2*x**2*y**6: Rational(489262, 9)}
+
+    #  Tests when all integral of all monomials up to a max_degree is to be
+    #  calculated.
+    assert polytope_integrate(Polygon(Point(0, 0), Point(0, 1),
+                                      Point(1, 1), Point(1, 0)),
+                              max_degree=4) == {0: 0, 1: 1, x: S.Half,
+                                                x ** 2 * y ** 2: S.One / 9,
+                                                x ** 4: S.One / 5,
+                                                y ** 4: S.One / 5,
+                                                y: S.Half,
+                                                x * y ** 2: S.One / 6,
+                                                y ** 2: S.One / 3,
+                                                x ** 3: S.One / 4,
+                                                x ** 2 * y: S.One / 6,
+                                                x ** 3 * y: S.One / 8,
+                                                x * y: S.One / 4,
+                                                y ** 3: S.One / 4,
+                                                x ** 2: S.One / 3,
+                                                x * y ** 3: S.One / 8}
+
+    #  Tests for 3D polytopes
+    cube1 = [[(0, 0, 0), (0, 6, 6), (6, 6, 6), (3, 6, 0),
+              (0, 6, 0), (6, 0, 6), (3, 0, 0), (0, 0, 6)],
+             [1, 2, 3, 4], [3, 2, 5, 6], [1, 7, 5, 2], [0, 6, 5, 7],
+             [1, 4, 0, 7], [0, 4, 3, 6]]
+    assert polytope_integrate(cube1, 1) == S(162)
+
+    #  3D Test cases in Chin et al(2015)
+    cube2 = [[(0, 0, 0), (0, 0, 5), (0, 5, 0), (0, 5, 5), (5, 0, 0),
+             (5, 0, 5), (5, 5, 0), (5, 5, 5)],
+             [3, 7, 6, 2], [1, 5, 7, 3], [5, 4, 6, 7], [0, 4, 5, 1],
+             [2, 0, 1, 3], [2, 6, 4, 0]]
+
+    cube3 = [[(0, 0, 0), (5, 0, 0), (5, 4, 0), (3, 2, 0), (3, 5, 0),
+              (0, 5, 0), (0, 0, 5), (5, 0, 5), (5, 4, 5), (3, 2, 5),
+              (3, 5, 5), (0, 5, 5)],
+             [6, 11, 5, 0], [1, 7, 6, 0], [5, 4, 3, 2, 1, 0], [11, 10, 4, 5],
+             [10, 9, 3, 4], [9, 8, 2, 3], [8, 7, 1, 2], [7, 8, 9, 10, 11, 6]]
+
+    cube4 = [[(0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1),
+              (S.One / 4, S.One / 4, S.One / 4)],
+             [0, 2, 1], [1, 3, 0], [4, 2, 3], [4, 3, 1],
+             [0, 1, 2], [2, 4, 1], [0, 3, 2]]
+
+    assert polytope_integrate(cube2, x ** 2 + y ** 2 + x * y + z ** 2) ==\
+           Rational(15625, 4)
+    assert polytope_integrate(cube3, x ** 2 + y ** 2 + x * y + z ** 2) ==\
+           S(33835) / 12
+    assert polytope_integrate(cube4, x ** 2 + y ** 2 + x * y + z ** 2) ==\
+           S(37) / 960
+
+    #  Test cases from Mathematica's PolyhedronData library
+    octahedron = [[(S.NegativeOne / sqrt(2), 0, 0), (0, S.One / sqrt(2), 0),
+                   (0, 0, S.NegativeOne / sqrt(2)), (0, 0, S.One / sqrt(2)),
+                   (0, S.NegativeOne / sqrt(2), 0), (S.One / sqrt(2), 0, 0)],
+                  [3, 4, 5], [3, 5, 1], [3, 1, 0], [3, 0, 4], [4, 0, 2],
+                  [4, 2, 5], [2, 0, 1], [5, 2, 1]]
+
+    assert polytope_integrate(octahedron, 1) == sqrt(2) / 3
+
+    great_stellated_dodecahedron =\
+        [[(-0.32491969623290634095, 0, 0.42532540417601993887),
+          (0.32491969623290634095, 0, -0.42532540417601993887),
+          (-0.52573111211913359231, 0, 0.10040570794311363956),
+          (0.52573111211913359231, 0, -0.10040570794311363956),
+          (-0.10040570794311363956, -0.3090169943749474241, 0.42532540417601993887),
+          (-0.10040570794311363956, 0.30901699437494742410, 0.42532540417601993887),
+          (0.10040570794311363956, -0.3090169943749474241, -0.42532540417601993887),
+          (0.10040570794311363956, 0.30901699437494742410, -0.42532540417601993887),
+          (-0.16245984811645317047, -0.5, 0.10040570794311363956),
+          (-0.16245984811645317047,  0.5, 0.10040570794311363956),
+          (0.16245984811645317047,  -0.5, -0.10040570794311363956),
+          (0.16245984811645317047,   0.5, -0.10040570794311363956),
+          (-0.42532540417601993887, -0.3090169943749474241, -0.10040570794311363956),
+          (-0.42532540417601993887, 0.30901699437494742410, -0.10040570794311363956),
+          (-0.26286555605956679615, 0.1909830056250525759, -0.42532540417601993887),
+          (-0.26286555605956679615, -0.1909830056250525759, -0.42532540417601993887),
+          (0.26286555605956679615, 0.1909830056250525759, 0.42532540417601993887),
+          (0.26286555605956679615, -0.1909830056250525759, 0.42532540417601993887),
+          (0.42532540417601993887, -0.3090169943749474241, 0.10040570794311363956),
+          (0.42532540417601993887, 0.30901699437494742410, 0.10040570794311363956)],
+         [12, 3, 0, 6, 16], [17, 7, 0, 3, 13],
+         [9, 6, 0, 7, 8], [18, 2, 1, 4, 14],
+         [15, 5, 1, 2, 19], [11, 4, 1, 5, 10],
+         [8, 19, 2, 18, 9], [10, 13, 3, 12, 11],
+         [16, 14, 4, 11, 12], [13, 10, 5, 15, 17],
+         [14, 16, 6, 9, 18], [19, 8, 7, 17, 15]]
+    #  Actual volume is : 0.163118960624632
+    assert Abs(polytope_integrate(great_stellated_dodecahedron, 1) -\
+        0.163118960624632) < 1e-12
+
+    expr = x **2 + y ** 2 + z ** 2
+    octahedron_five_compound = [[(0, -0.7071067811865475244, 0),
+                                 (0, 0.70710678118654752440, 0),
+                                 (0.1148764602736805918,
+                                  -0.35355339059327376220, -0.60150095500754567366),
+                                 (0.1148764602736805918, 0.35355339059327376220,
+                                     -0.60150095500754567366),
+                                 (0.18587401723009224507,
+                                  -0.57206140281768429760, 0.37174803446018449013),
+                                 (0.18587401723009224507,  0.57206140281768429760,
+                                  0.37174803446018449013),
+                                 (0.30075047750377283683, -0.21850801222441053540,
+                                  0.60150095500754567366),
+                                 (0.30075047750377283683, 0.21850801222441053540,
+                                  0.60150095500754567366),
+                                 (0.48662449473386508189, -0.35355339059327376220,
+                                  -0.37174803446018449013),
+                                 (0.48662449473386508189, 0.35355339059327376220,
+                                  -0.37174803446018449013),
+                                 (-0.60150095500754567366, 0, -0.37174803446018449013),
+                                 (-0.30075047750377283683, -0.21850801222441053540,
+                                  -0.60150095500754567366),
+                                 (-0.30075047750377283683, 0.21850801222441053540,
+                                  -0.60150095500754567366),
+                                 (0.60150095500754567366, 0, 0.37174803446018449013),
+                                 (0.4156269377774534286, -0.57206140281768429760, 0),
+                                 (0.4156269377774534286, 0.57206140281768429760, 0),
+                                 (0.37174803446018449013, 0, -0.60150095500754567366),
+                                 (-0.4156269377774534286, -0.57206140281768429760, 0),
+                                 (-0.4156269377774534286, 0.57206140281768429760, 0),
+                                 (-0.67249851196395732696, -0.21850801222441053540, 0),
+                                 (-0.67249851196395732696, 0.21850801222441053540, 0),
+                                 (0.67249851196395732696, -0.21850801222441053540, 0),
+                                 (0.67249851196395732696, 0.21850801222441053540, 0),
+                                 (-0.37174803446018449013, 0, 0.60150095500754567366),
+                                 (-0.48662449473386508189, -0.35355339059327376220,
+                                 0.37174803446018449013),
+                                 (-0.48662449473386508189, 0.35355339059327376220,
+                                  0.37174803446018449013),
+                                 (-0.18587401723009224507, -0.57206140281768429760,
+                                  -0.37174803446018449013),
+                                 (-0.18587401723009224507, 0.57206140281768429760,
+                                  -0.37174803446018449013),
+                                 (-0.11487646027368059176, -0.35355339059327376220,
+                                  0.60150095500754567366),
+                                 (-0.11487646027368059176, 0.35355339059327376220,
+                                 0.60150095500754567366)],
+                                 [0, 10, 16], [23, 10, 0], [16, 13, 0],
+                                 [0, 13, 23], [16, 10, 1], [1, 10, 23],
+                                 [1, 13, 16], [23, 13, 1], [2, 4, 19],
+                                 [22, 4, 2], [2, 19, 27], [27, 22, 2],
+                                 [20, 5, 3], [3, 5, 21], [26, 20, 3],
+                                 [3, 21, 26], [29, 19, 4], [4, 22, 29],
+                                 [5, 20, 28], [28, 21, 5], [6, 8, 15],
+                                 [17, 8, 6], [6, 15, 25], [25, 17, 6],
+                                 [14, 9, 7], [7, 9, 18], [24, 14, 7],
+                                 [7, 18, 24], [8, 12, 15], [17, 12, 8],
+                                 [14, 11, 9], [9, 11, 18], [11, 14, 24],
+                                 [24, 18, 11], [25, 15, 12], [12, 17, 25],
+                                 [29, 27, 19], [20, 26, 28], [28, 26, 21],
+                                 [22, 27, 29]]
+    assert Abs(polytope_integrate(octahedron_five_compound, expr)) - 0.353553\
+        < 1e-6
+
+    cube_five_compound = [[(-0.1624598481164531631, -0.5, -0.6881909602355867691),
+                           (-0.1624598481164531631, 0.5, -0.6881909602355867691),
+                           (0.1624598481164531631, -0.5, 0.68819096023558676910),
+                           (0.1624598481164531631, 0.5, 0.68819096023558676910),
+                          (-0.52573111211913359231, 0, -0.6881909602355867691),
+                          (0.52573111211913359231, 0, 0.68819096023558676910),
+                          (-0.26286555605956679615, -0.8090169943749474241,
+                           -0.1624598481164531631),
+                          (-0.26286555605956679615, 0.8090169943749474241,
+                           -0.1624598481164531631),
+                          (0.26286555605956680301, -0.8090169943749474241,
+                           0.1624598481164531631),
+                          (0.26286555605956680301, 0.8090169943749474241,
+                           0.1624598481164531631),
+                          (-0.42532540417601993887, -0.3090169943749474241,
+                           0.68819096023558676910),
+                          (-0.42532540417601993887, 0.30901699437494742410,
+                           0.68819096023558676910),
+                          (0.42532540417601996609, -0.3090169943749474241,
+                           -0.6881909602355867691),
+                          (0.42532540417601996609, 0.30901699437494742410,
+                           -0.6881909602355867691),
+                          (-0.6881909602355867691, -0.5, 0.1624598481164531631),
+                          (-0.6881909602355867691, 0.5,  0.1624598481164531631),
+                          (0.68819096023558676910, -0.5, -0.1624598481164531631),
+                          (0.68819096023558676910, 0.5, -0.1624598481164531631),
+                          (-0.85065080835203998877, 0, -0.1624598481164531631),
+                          (0.85065080835203993218, 0, 0.1624598481164531631)],
+                          [18, 10, 3, 7], [13, 19, 8, 0], [18, 0, 8, 10],
+                          [3, 19, 13, 7], [18, 7, 13, 0], [8, 19, 3, 10],
+                          [6, 2, 11, 18], [1, 9, 19, 12], [11, 9, 1, 18],
+                          [6, 12, 19, 2], [1, 12, 6, 18], [11, 2, 19, 9],
+                          [4, 14, 11, 7], [17, 5, 8, 12], [4, 12, 8, 14],
+                          [11, 5, 17, 7], [4, 7, 17, 12], [8, 5, 11, 14],
+                          [6, 10, 15, 4], [13, 9, 5, 16], [15, 9, 13, 4],
+                          [6, 16, 5, 10], [13, 16, 6, 4], [15, 10, 5, 9],
+                          [14, 15, 1, 0], [16, 17, 3, 2], [14, 2, 3, 15],
+                          [1, 17, 16, 0], [14, 0, 16, 2], [3, 17, 1, 15]]
+    assert Abs(polytope_integrate(cube_five_compound, expr) - 1.25) < 1e-12
+
+    echidnahedron = [[(0, 0, -2.4898982848827801995),
+                      (0, 0, 2.4898982848827802734),
+                      (0, -4.2360679774997896964, -2.4898982848827801995),
+                      (0, -4.2360679774997896964, 2.4898982848827802734),
+                      (0, 4.2360679774997896964, -2.4898982848827801995),
+                      (0, 4.2360679774997896964, 2.4898982848827802734),
+                      (-4.0287400534704067567, -1.3090169943749474241, -2.4898982848827801995),
+                      (-4.0287400534704067567, -1.3090169943749474241, 2.4898982848827802734),
+                      (-4.0287400534704067567, 1.3090169943749474241, -2.4898982848827801995),
+                      (-4.0287400534704067567, 1.3090169943749474241, 2.4898982848827802734),
+                      (4.0287400534704069747, -1.3090169943749474241, -2.4898982848827801995),
+                      (4.0287400534704069747, -1.3090169943749474241, 2.4898982848827802734),
+                      (4.0287400534704069747, 1.3090169943749474241, -2.4898982848827801995),
+                      (4.0287400534704069747, 1.3090169943749474241, 2.4898982848827802734),
+                      (-2.4898982848827801995, -3.4270509831248422723, -2.4898982848827801995),
+                      (-2.4898982848827801995, -3.4270509831248422723, 2.4898982848827802734),
+                      (-2.4898982848827801995, 3.4270509831248422723, -2.4898982848827801995),
+                      (-2.4898982848827801995, 3.4270509831248422723, 2.4898982848827802734),
+                      (2.4898982848827802734, -3.4270509831248422723, -2.4898982848827801995),
+                      (2.4898982848827802734, -3.4270509831248422723, 2.4898982848827802734),
+                      (2.4898982848827802734, 3.4270509831248422723, -2.4898982848827801995),
+                      (2.4898982848827802734, 3.4270509831248422723, 2.4898982848827802734),
+                      (-4.7169310137059934362, -0.8090169943749474241, -1.1135163644116066184),
+                      (-4.7169310137059934362, 0.8090169943749474241, -1.1135163644116066184),
+                      (4.7169310137059937438, -0.8090169943749474241, 1.11351636441160673519),
+                      (4.7169310137059937438, 0.8090169943749474241, 1.11351636441160673519),
+                      (-4.2916056095299737777, -2.1180339887498948482, 1.11351636441160673519),
+                      (-4.2916056095299737777, 2.1180339887498948482, 1.11351636441160673519),
+                      (4.2916056095299737777, -2.1180339887498948482, -1.1135163644116066184),
+                      (4.2916056095299737777, 2.1180339887498948482, -1.1135163644116066184),
+                      (-3.6034146492943870399, 0, -3.3405490932348205213),
+                      (3.6034146492943870399, 0, 3.3405490932348202056),
+                      (-3.3405490932348205213, -3.4270509831248422723, 1.11351636441160673519),
+                      (-3.3405490932348205213, 3.4270509831248422723, 1.11351636441160673519),
+                      (3.3405490932348202056, -3.4270509831248422723, -1.1135163644116066184),
+                      (3.3405490932348202056, 3.4270509831248422723, -1.1135163644116066184),
+                      (-2.9152236890588002395, -2.1180339887498948482, 3.3405490932348202056),
+                      (-2.9152236890588002395, 2.1180339887498948482, 3.3405490932348202056),
+                      (2.9152236890588002395, -2.1180339887498948482, -3.3405490932348205213),
+                      (2.9152236890588002395, 2.1180339887498948482, -3.3405490932348205213),
+                      (-2.2270327288232132368, 0, -1.1135163644116066184),
+                      (-2.2270327288232132368, -4.2360679774997896964, -1.1135163644116066184),
+                      (-2.2270327288232132368, 4.2360679774997896964, -1.1135163644116066184),
+                      (2.2270327288232134704, 0, 1.11351636441160673519),
+                      (2.2270327288232134704, -4.2360679774997896964, 1.11351636441160673519),
+                      (2.2270327288232134704, 4.2360679774997896964, 1.11351636441160673519),
+                      (-1.8017073246471935200, -1.3090169943749474241, 1.11351636441160673519),
+                      (-1.8017073246471935200, 1.3090169943749474241, 1.11351636441160673519),
+                      (1.8017073246471935043, -1.3090169943749474241, -1.1135163644116066184),
+                      (1.8017073246471935043, 1.3090169943749474241, -1.1135163644116066184),
+                      (-1.3763819204711735382, 0, -4.7169310137059934362),
+                      (-1.3763819204711735382, 0, 0.26286555605956679615),
+                      (1.37638192047117353821, 0, 4.7169310137059937438),
+                      (1.37638192047117353821, 0, -0.26286555605956679615),
+                      (-1.1135163644116066184, -3.4270509831248422723, -3.3405490932348205213),
+                      (-1.1135163644116066184, -0.8090169943749474241, 4.7169310137059937438),
+                      (-1.1135163644116066184, -0.8090169943749474241, -0.26286555605956679615),
+                      (-1.1135163644116066184, 0.8090169943749474241, 4.7169310137059937438),
+                      (-1.1135163644116066184, 0.8090169943749474241, -0.26286555605956679615),
+                      (-1.1135163644116066184, 3.4270509831248422723, -3.3405490932348205213),
+                      (1.11351636441160673519, -3.4270509831248422723, 3.3405490932348202056),
+                      (1.11351636441160673519, -0.8090169943749474241, -4.7169310137059934362),
+                      (1.11351636441160673519, -0.8090169943749474241, 0.26286555605956679615),
+                      (1.11351636441160673519, 0.8090169943749474241, -4.7169310137059934362),
+                      (1.11351636441160673519, 0.8090169943749474241, 0.26286555605956679615),
+                      (1.11351636441160673519, 3.4270509831248422723, 3.3405490932348202056),
+                      (-0.85065080835203998877, 0, 1.11351636441160673519),
+                      (0.85065080835203993218, 0, -1.1135163644116066184),
+                      (-0.6881909602355867691, -0.5, -1.1135163644116066184),
+                      (-0.6881909602355867691, 0.5, -1.1135163644116066184),
+                      (-0.6881909602355867691, -4.7360679774997896964, -1.1135163644116066184),
+                      (-0.6881909602355867691, -2.1180339887498948482, -1.1135163644116066184),
+                      (-0.6881909602355867691, 2.1180339887498948482, -1.1135163644116066184),
+                      (-0.6881909602355867691, 4.7360679774997896964, -1.1135163644116066184),
+                      (0.68819096023558676910, -0.5, 1.11351636441160673519),
+                      (0.68819096023558676910, 0.5, 1.11351636441160673519),
+                      (0.68819096023558676910, -4.7360679774997896964, 1.11351636441160673519),
+                      (0.68819096023558676910, -2.1180339887498948482, 1.11351636441160673519),
+                      (0.68819096023558676910, 2.1180339887498948482, 1.11351636441160673519),
+                      (0.68819096023558676910, 4.7360679774997896964, 1.11351636441160673519),
+                      (-0.42532540417601993887, -1.3090169943749474241, -4.7169310137059934362),
+                      (-0.42532540417601993887, -1.3090169943749474241, 0.26286555605956679615),
+                      (-0.42532540417601993887, 1.3090169943749474241, -4.7169310137059934362),
+                      (-0.42532540417601993887, 1.3090169943749474241, 0.26286555605956679615),
+                      (-0.26286555605956679615, -0.8090169943749474241, 1.11351636441160673519),
+                      (-0.26286555605956679615, 0.8090169943749474241, 1.11351636441160673519),
+                      (0.26286555605956679615, -0.8090169943749474241, -1.1135163644116066184),
+                      (0.26286555605956679615, 0.8090169943749474241, -1.1135163644116066184),
+                      (0.42532540417601996609, -1.3090169943749474241, 4.7169310137059937438),
+                      (0.42532540417601996609, -1.3090169943749474241, -0.26286555605956679615),
+                      (0.42532540417601996609, 1.3090169943749474241, 4.7169310137059937438),
+                      (0.42532540417601996609, 1.3090169943749474241, -0.26286555605956679615)],
+                      [9, 66, 47], [44, 62, 77], [20, 91, 49], [33, 47, 83],
+                      [3, 77, 84], [12, 49, 53], [36, 84, 66], [28, 53, 62],
+                      [73, 83, 91], [15, 84, 46], [25, 64, 43], [16, 58, 72],
+                      [26, 46, 51], [11, 43, 74], [4, 72, 91], [60, 74, 84],
+                      [35, 91, 64], [23, 51, 58], [19, 74, 77], [79, 83, 78],
+                      [6, 56, 40], [76, 77, 81], [21, 78, 75], [8, 40, 58],
+                      [31, 75, 74], [42, 58, 83], [41, 81, 56], [13, 75, 43],
+                      [27, 51, 47], [2, 89, 71], [24, 43, 62], [17, 47, 85],
+                      [14, 71, 56], [65, 85, 75], [22, 56, 51], [34, 62, 89],
+                      [5, 85, 78], [32, 81, 46], [10, 53, 48], [45, 78, 64],
+                      [7, 46, 66], [18, 48, 89], [37, 66, 85], [70, 89, 81],
+                      [29, 64, 53], [88, 74, 1], [38, 67, 48], [42, 83, 72],
+                      [57, 1, 85], [34, 48, 62], [59, 72, 87], [19, 62, 74],
+                      [63, 87, 67], [17, 85, 83], [52, 75, 1], [39, 87, 49],
+                      [22, 51, 40], [55, 1, 66], [29, 49, 64], [30, 40, 69],
+                      [13, 64, 75], [82, 69, 87], [7, 66, 51], [90, 85, 1],
+                      [59, 69, 72], [70, 81, 71], [88, 1, 84], [73, 72, 83],
+                      [54, 71, 68], [5, 83, 85], [50, 68, 69], [3, 84, 81],
+                      [57, 66, 1], [30, 68, 40], [28, 62, 48], [52, 1, 74],
+                      [23, 40, 51], [38, 48, 86], [9, 51, 66], [80, 86, 68],
+                      [11, 74, 62], [55, 84, 1], [54, 86, 71], [35, 64, 49],
+                      [90, 1, 75], [41, 71, 81], [39, 49, 67], [15, 81, 84],
+                      [61, 67, 86], [21, 75, 64], [24, 53, 43], [50, 69, 0],
+                      [37, 85, 47], [31, 43, 75], [61, 0, 67], [27, 47, 58],
+                      [10, 67, 53], [8, 58, 69], [90, 75, 85], [45, 91, 78],
+                      [80, 68, 0], [36, 66, 46], [65, 78, 85], [63, 0, 87],
+                      [32, 46, 56], [20, 87, 91], [14, 56, 68], [57, 85, 66],
+                      [33, 58, 47], [61, 86, 0], [60, 84, 77], [37, 47, 66],
+                      [82, 0, 69], [44, 77, 89], [16, 69, 58], [18, 89, 86],
+                      [55, 66, 84], [26, 56, 46], [63, 67, 0], [31, 74, 43],
+                      [36, 46, 84], [50, 0, 68], [25, 43, 53], [6, 68, 56],
+                      [12, 53, 67], [88, 84, 74], [76, 89, 77], [82, 87, 0],
+                      [65, 75, 78], [60, 77, 74], [80, 0, 86], [79, 78, 91],
+                      [2, 86, 89], [4, 91, 87], [52, 74, 75], [21, 64, 78],
+                      [18, 86, 48], [23, 58, 40], [5, 78, 83], [28, 48, 53],
+                      [6, 40, 68], [25, 53, 64], [54, 68, 86], [33, 83, 58],
+                      [17, 83, 47], [12, 67, 49], [41, 56, 71], [9, 47, 51],
+                      [35, 49, 91], [2, 71, 86], [79, 91, 83], [38, 86, 67],
+                      [26, 51, 56], [7, 51, 46], [4, 87, 72], [34, 89, 48],
+                      [15, 46, 81], [42, 72, 58], [10, 48, 67], [27, 58, 51],
+                      [39, 67, 87], [76, 81, 89], [3, 81, 77], [8, 69, 40],
+                      [29, 53, 49], [19, 77, 62], [22, 40, 56], [20, 49, 87],
+                      [32, 56, 81], [59, 87, 69], [24, 62, 53], [11, 62, 43],
+                      [14, 68, 71], [73, 91, 72], [13, 43, 64], [70, 71, 89],
+                      [16, 72, 69], [44, 89, 62], [30, 69, 68], [45, 64, 91]]
+    #  Actual volume is : 51.405764746872634
+    assert Abs(polytope_integrate(echidnahedron, 1) - 51.4057647468726) < 1e-12
+    assert Abs(polytope_integrate(echidnahedron, expr) - 253.569603474519) <\
+    1e-12
+
+    #  Tests for many polynomials with maximum degree given(2D case).
+    assert polytope_integrate(cube2, [x**2, y*z], max_degree=2) == \
+        {y * z: 3125 / S(4), x ** 2: 3125 / S(3)}
+
+    assert polytope_integrate(cube2, max_degree=2) == \
+        {1: 125, x: 625 / S(2), x * z: 3125 / S(4), y: 625 / S(2),
+         y * z: 3125 / S(4), z ** 2: 3125 / S(3), y ** 2: 3125 / S(3),
+         z: 625 / S(2), x * y: 3125 / S(4), x ** 2: 3125 / S(3)}
+
+def test_point_sort():
+    assert point_sort([Point(0, 0), Point(1, 0), Point(1, 1)]) == \
+        [Point2D(1, 1), Point2D(1, 0), Point2D(0, 0)]
+
+    fig6 = Polygon((0, 0), (1, 0), (1, 1))
+    assert polytope_integrate(fig6, x*y) == Rational(-1, 8)
+    assert polytope_integrate(fig6, x*y, clockwise = True) == Rational(1, 8)
+
+
+def test_polytopes_intersecting_sides():
+    fig5 = Polygon(Point(-4.165, -0.832), Point(-3.668, 1.568),
+                   Point(-3.266, 1.279), Point(-1.090, -2.080),
+                   Point(3.313, -0.683), Point(3.033, -4.845),
+                   Point(-4.395, 4.840), Point(-1.007, -3.328))
+    assert polytope_integrate(fig5, x**2 + x*y + y**2) ==\
+        S(1633405224899363)/(24*10**12)
+
+    fig6 = Polygon(Point(-3.018, -4.473), Point(-0.103, 2.378),
+                   Point(-1.605, -2.308), Point(4.516, -0.771),
+                   Point(4.203, 0.478))
+    assert polytope_integrate(fig6, x**2 + x*y + y**2) ==\
+        S(88161333955921)/(3*10**12)
+
+
+def test_max_degree():
+    polygon = Polygon((0, 0), (0, 1), (1, 1), (1, 0))
+    polys = [1, x, y, x*y, x**2*y, x*y**2]
+    assert polytope_integrate(polygon, polys, max_degree=3) == \
+        {1: 1, x: S.Half, y: S.Half, x*y: Rational(1, 4), x**2*y: Rational(1, 6), x*y**2: Rational(1, 6)}
+    assert polytope_integrate(polygon, polys, max_degree=2) == \
+        {1: 1, x: S.Half, y: S.Half, x*y: Rational(1, 4)}
+    assert polytope_integrate(polygon, polys, max_degree=1) == \
+        {1: 1, x: S.Half, y: S.Half}
+
+
+def test_main_integrate3d():
+    cube = [[(0, 0, 0), (0, 0, 5), (0, 5, 0), (0, 5, 5), (5, 0, 0),\
+             (5, 0, 5), (5, 5, 0), (5, 5, 5)],\
+            [2, 6, 7, 3], [3, 7, 5, 1], [7, 6, 4, 5], [1, 5, 4, 0],\
+            [3, 1, 0, 2], [0, 4, 6, 2]]
+    vertices = cube[0]
+    faces = cube[1:]
+    hp_params = hyperplane_parameters(faces, vertices)
+    assert main_integrate3d(1, faces, vertices, hp_params) == -125
+    assert main_integrate3d(1, faces, vertices, hp_params, max_degree=1) == \
+        {1: -125, y: Rational(-625, 2), z: Rational(-625, 2), x: Rational(-625, 2)}
+
+
+def test_main_integrate():
+    triangle = Polygon((0, 3), (5, 3), (1, 1))
+    facets = triangle.sides
+    hp_params = hyperplane_parameters(triangle)
+    assert main_integrate(x**2 + y**2, facets, hp_params) == Rational(325, 6)
+    assert main_integrate(x**2 + y**2, facets, hp_params, max_degree=1) == \
+        {0: 0, 1: 5, y: Rational(35, 3), x: 10}
+
+
+def test_polygon_integrate():
+    cube = [[(0, 0, 0), (0, 0, 5), (0, 5, 0), (0, 5, 5), (5, 0, 0),\
+             (5, 0, 5), (5, 5, 0), (5, 5, 5)],\
+            [2, 6, 7, 3], [3, 7, 5, 1], [7, 6, 4, 5], [1, 5, 4, 0],\
+            [3, 1, 0, 2], [0, 4, 6, 2]]
+    facet = cube[1]
+    facets = cube[1:]
+    vertices = cube[0]
+    assert polygon_integrate(facet, [(0, 1, 0), 5], 0, facets, vertices, 1, 0) == -25
+
+
+def test_distance_to_side():
+    point = (0, 0, 0)
+    assert distance_to_side(point, [(0, 0, 1), (0, 1, 0)], (1, 0, 0)) == -sqrt(2)/2
+
+
+def test_lineseg_integrate():
+    polygon = [(0, 5, 0), (5, 5, 0), (5, 5, 5), (0, 5, 5)]
+    line_seg = [(0, 5, 0), (5, 5, 0)]
+    assert lineseg_integrate(polygon, 0, line_seg, 1, 0) == 5
+    assert lineseg_integrate(polygon, 0, line_seg, 0, 0) == 0
+
+
+def test_integration_reduction():
+    triangle = Polygon(Point(0, 3), Point(5, 3), Point(1, 1))
+    facets = triangle.sides
+    a, b = hyperplane_parameters(triangle)[0]
+    assert integration_reduction(facets, 0, a, b, 1, (x, y), 0) == 5
+    assert integration_reduction(facets, 0, a, b, 0, (x, y), 0) == 0
+
+
+def test_integration_reduction_dynamic():
+    triangle = Polygon(Point(0, 3), Point(5, 3), Point(1, 1))
+    facets = triangle.sides
+    a, b = hyperplane_parameters(triangle)[0]
+    x0 = facets[0].points[0]
+    monomial_values = [[0, 0, 0, 0], [1, 0, 0, 5],\
+                       [y, 0, 1, 15], [x, 1, 0, None]]
+
+    assert integration_reduction_dynamic(facets, 0, a, b, x, 1, (x, y), 1,\
+                                         0, 1, x0, monomial_values, 3) == Rational(25, 2)
+    assert integration_reduction_dynamic(facets, 0, a, b, 0, 1, (x, y), 1,\
+                                         0, 1, x0, monomial_values, 3) == 0
+
+
+def test_is_vertex():
+    assert is_vertex(2) is False
+    assert is_vertex((2, 3)) is True
+    assert is_vertex(Point(2, 3)) is True
+    assert is_vertex((2, 3, 4)) is True
+    assert is_vertex((2, 3, 4, 5)) is False
+
+
+def test_issue_19234():
+    polygon = Polygon(Point(0, 0), Point(0, 1), Point(1, 1), Point(1, 0))
+    polys =  [ 1, x, y, x*y, x**2*y, x*y**2]
+    assert polytope_integrate(polygon, polys) == \
+        {1: 1, x: S.Half, y: S.Half, x*y: Rational(1, 4), x**2*y: Rational(1, 6), x*y**2: Rational(1, 6)}
+    polys =  [ 1, x, y, x*y, 3 + x**2*y, x + x*y**2]
+    assert polytope_integrate(polygon, polys) == \
+        {1: 1, x: S.Half, y: S.Half, x*y: Rational(1, 4), x**2*y + 3: Rational(19, 6), x*y**2 + x: Rational(2, 3)}

rl/Lib/site-packages/sympy/integrals/tests/test_laplace.py

@@ -0,0 +1,769 @@
+from sympy.integrals.laplace import (
+    laplace_transform, inverse_laplace_transform,
+    LaplaceTransform, InverseLaplaceTransform,
+    _laplace_deep_collect, laplace_correspondence,
+    laplace_initial_conds)
+from sympy.core.function import Function, expand_mul
+from sympy.core import EulerGamma, Subs, Derivative, diff
+from sympy.core.exprtools import factor_terms
+from sympy.core.numbers import I, oo, pi
+from sympy.core.relational import Eq
+from sympy.core.singleton import S
+from sympy.core.symbol import Symbol, symbols
+from sympy.simplify.simplify import simplify
+from sympy.functions.elementary.complexes import Abs, re
+from sympy.functions.elementary.exponential import exp, log, exp_polar
+from sympy.functions.elementary.hyperbolic import cosh, sinh, coth, asinh
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.piecewise import Piecewise
+from sympy.functions.elementary.trigonometric import atan, cos, sin
+from sympy.logic.boolalg import And
+from sympy.functions.special.gamma_functions import (
+    lowergamma, gamma, uppergamma)
+from sympy.functions.special.delta_functions import DiracDelta, Heaviside
+from sympy.functions.special.singularity_functions import SingularityFunction
+from sympy.functions.special.zeta_functions import lerchphi
+from sympy.functions.special.error_functions import (
+    fresnelc, fresnels, erf, erfc, Ei, Ci, expint, E1)
+from sympy.functions.special.bessel import besseli, besselj, besselk, bessely
+from sympy.testing.pytest import slow, warns_deprecated_sympy
+from sympy.matrices import Matrix, eye
+from sympy.abc import s
+
+
+@slow
+def test_laplace_transform():
+    LT = laplace_transform
+    ILT = inverse_laplace_transform
+    a, b, c = symbols('a, b, c', positive=True)
+    np = symbols('np', integer=True, positive=True)
+    t, w, x = symbols('t, w, x')
+    f = Function('f')
+    F = Function('F')
+    g = Function('g')
+    y = Function('y')
+    Y = Function('Y')
+
+    # Test helper functions
+    assert (
+        _laplace_deep_collect(exp((t+a)*(t+b)) +
+                              besselj(2, exp((t+a)*(t+b)-t**2)), t) ==
+        exp(a*b + t**2 + t*(a + b)) + besselj(2, exp(a*b + t*(a + b))))
+    L = laplace_transform(diff(y(t), t, 3), t, s, noconds=True)
+    L = laplace_correspondence(L, {y: Y})
+    L = laplace_initial_conds(L, t, {y: [2, 4, 8, 16, 32]})
+    assert L == s**3*Y(s) - 2*s**2 - 4*s - 8
+    # Test whether `noconds=True` in `doit`:
+    assert (2*LaplaceTransform(exp(t), t, s) - 1).doit() == -1 + 2/(s - 1)
+    assert (LT(a*t+t**2+t**(S(5)/2), t, s) ==
+            (a/s**2 + 2/s**3 + 15*sqrt(pi)/(8*s**(S(7)/2)), 0, True))
+    assert LT(b/(t+a), t, s) == (-b*exp(-a*s)*Ei(-a*s), 0, True)
+    assert (LT(1/sqrt(t+a), t, s) ==
+            (sqrt(pi)*sqrt(1/s)*exp(a*s)*erfc(sqrt(a)*sqrt(s)), 0, True))
+    assert (LT(sqrt(t)/(t+a), t, s) ==
+            (-pi*sqrt(a)*exp(a*s)*erfc(sqrt(a)*sqrt(s)) + sqrt(pi)*sqrt(1/s),
+             0, True))
+    assert (LT((t+a)**(-S(3)/2), t, s) ==
+            (-2*sqrt(pi)*sqrt(s)*exp(a*s)*erfc(sqrt(a)*sqrt(s)) + 2/sqrt(a),
+             0, True))
+    assert (LT(t**(S(1)/2)*(t+a)**(-1), t, s) ==
+            (-pi*sqrt(a)*exp(a*s)*erfc(sqrt(a)*sqrt(s)) + sqrt(pi)*sqrt(1/s),
+             0, True))
+    assert (LT(1/(a*sqrt(t) + t**(3/2)), t, s) ==
+            (pi*sqrt(a)*exp(a*s)*erfc(sqrt(a)*sqrt(s)), 0, True))
+    assert (LT((t+a)**b, t, s) ==
+            (s**(-b - 1)*exp(-a*s)*uppergamma(b + 1, a*s), 0, True))
+    assert LT(t**5/(t+a), t, s) == (120*a**5*uppergamma(-5, a*s), 0, True)
+    assert LT(exp(t), t, s) == (1/(s - 1), 1, True)
+    assert LT(exp(2*t), t, s) == (1/(s - 2), 2, True)
+    assert LT(exp(a*t), t, s) == (1/(s - a), a, True)
+    assert LT(exp(a*(t-b)), t, s) == (exp(-a*b)/(-a + s), a, True)
+    assert LT(t*exp(-a*(t)), t, s) == ((a + s)**(-2), -a, True)
+    assert LT(t*exp(-a*(t-b)), t, s) == (exp(a*b)/(a + s)**2, -a, True)
+    assert LT(b*t*exp(-a*t), t, s) == (b/(a + s)**2, -a, True)
+    assert LT(exp(-a*exp(-t)), t, s) == (lowergamma(s, a)/a**s, 0, True)
+    assert LT(exp(-a*exp(t)), t, s) == (a**s*uppergamma(-s, a), 0, True)
+    assert (LT(t**(S(7)/4)*exp(-8*t)/gamma(S(11)/4), t, s) ==
+            ((s + 8)**(-S(11)/4), -8, True))
+    assert (LT(t**(S(3)/2)*exp(-8*t), t, s) ==
+            (3*sqrt(pi)/(4*(s + 8)**(S(5)/2)), -8, True))
+    assert LT(t**a*exp(-a*t), t, s) == ((a+s)**(-a-1)*gamma(a+1), -a, True)
+    assert (LT(b*exp(-a*t**2), t, s) ==
+            (sqrt(pi)*b*exp(s**2/(4*a))*erfc(s/(2*sqrt(a)))/(2*sqrt(a)),
+             0, True))
+    assert (LT(exp(-2*t**2), t, s) ==
+            (sqrt(2)*sqrt(pi)*exp(s**2/8)*erfc(sqrt(2)*s/4)/4, 0, True))
+    assert (LT(b*exp(2*t**2), t, s) ==
+            (b*LaplaceTransform(exp(2*t**2), t, s), -oo, True))
+    assert (LT(t*exp(-a*t**2), t, s) ==
+            (1/(2*a) - s*erfc(s/(2*sqrt(a)))/(4*sqrt(pi)*a**(S(3)/2)),
+             0, True))
+    assert (LT(exp(-a/t), t, s) ==
+            (2*sqrt(a)*sqrt(1/s)*besselk(1, 2*sqrt(a)*sqrt(s)), 0, True))
+    assert LT(sqrt(t)*exp(-a/t), t, s, simplify=True) == (
+        sqrt(pi)*(sqrt(a)*sqrt(s) + 1/S(2))*sqrt(s**(-3)) *
+        exp(-2*sqrt(a)*sqrt(s)), 0, True)
+    assert (LT(exp(-a/t)/sqrt(t), t, s) ==
+            (sqrt(pi)*sqrt(1/s)*exp(-2*sqrt(a)*sqrt(s)), 0, True))
+    assert (LT(exp(-a/t)/(t*sqrt(t)), t, s) ==
+            (sqrt(pi)*sqrt(1/a)*exp(-2*sqrt(a)*sqrt(s)), 0, True))
+    assert (
+        LT(exp(-2*sqrt(a*t)), t, s) ==
+        (1/s - sqrt(pi)*sqrt(a) * exp(a/s)*erfc(sqrt(a)*sqrt(1/s)) /
+         s**(S(3)/2), 0, True))
+    assert LT(exp(-2*sqrt(a*t))/sqrt(t), t, s) == (
+        exp(a/s)*erfc(sqrt(a) * sqrt(1/s))*(sqrt(pi)*sqrt(1/s)), 0, True)
+    assert (LT(t**4*exp(-2/t), t, s) ==
+            (8*sqrt(2)*(1/s)**(S(5)/2)*besselk(5, 2*sqrt(2)*sqrt(s)),
+             0, True))
+    assert LT(sinh(a*t), t, s) == (a/(-a**2 + s**2), a, True)
+    assert (LT(b*sinh(a*t)**2, t, s) ==
+            (2*a**2*b/(-4*a**2*s + s**3), 2*a, True))
+    assert (LT(b*sinh(a*t)**2, t, s, simplify=True) ==
+            (2*a**2*b/(s*(-4*a**2 + s**2)), 2*a, True))
+    # The following line confirms that issue #21202 is solved
+    assert LT(cosh(2*t), t, s) == (s/(-4 + s**2), 2, True)
+    assert LT(cosh(a*t), t, s) == (s/(-a**2 + s**2), a, True)
+    assert (LT(cosh(a*t)**2, t, s, simplify=True) ==
+            ((2*a**2 - s**2)/(s*(4*a**2 - s**2)), 2*a, True))
+    assert (LT(sinh(x+3), x, s, simplify=True) ==
+            ((s*sinh(3) + cosh(3))/(s**2 - 1), 1, True))
+    L, _, _ = LT(42*sin(w*t+x)**2, t, s)
+    assert (
+        L -
+        21*(s**2 + s*(-s*cos(2*x) + 2*w*sin(2*x)) +
+            4*w**2)/(s*(s**2 + 4*w**2))).simplify() == 0
+    # The following line replaces the old test test_issue_7173()
+    assert LT(sinh(a*t)*cosh(a*t), t, s, simplify=True) == (a/(-4*a**2 + s**2),
+                                                            2*a, True)
+    assert LT(sinh(a*t)/t, t, s) == (log((a + s)/(-a + s))/2, a, True)
+    assert (LT(t**(-S(3)/2)*sinh(a*t), t, s) ==
+            (-sqrt(pi)*(sqrt(-a + s) - sqrt(a + s)), a, True))
+    assert (LT(sinh(2*sqrt(a*t)), t, s) ==
+            (sqrt(pi)*sqrt(a)*exp(a/s)/s**(S(3)/2), 0, True))
+    assert (LT(sqrt(t)*sinh(2*sqrt(a*t)), t, s, simplify=True) ==
+            ((-sqrt(a)*s**(S(5)/2) + sqrt(pi)*s**2*(2*a + s)*exp(a/s) *
+              erf(sqrt(a)*sqrt(1/s))/2)/s**(S(9)/2), 0, True))
+    assert (LT(sinh(2*sqrt(a*t))/sqrt(t), t, s) ==
+            (sqrt(pi)*exp(a/s)*erf(sqrt(a)*sqrt(1/s))/sqrt(s), 0, True))
+    assert (LT(sinh(sqrt(a*t))**2/sqrt(t), t, s) ==
+            (sqrt(pi)*(exp(a/s) - 1)/(2*sqrt(s)), 0, True))
+    assert (LT(t**(S(3)/7)*cosh(a*t), t, s) ==
+            (((a + s)**(-S(10)/7) + (-a+s)**(-S(10)/7))*gamma(S(10)/7)/2,
+             a, True))
+    assert (LT(cosh(2*sqrt(a*t)), t, s) ==
+            (sqrt(pi)*sqrt(a)*exp(a/s)*erf(sqrt(a)*sqrt(1/s))/s**(S(3)/2) +
+             1/s, 0, True))
+    assert (LT(sqrt(t)*cosh(2*sqrt(a*t)), t, s) ==
+            (sqrt(pi)*(a + s/2)*exp(a/s)/s**(S(5)/2), 0, True))
+    assert (LT(cosh(2*sqrt(a*t))/sqrt(t), t, s) ==
+            (sqrt(pi)*exp(a/s)/sqrt(s), 0, True))
+    assert (LT(cosh(sqrt(a*t))**2/sqrt(t), t, s) ==
+            (sqrt(pi)*(exp(a/s) + 1)/(2*sqrt(s)), 0, True))
+    assert LT(log(t), t, s, simplify=True) == (
+        (-log(s) - EulerGamma)/s, 0, True)
+    assert (LT(-log(t/a), t, s, simplify=True) ==
+            ((log(a) + log(s) + EulerGamma)/s, 0, True))
+    assert LT(log(1+a*t), t, s) == (-exp(s/a)*Ei(-s/a)/s, 0, True)
+    assert (LT(log(t+a), t, s, simplify=True) ==
+            ((s*log(a) - exp(s/a)*Ei(-s/a))/s**2, 0, True))
+    assert (LT(log(t)/sqrt(t), t, s, simplify=True) ==
+            (sqrt(pi)*(-log(s) - log(4) - EulerGamma)/sqrt(s), 0, True))
+    assert (LT(t**(S(5)/2)*log(t), t, s, simplify=True) ==
+            (sqrt(pi)*(-15*log(s) - log(1073741824) - 15*EulerGamma + 46) /
+             (8*s**(S(7)/2)), 0, True))
+    assert (LT(t**3*log(t), t, s, noconds=True, simplify=True) -
+            6*(-log(s) - S.EulerGamma + S(11)/6)/s**4).simplify() == S.Zero
+    assert (LT(log(t)**2, t, s, simplify=True) ==
+            (((log(s) + EulerGamma)**2 + pi**2/6)/s, 0, True))
+    assert (LT(exp(-a*t)*log(t), t, s, simplify=True) ==
+            ((-log(a + s) - EulerGamma)/(a + s), -a, True))
+    assert LT(sin(a*t), t, s) == (a/(a**2 + s**2), 0, True)
+    assert (LT(Abs(sin(a*t)), t, s) ==
+            (a*coth(pi*s/(2*a))/(a**2 + s**2), 0, True))
+    assert LT(sin(a*t)/t, t, s) == (atan(a/s), 0, True)
+    assert LT(sin(a*t)**2/t, t, s) == (log(4*a**2/s**2 + 1)/4, 0, True)
+    assert (LT(sin(a*t)**2/t**2, t, s) ==
+            (a*atan(2*a/s) - s*log(4*a**2/s**2 + 1)/4, 0, True))
+    assert (LT(sin(2*sqrt(a*t)), t, s) ==
+            (sqrt(pi)*sqrt(a)*exp(-a/s)/s**(S(3)/2), 0, True))
+    assert LT(sin(2*sqrt(a*t))/t, t, s) == (pi*erf(sqrt(a)*sqrt(1/s)), 0, True)
+    assert LT(cos(a*t), t, s) == (s/(a**2 + s**2), 0, True)
+    assert (LT(cos(a*t)**2, t, s) ==
+            ((2*a**2 + s**2)/(s*(4*a**2 + s**2)), 0, True))
+    assert (LT(sqrt(t)*cos(2*sqrt(a*t)), t, s, simplify=True) ==
+            (sqrt(pi)*(-a + s/2)*exp(-a/s)/s**(S(5)/2), 0, True))
+    assert (LT(cos(2*sqrt(a*t))/sqrt(t), t, s) ==
+            (sqrt(pi)*sqrt(1/s)*exp(-a/s), 0, True))
+    assert (LT(sin(a*t)*sin(b*t), t, s) ==
+            (2*a*b*s/((s**2 + (a - b)**2)*(s**2 + (a + b)**2)), 0, True))
+    assert (LT(cos(a*t)*sin(b*t), t, s) ==
+            (b*(-a**2 + b**2 + s**2)/((s**2 + (a - b)**2)*(s**2 + (a + b)**2)),
+             0, True))
+    assert (LT(cos(a*t)*cos(b*t), t, s) ==
+            (s*(a**2 + b**2 + s**2)/((s**2 + (a - b)**2)*(s**2 + (a + b)**2)),
+             0, True))
+    assert (LT(-a*t*cos(a*t) + sin(a*t), t, s, simplify=True) ==
+            (2*a**3/(a**4 + 2*a**2*s**2 + s**4), 0, True))
+    assert LT(c*exp(-b*t)*sin(a*t), t, s) == (a *
+                                              c/(a**2 + (b + s)**2), -b, True)
+    assert LT(c*exp(-b*t)*cos(a*t), t, s) == (c*(b + s)/(a**2 + (b + s)**2),
+                                              -b, True)
+    L, plane, cond = LT(cos(x + 3), x, s, simplify=True)
+    assert plane == 0
+    assert L - (s*cos(3) - sin(3))/(s**2 + 1) == 0
+    # Error functions (laplace7.pdf)
+    assert LT(erf(a*t), t, s) == (exp(s**2/(4*a**2))*erfc(s/(2*a))/s, 0, True)
+    assert LT(erf(sqrt(a*t)), t, s) == (sqrt(a)/(s*sqrt(a + s)), 0, True)
+    assert (LT(exp(a*t)*erf(sqrt(a*t)), t, s, simplify=True) ==
+            (-sqrt(a)/(sqrt(s)*(a - s)), a, True))
+    assert (LT(erf(sqrt(a/t)/2), t, s, simplify=True) ==
+            (1/s - exp(-sqrt(a)*sqrt(s))/s, 0, True))
+    assert (LT(erfc(sqrt(a*t)), t, s, simplify=True) ==
+            (-sqrt(a)/(s*sqrt(a + s)) + 1/s, -a, True))
+    assert (LT(exp(a*t)*erfc(sqrt(a*t)), t, s) ==
+            (1/(sqrt(a)*sqrt(s) + s), 0, True))
+    assert LT(erfc(sqrt(a/t)/2), t, s) == (exp(-sqrt(a)*sqrt(s))/s, 0, True)
+    # Bessel functions (laplace8.pdf)
+    assert LT(besselj(0, a*t), t, s) == (1/sqrt(a**2 + s**2), 0, True)
+    assert (LT(besselj(1, a*t), t, s, simplify=True) ==
+            (a/(a**2 + s**2 + s*sqrt(a**2 + s**2)), 0, True))
+    assert (LT(besselj(2, a*t), t, s, simplify=True) ==
+            (a**2/(sqrt(a**2 + s**2)*(s + sqrt(a**2 + s**2))**2), 0, True))
+    assert (LT(t*besselj(0, a*t), t, s) ==
+            (s/(a**2 + s**2)**(S(3)/2), 0, True))
+    assert (LT(t*besselj(1, a*t), t, s) ==
+            (a/(a**2 + s**2)**(S(3)/2), 0, True))
+    assert (LT(t**2*besselj(2, a*t), t, s) ==
+            (3*a**2/(a**2 + s**2)**(S(5)/2), 0, True))
+    assert LT(besselj(0, 2*sqrt(a*t)), t, s) == (exp(-a/s)/s, 0, True)
+    assert (LT(t**(S(3)/2)*besselj(3, 2*sqrt(a*t)), t, s) ==
+            (a**(S(3)/2)*exp(-a/s)/s**4, 0, True))
+    assert (LT(besselj(0, a*sqrt(t**2+b*t)), t, s, simplify=True) ==
+            (exp(b*(s - sqrt(a**2 + s**2)))/sqrt(a**2 + s**2), 0, True))
+    assert LT(besseli(0, a*t), t, s) == (1/sqrt(-a**2 + s**2), a, True)
+    assert (LT(besseli(1, a*t), t, s, simplify=True) ==
+            (a/(-a**2 + s**2 + s*sqrt(-a**2 + s**2)), a, True))
+    assert (LT(besseli(2, a*t), t, s, simplify=True) ==
+            (a**2/(sqrt(-a**2 + s**2)*(s + sqrt(-a**2 + s**2))**2), a, True))
+    assert LT(t*besseli(0, a*t), t, s) == (s/(-a**2 + s**2)**(S(3)/2), a, True)
+    assert LT(t*besseli(1, a*t), t, s) == (a/(-a**2 + s**2)**(S(3)/2), a, True)
+    assert (LT(t**2*besseli(2, a*t), t, s) ==
+            (3*a**2/(-a**2 + s**2)**(S(5)/2), a, True))
+    assert (LT(t**(S(3)/2)*besseli(3, 2*sqrt(a*t)), t, s) ==
+            (a**(S(3)/2)*exp(a/s)/s**4, 0, True))
+    assert (LT(bessely(0, a*t), t, s) ==
+            (-2*asinh(s/a)/(pi*sqrt(a**2 + s**2)), 0, True))
+    assert (LT(besselk(0, a*t), t, s) ==
+            (log((s + sqrt(-a**2 + s**2))/a)/sqrt(-a**2 + s**2), -a, True))
+    assert (LT(sin(a*t)**4, t, s, simplify=True) ==
+            (24*a**4/(s*(64*a**4 + 20*a**2*s**2 + s**4)), 0, True))
+    # Test general rules and unevaluated forms
+    # These all also test whether issue #7219 is solved.
+    assert LT(Heaviside(t-1)*cos(t-1), t, s) == (s*exp(-s)/(s**2 + 1), 0, True)
+    assert LT(a*f(t), t, w) == (a*LaplaceTransform(f(t), t, w), -oo, True)
+    assert (LT(a*Heaviside(t+1)*f(t+1), t, s) ==
+            (a*LaplaceTransform(f(t + 1), t, s), -oo, True))
+    assert (LT(a*Heaviside(t-1)*f(t-1), t, s) ==
+            (a*LaplaceTransform(f(t), t, s)*exp(-s), -oo, True))
+    assert (LT(b*f(t/a), t, s) ==
+            (a*b*LaplaceTransform(f(t), t, a*s), -oo, True))
+    assert LT(exp(-f(x)*t), t, s) == (1/(s + f(x)), -re(f(x)), True)
+    assert (LT(exp(-a*t)*f(t), t, s) ==
+            (LaplaceTransform(f(t), t, a + s), -oo, True))
+    assert (LT(exp(-a*t)*erfc(sqrt(b/t)/2), t, s) ==
+            (exp(-sqrt(b)*sqrt(a + s))/(a + s), -a, True))
+    assert (LT(sinh(a*t)*f(t), t, s) ==
+            (LaplaceTransform(f(t), t, -a + s)/2 -
+             LaplaceTransform(f(t), t, a + s)/2, -oo, True))
+    assert (LT(sinh(a*t)*t, t, s, simplify=True) ==
+            (2*a*s/(a**4 - 2*a**2*s**2 + s**4), a, True))
+    assert (LT(cosh(a*t)*f(t), t, s) ==
+            (LaplaceTransform(f(t), t, -a + s)/2 +
+             LaplaceTransform(f(t), t, a + s)/2, -oo, True))
+    assert (LT(cosh(a*t)*t, t, s, simplify=True) ==
+            (1/(2*(a + s)**2) + 1/(2*(a - s)**2), a, True))
+    assert (LT(sin(a*t)*f(t), t, s, simplify=True) ==
+            (I*(-LaplaceTransform(f(t), t, -I*a + s) +
+                LaplaceTransform(f(t), t, I*a + s))/2, -oo, True))
+    assert (LT(sin(f(t)), t, s) ==
+            (LaplaceTransform(sin(f(t)), t, s), -oo, True))
+    assert (LT(sin(a*t)*t, t, s, simplify=True) ==
+            (2*a*s/(a**4 + 2*a**2*s**2 + s**4), 0, True))
+    assert (LT(cos(a*t)*f(t), t, s) ==
+            (LaplaceTransform(f(t), t, -I*a + s)/2 +
+             LaplaceTransform(f(t), t, I*a + s)/2, -oo, True))
+    assert (LT(cos(a*t)*t, t, s, simplify=True) ==
+            ((-a**2 + s**2)/(a**4 + 2*a**2*s**2 + s**4), 0, True))
+    L, plane, _ = LT(sin(a*t+b)**2*f(t), t, s)
+    assert plane == -oo
+    assert (
+        -L + (
+            LaplaceTransform(f(t), t, s)/2 -
+            LaplaceTransform(f(t), t, -2*I*a + s)*exp(2*I*b)/4 -
+            LaplaceTransform(f(t), t, 2*I*a + s)*exp(-2*I*b)/4)) == 0
+    L = LT(sin(a*t+b)**2*f(t), t, s, noconds=True)
+    assert (
+        laplace_correspondence(L, {f: F}) ==
+        F(s)/2 - F(-2*I*a + s)*exp(2*I*b)/4 -
+        F(2*I*a + s)*exp(-2*I*b)/4)
+    L, plane, _ = LT(sin(a*t)**3*cosh(b*t), t, s)
+    assert plane == b
+    assert (
+        -L - 3*a/(8*(9*a**2 + b**2 + 2*b*s + s**2)) -
+        3*a/(8*(9*a**2 + b**2 - 2*b*s + s**2)) +
+        3*a/(8*(a**2 + b**2 + 2*b*s + s**2)) +
+        3*a/(8*(a**2 + b**2 - 2*b*s + s**2))).simplify() == 0
+    assert (LT(t**2*exp(-t**2), t, s) ==
+            (sqrt(pi)*s**2*exp(s**2/4)*erfc(s/2)/8 - s/4 +
+             sqrt(pi)*exp(s**2/4)*erfc(s/2)/4, 0, True))
+    assert (LT((a*t**2 + b*t + c)*f(t), t, s) ==
+            (a*Derivative(LaplaceTransform(f(t), t, s), (s, 2)) -
+             b*Derivative(LaplaceTransform(f(t), t, s), s) +
+            c*LaplaceTransform(f(t), t, s), -oo, True))
+    assert (LT(t**np*g(t), t, s) ==
+            ((-1)**np*Derivative(LaplaceTransform(g(t), t, s), (s, np)),
+             -oo, True))
+    # The following tests check whether _piecewise_to_heaviside works:
+    x1 = Piecewise((0, t <= 0), (1, t <= 1), (0, True))
+    X1 = LT(x1, t, s)[0]
+    assert X1 == 1/s - exp(-s)/s
+    y1 = ILT(X1, s, t)
+    assert y1 == Heaviside(t) - Heaviside(t - 1)
+    x1 = Piecewise((0, t <= 0), (t, t <= 1), (2-t, t <= 2), (0, True))
+    X1 = LT(x1, t, s)[0].simplify()
+    assert X1 == (exp(2*s) - 2*exp(s) + 1)*exp(-2*s)/s**2
+    y1 = ILT(X1, s, t)
+    assert (
+        -y1 + t*Heaviside(t) + (t - 2)*Heaviside(t - 2) -
+        2*(t - 1)*Heaviside(t - 1)).simplify() == 0
+    x1 = Piecewise((exp(t), t <= 0), (1, t <= 1), (exp(-(t)), True))
+    X1 = LT(x1, t, s)[0]
+    assert X1 == exp(-1)*exp(-s)/(s + 1) + 1/s - exp(-s)/s
+    y1 = ILT(X1, s, t)
+    assert y1 == (
+        exp(-1)*exp(1 - t)*Heaviside(t - 1) + Heaviside(t) - Heaviside(t - 1))
+    x1 = Piecewise((0, x <= 0), (1, x <= 1), (0, True))
+    X1 = LT(x1, t, s)[0]
+    assert X1 == Piecewise((0, x <= 0), (1, x <= 1), (0, True))/s
+    x1 = [
+        a*Piecewise((1, And(t > 1, t <= 3)), (2, True)),
+        a*Piecewise((1, And(t >= 1, t <= 3)), (2, True)),
+        a*Piecewise((1, And(t >= 1, t < 3)), (2, True)),
+        a*Piecewise((1, And(t > 1, t < 3)), (2, True))]
+    for x2 in x1:
+        assert LT(x2, t, s)[0].expand() == 2*a/s - a*exp(-s)/s + a*exp(-3*s)/s
+    assert (
+        LT(Piecewise((1, Eq(t, 1)), (2, True)), t, s)[0] ==
+        LaplaceTransform(Piecewise((1, Eq(t, 1)), (2, True)), t, s))
+    # The following lines test whether _laplace_transform successfully
+    # removes Heaviside(1) before processing espressions. It fails if
+    # Heaviside(t) remains because then meijerg functions will appear.
+    X1 = 1/sqrt(a*s**2-b)
+    x1 = ILT(X1, s, t)
+    Y1 = LT(x1, t, s)[0]
+    Z1 = (Y1**2/X1**2).simplify()
+    assert Z1 == 1
+    # The following two lines test whether issues #5813 and #7176 are solved.
+    assert (LT(diff(f(t), (t, 1)), t, s, noconds=True) ==
+            s*LaplaceTransform(f(t), t, s) - f(0))
+    assert (LT(diff(f(t), (t, 3)), t, s, noconds=True) ==
+            s**3*LaplaceTransform(f(t), t, s) - s**2*f(0) -
+            s*Subs(Derivative(f(t), t), t, 0) -
+            Subs(Derivative(f(t), (t, 2)), t, 0))
+    # Issue #7219
+    assert (LT(diff(f(x, t, w), t, 2), t, s) ==
+            (s**2*LaplaceTransform(f(x, t, w), t, s) - s*f(x, 0, w) -
+             Subs(Derivative(f(x, t, w), t), t, 0), -oo, True))
+    # Issue #23307
+    assert (LT(10*diff(f(t), (t, 1)), t, s, noconds=True) ==
+            10*s*LaplaceTransform(f(t), t, s) - 10*f(0))
+    assert (LT(a*f(b*t)+g(c*t), t, s, noconds=True) ==
+            a*LaplaceTransform(f(t), t, s/b)/b +
+            LaplaceTransform(g(t), t, s/c)/c)
+    assert inverse_laplace_transform(
+        f(w), w, t, plane=0) == InverseLaplaceTransform(f(w), w, t, 0)
+    assert (LT(f(t)*g(t), t, s, noconds=True) ==
+            LaplaceTransform(f(t)*g(t), t, s))
+    # Issue #24294
+    assert (LT(b*f(a*t), t, s, noconds=True) ==
+            b*LaplaceTransform(f(t), t, s/a)/a)
+    assert LT(3*exp(t)*Heaviside(t), t, s) == (3/(s - 1), 1, True)
+    assert (LT(2*sin(t)*Heaviside(t), t, s, simplify=True) ==
+            (2/(s**2 + 1), 0, True))
+    # Issue #25293
+    assert (
+        LT((1/(t-1))*sin(4*pi*(t-1))*DiracDelta(t-1) *
+           (Heaviside(t-1/4) - Heaviside(t-2)), t, s)[0] == 4*pi*exp(-s))
+    # additional basic tests from wikipedia
+    assert (LT((t - a)**b*exp(-c*(t - a))*Heaviside(t - a), t, s) ==
+            ((c + s)**(-b - 1)*exp(-a*s)*gamma(b + 1), -c, True))
+    assert (
+        LT((exp(2*t)-1)*exp(-b-t)*Heaviside(t)/2, t, s, noconds=True,
+           simplify=True) ==
+        exp(-b)/(s**2 - 1))
+    # DiracDelta function: standard cases
+    assert LT(DiracDelta(t), t, s) == (1, -oo, True)
+    assert LT(DiracDelta(a*t), t, s) == (1/a, -oo, True)
+    assert LT(DiracDelta(t/42), t, s) == (42, -oo, True)
+    assert LT(DiracDelta(t+42), t, s) == (0, -oo, True)
+    assert (LT(DiracDelta(t)+DiracDelta(t-42), t, s) ==
+            (1 + exp(-42*s), -oo, True))
+    assert (LT(DiracDelta(t)-a*exp(-a*t), t, s, simplify=True) ==
+            (s/(a + s), -a, True))
+    assert (
+        LT(exp(-t)*(DiracDelta(t)+DiracDelta(t-42)), t, s, simplify=True) ==
+        (exp(-42*s - 42) + 1, -oo, True))
+    assert LT(f(t)*DiracDelta(t-42), t, s) == (f(42)*exp(-42*s), -oo, True)
+    assert LT(f(t)*DiracDelta(b*t-a), t, s) == (f(a/b)*exp(-a*s/b)/b,
+                                                -oo, True)
+    assert LT(f(t)*DiracDelta(b*t+a), t, s) == (0, -oo, True)
+    # SingularityFunction
+    assert LT(SingularityFunction(t, a, -1), t, s)[0] == exp(-a*s)
+    assert LT(SingularityFunction(t, a, 1), t, s)[0] == exp(-a*s)/s**2
+    assert LT(SingularityFunction(t, a, x), t, s)[0] == (
+        LaplaceTransform(SingularityFunction(t, a, x), t, s))
+    # Collection of cases that cannot be fully evaluated and/or would catch
+    # some common implementation errors
+    assert (LT(DiracDelta(t**2), t, s, noconds=True) ==
+            LaplaceTransform(DiracDelta(t**2), t, s))
+    assert LT(DiracDelta(t**2 - 1), t, s) == (exp(-s)/2, -oo, True)
+    assert LT(DiracDelta(t*(1 - t)), t, s) == (1 - exp(-s), -oo, True)
+    assert (LT((DiracDelta(t) + 1)*(DiracDelta(t - 1) + 1), t, s) ==
+            (LaplaceTransform(DiracDelta(t)*DiracDelta(t - 1), t, s) +
+             1 + exp(-s) + 1/s, 0, True))
+    assert LT(DiracDelta(2*t-2*exp(a)), t, s) == (exp(-s*exp(a))/2, -oo, True)
+    assert LT(DiracDelta(-2*t+2*exp(a)), t, s) == (exp(-s*exp(a))/2, -oo, True)
+    # Heaviside tests
+    assert LT(Heaviside(t), t, s) == (1/s, 0, True)
+    assert LT(Heaviside(t - a), t, s) == (exp(-a*s)/s, 0, True)
+    assert LT(Heaviside(t-1), t, s) == (exp(-s)/s, 0, True)
+    assert LT(Heaviside(2*t-4), t, s) == (exp(-2*s)/s, 0, True)
+    assert LT(Heaviside(2*t+4), t, s) == (1/s, 0, True)
+    assert (LT(Heaviside(-2*t+4), t, s, simplify=True) ==
+            (1/s - exp(-2*s)/s, 0, True))
+    assert (LT(g(t)*Heaviside(t - w), t, s) ==
+            (LaplaceTransform(g(t)*Heaviside(t - w), t, s), -oo, True))
+    assert (
+        LT(Heaviside(t-a)*g(t), t, s) ==
+        (LaplaceTransform(g(a + t), t, s)*exp(-a*s), -oo, True))
+    assert (
+        LT(Heaviside(t+a)*g(t), t, s) ==
+        (LaplaceTransform(g(t), t, s), -oo, True))
+    assert (
+        LT(Heaviside(-t+a)*g(t), t, s) ==
+        (LaplaceTransform(g(t), t, s) -
+         LaplaceTransform(g(a + t), t, s)*exp(-a*s), -oo, True))
+    assert (
+        LT(Heaviside(-t-a)*g(t), t, s) == (0, 0, True))
+    # Fresnel functions
+    assert (laplace_transform(fresnels(t), t, s, simplify=True) ==
+            ((-sin(s**2/(2*pi))*fresnels(s/pi) +
+              sqrt(2)*sin(s**2/(2*pi) + pi/4)/2 -
+              cos(s**2/(2*pi))*fresnelc(s/pi))/s, 0, True))
+    assert (laplace_transform(fresnelc(t), t, s, simplify=True) ==
+            ((sin(s**2/(2*pi))*fresnelc(s/pi) -
+              cos(s**2/(2*pi))*fresnels(s/pi) +
+              sqrt(2)*cos(s**2/(2*pi) + pi/4)/2)/s, 0, True))
+    # Matrix tests
+    Mt = Matrix([[exp(t), t*exp(-t)], [t*exp(-t), exp(t)]])
+    Ms = Matrix([[1/(s - 1), (s + 1)**(-2)],
+                 [(s + 1)**(-2),     1/(s - 1)]])
+    # The default behaviour for Laplace transform of a Matrix returns a Matrix
+    # of Tuples and is deprecated:
+    with warns_deprecated_sympy():
+        Ms_conds = Matrix(
+            [[(1/(s - 1), 1, True), ((s + 1)**(-2), -1, True)],
+             [((s + 1)**(-2), -1, True), (1/(s - 1), 1, True)]])
+    with warns_deprecated_sympy():
+        assert LT(Mt, t, s) == Ms_conds
+    # The new behavior is to return a tuple of a Matrix and the convergence
+    # conditions for the matrix as a whole:
+    assert LT(Mt, t, s, legacy_matrix=False) == (Ms, 1, True)
+    # With noconds=True the transformed matrix is returned without conditions
+    # either way:
+    assert LT(Mt, t, s, noconds=True) == Ms
+    assert LT(Mt, t, s, legacy_matrix=False, noconds=True) == Ms
+
+
+@slow
+def test_inverse_laplace_transform():
+    s = symbols('s')
+    k, n, t = symbols('k, n, t', real=True)
+    a, b, c, d = symbols('a, b, c, d', positive=True)
+    f = Function('f')
+    F = Function('F')
+
+    def ILT(g):
+        return inverse_laplace_transform(g, s, t)
+
+    def ILTS(g):
+        return inverse_laplace_transform(g, s, t, simplify=True)
+
+    def ILTF(g):
+        return laplace_correspondence(
+            inverse_laplace_transform(g, s, t), {f: F})
+
+    # Tests for the rules in Bateman54.
+
+    # Section 4.1: Some of the Laplace transform rules can also be used well
+    #     in the inverse transform.
+    assert ILTF(exp(-a*s)*F(s)) == f(-a + t)
+    assert ILTF(k*F(s-a)) == k*f(t)*exp(-a*t)
+    assert ILTF(diff(F(s), s, 3)) == -t**3*f(t)
+    assert ILTF(diff(F(s), s, 4)) == t**4*f(t)
+
+    # Section 5.1: Most rules are impractical for a computer algebra system.
+
+    # Section 5.2: Rational functions
+    assert ILT(2) == 2*DiracDelta(t)
+    assert ILT(1/s) == Heaviside(t)
+    assert ILT(1/s**2) == t*Heaviside(t)
+    assert ILT(1/s**5) == t**4*Heaviside(t)/24
+    assert ILT(1/s**n) == t**(n - 1)*Heaviside(t)/gamma(n)
+    assert ILT(a/(a + s)) == a*exp(-a*t)*Heaviside(t)
+    assert ILT(s/(a + s)) == -a*exp(-a*t)*Heaviside(t) + DiracDelta(t)
+    assert (ILT(b*s/(s+a)**2) ==
+            b*(-a*t*exp(-a*t)*Heaviside(t) + exp(-a*t)*Heaviside(t)))
+    assert (ILTS(c/((s+a)*(s+b))) ==
+            c*(exp(a*t) - exp(b*t))*exp(-t*(a + b))*Heaviside(t)/(a - b))
+    assert (ILTS(c*s/((s+a)*(s+b))) ==
+            c*(a*exp(b*t) - b*exp(a*t))*exp(-t*(a + b))*Heaviside(t)/(a - b))
+    assert ILTS(s/(a + s)**3) == t*(-a*t + 2)*exp(-a*t)*Heaviside(t)/2
+    assert ILTS(1/(s*(a + s)**3)) == (
+        -a**2*t**2 - 2*a*t + 2*exp(a*t) - 2)*exp(-a*t)*Heaviside(t)/(2*a**3)
+    assert ILT(1/(s*(a + s)**n)) == (
+        Heaviside(t)*lowergamma(n, a*t)/(a**n*gamma(n)))
+    assert ILT((s-a)**(-b)) == t**(b - 1)*exp(a*t)*Heaviside(t)/gamma(b)
+    assert ILT((a + s)**(-2)) == t*exp(-a*t)*Heaviside(t)
+    assert ILT((a + s)**(-5)) == t**4*exp(-a*t)*Heaviside(t)/24
+    assert ILT(s**2/(s**2 + 1)) == -sin(t)*Heaviside(t) + DiracDelta(t)
+    assert ILT(1 - 1/(s**2 + 1)) == -sin(t)*Heaviside(t) + DiracDelta(t)
+    assert ILT(a/(a**2 + s**2)) == sin(a*t)*Heaviside(t)
+    assert ILT(s/(s**2 + a**2)) == cos(a*t)*Heaviside(t)
+    assert ILT(b/(b**2 + (a + s)**2)) == exp(-a*t)*sin(b*t)*Heaviside(t)
+    assert (ILT(b*s/(b**2 + (a + s)**2)) ==
+            b*(-a*exp(-a*t)*sin(b*t)/b + exp(-a*t)*cos(b*t))*Heaviside(t))
+    assert ILT(1/(s**2*(s**2 + 1))) == t*Heaviside(t) - sin(t)*Heaviside(t)
+    assert (ILTS(c*s/(d**2*(s+a)**2+b**2)) ==
+            c*(-a*d*sin(b*t/d) + b*cos(b*t/d))*exp(-a*t)*Heaviside(t)/(b*d**2))
+    assert ILTS((b*s**2 + d)/(a**2 + s**2)**2) == (
+        2*a**2*b*sin(a*t) + (a**2*b - d)*(a*t*cos(a*t) -
+                                          sin(a*t)))*Heaviside(t)/(2*a**3)
+    assert ILTS(b/(s**2-a**2)) == b*sinh(a*t)*Heaviside(t)/a
+    assert (ILT(b/(s**2-a**2)) ==
+            b*(exp(a*t)*Heaviside(t)/(2*a) - exp(-a*t)*Heaviside(t)/(2*a)))
+    assert ILTS(b*s/(s**2-a**2)) == b*cosh(a*t)*Heaviside(t)
+    assert (ILT(b/(s*(s+a))) ==
+            b*(Heaviside(t)/a - exp(-a*t)*Heaviside(t)/a))
+    # Issue #24424
+    assert (ILTS((s + 8)/((s + 2)*(s**2 + 2*s + 10))) ==
+            ((8*sin(3*t) - 9*cos(3*t))*exp(t) + 9)*exp(-2*t)*Heaviside(t)/15)
+    # Issue #8514; this is not important anymore, since this function
+    # is not solved by integration anymore
+    assert (ILT(1/(a*s**2+b*s+c)) ==
+            2*exp(-b*t/(2*a))*sin(t*sqrt(4*a*c - b**2)/(2*a)) *
+            Heaviside(t)/sqrt(4*a*c - b**2))
+
+    # Section 5.3: Irrational algebraic functions
+    assert (  # (1)
+        ILT(1/sqrt(s)/(b*s-a)) ==
+        exp(a*t/b)*Heaviside(t)*erf(sqrt(a)*sqrt(t)/sqrt(b))/(sqrt(a)*sqrt(b)))
+    assert (  # (2)
+        ILT(1/sqrt(k*s)/(c*s-a)/s) ==
+        (-2*c*sqrt(t)/(sqrt(pi)*a) +
+         c**(S(3)/2)*exp(a*t/c)*erf(sqrt(a)*sqrt(t)/sqrt(c))/a**(S(3)/2)) *
+        Heaviside(t)/(c*sqrt(k)))
+    assert (  # (4)
+        ILT(1/(sqrt(c*s)+a)) == (-a*exp(a**2*t/c)*erfc(a*sqrt(t)/sqrt(c))/c +
+                                 1/(sqrt(pi)*sqrt(c)*sqrt(t)))*Heaviside(t))
+    assert (  # (5)
+        ILT(a/s/(b*sqrt(s)+a)) ==
+        (-exp(a**2*t/b**2)*erfc(a*sqrt(t)/b) + 1)*Heaviside(t))
+    assert (  # (6)
+            ILT((a-b)*sqrt(s)/(sqrt(s)+sqrt(a))/(s-b)) ==
+            (sqrt(a)*sqrt(b)*exp(b*t)*erfc(sqrt(b)*sqrt(t)) +
+             a*exp(a*t)*erfc(sqrt(a)*sqrt(t)) - b*exp(b*t))*Heaviside(t))
+    assert (  # (7)
+            ILT(1/sqrt(s)/(sqrt(b*s)+a)) ==
+            exp(a**2*t/b)*Heaviside(t)*erfc(a*sqrt(t)/sqrt(b))/sqrt(b))
+    assert (  # (8)
+            ILT(a**2/(sqrt(s)+a)/s**(S(3)/2)) ==
+            (2*a*sqrt(t)/sqrt(pi) + exp(a**2*t)*erfc(a*sqrt(t)) - 1) *
+            Heaviside(t))
+    assert (  # (9)
+            ILT((a-b)*sqrt(b)/(s-b)/sqrt(s)/(sqrt(s)+sqrt(a))) ==
+            (sqrt(a)*exp(b*t)*erf(sqrt(b)*sqrt(t)) +
+             sqrt(b)*exp(a*t)*erfc(sqrt(a)*sqrt(t)) -
+             sqrt(b)*exp(b*t))*Heaviside(t))
+    assert (  # (10)
+            ILT(1/(sqrt(s)+sqrt(a))**2) ==
+            (-2*sqrt(a)*sqrt(t)/sqrt(pi) +
+             (-2*a*t + 1)*(erf(sqrt(a)*sqrt(t)) -
+                           1)*exp(a*t) + 1)*Heaviside(t))
+    assert (  # (11)
+            ILT(1/(sqrt(s)+sqrt(a))**2/s) ==
+            ((2*t - 1/a)*exp(a*t)*erfc(sqrt(a)*sqrt(t)) + 1/a -
+             2*sqrt(t)/(sqrt(pi)*sqrt(a)))*Heaviside(t))
+    assert (  # (12)
+            ILT(1/(sqrt(s)+a)**2/sqrt(s)) ==
+            (-2*a*t*exp(a**2*t)*erfc(a*sqrt(t)) +
+             2*sqrt(t)/sqrt(pi))*Heaviside(t))
+    assert (  # (13)
+            ILT(1/(sqrt(s)+a)**3) ==
+            (-a*t*(2*a**2*t + 3)*exp(a**2*t)*erfc(a*sqrt(t)) +
+             2*sqrt(t)*(a**2*t + 1)/sqrt(pi))*Heaviside(t))
+    x = (
+        - ILT(sqrt(s)/(sqrt(s)+a)**3) +
+        2*(sqrt(pi)*a**2*t*(-2*sqrt(pi)*erfc(a*sqrt(t)) +
+                            2*exp(-a**2*t)/(a*sqrt(t))) *
+           (-a**4*t**2 - 5*a**2*t/2 - S.Half) * exp(a**2*t)/2 +
+           sqrt(pi)*a*sqrt(t)*(a**2*t + 1)/2) *
+        Heaviside(t)/(pi*a**2*t)).simplify()
+    assert (  # (14)
+            x == 0)
+    x = (
+        - ILT(1/sqrt(s)/(sqrt(s)+a)**3) +
+        Heaviside(t)*(sqrt(t)*((2*a**2*t + 1) *
+                               (sqrt(pi)*a*sqrt(t)*exp(a**2*t) *
+                                erfc(a*sqrt(t)) - 1) + 1) /
+                      (sqrt(pi)*a))).simplify()
+    assert (  # (15)
+            x == 0)
+    assert (  # (16)
+            factor_terms(ILT(3/(sqrt(s)+a)**4)) ==
+            3*(-2*a**3*t**(S(5)/2)*(2*a**2*t + 5)/(3*sqrt(pi)) +
+               t*(4*a**4*t**2 + 12*a**2*t + 3)*exp(a**2*t) *
+               erfc(a*sqrt(t))/3)*Heaviside(t))
+    assert (  # (17)
+            ILT((sqrt(s)-a)/(s*(sqrt(s)+a))) ==
+            (2*exp(a**2*t)*erfc(a*sqrt(t))-1)*Heaviside(t))
+    assert (  # (18)
+            ILT((sqrt(s)-a)**2/(s*(sqrt(s)+a)**2)) == (
+                1 + 8*a**2*t*exp(a**2*t)*erfc(a*sqrt(t)) -
+                8/sqrt(pi)*a*sqrt(t))*Heaviside(t))
+    assert (  # (19)
+            ILT((sqrt(s)-a)**3/(s*(sqrt(s)+a)**3)) == Heaviside(t)*(
+                2*(8*a**4*t**2+8*a**2*t+1)*exp(a**2*t) *
+                erfc(a*sqrt(t))-8/sqrt(pi)*a*sqrt(t)*(2*a**2*t+1)-1))
+    assert (  # (22)
+            ILT(sqrt(s+a)/(s+b)) == Heaviside(t)*(
+                exp(-a*t)/sqrt(t)/sqrt(pi) +
+                sqrt(a-b)*exp(-b*t)*erf(sqrt(a-b)*sqrt(t))))
+    assert (  # (23)
+            ILT(1/sqrt(s+b)/(s+a)) == Heaviside(t)*(
+                1/sqrt(b-a)*exp(-a*t)*erf(sqrt(b-a)*sqrt(t))))
+    assert (  # (35)
+            ILT(1/sqrt(s**2+a**2)) == Heaviside(t)*(
+                besselj(0, a*t)))
+    assert (  # (44)
+            ILT(1/sqrt(s**2-a**2)) == Heaviside(t)*(
+                besseli(0, a*t)))
+
+    # Miscellaneous tests
+    # Can _inverse_laplace_time_shift deal with positive exponents?
+    assert (
+        - ILT((s**2*exp(2*s) + 4*exp(s) - 4)*exp(-2*s)/(s*(s**2 + 1))) +
+        cos(t)*Heaviside(t) + 4*cos(t - 2)*Heaviside(t - 2) -
+        4*cos(t - 1)*Heaviside(t - 1) - 4*Heaviside(t - 2) +
+        4*Heaviside(t - 1)).simplify() == 0
+
+
+@slow
+def test_inverse_laplace_transform_old():
+    from sympy.functions.special.delta_functions import DiracDelta
+    ILT = inverse_laplace_transform
+    a, b, c, d = symbols('a b c d', positive=True)
+    n, r = symbols('n, r', real=True)
+    t, z = symbols('t z')
+    f = Function('f')
+    F = Function('F')
+
+    def simp_hyp(expr):
+        return factor_terms(expand_mul(expr)).rewrite(sin)
+
+    L = ILT(F(s), s, t)
+    assert laplace_correspondence(L, {f: F}) == f(t)
+    assert ILT(exp(-a*s)/s, s, t) == Heaviside(-a + t)
+    assert ILT(exp(-a*s)/(b + s), s, t) == exp(-b*(-a + t))*Heaviside(-a + t)
+    assert (ILT((b + s)/(a**2 + (b + s)**2), s, t) ==
+            exp(-b*t)*cos(a*t)*Heaviside(t))
+    assert (ILT(exp(-a*s)/s**b, s, t) ==
+            (-a + t)**(b - 1)*Heaviside(-a + t)/gamma(b))
+    assert (ILT(exp(-a*s)/sqrt(s**2 + 1), s, t) ==
+            Heaviside(-a + t)*besselj(0, a - t))
+    assert ILT(1/(s*sqrt(s + 1)), s, t) == Heaviside(t)*erf(sqrt(t))
+    # TODO sinh/cosh shifted come out a mess. also delayed trig is a mess
+    # TODO should this simplify further?
+    assert (ILT(exp(-a*s)/s**b, s, t) ==
+            (t - a)**(b - 1)*Heaviside(t - a)/gamma(b))
+    assert (ILT(exp(-a*s)/sqrt(1 + s**2), s, t) ==
+            Heaviside(t - a)*besselj(0, a - t))  # note: besselj(0, x) is even
+    # XXX ILT turns these branch factor into trig functions ...
+    assert (
+        simplify(ILT(a**b*(s + sqrt(s**2 - a**2))**(-b)/sqrt(s**2 - a**2),
+                     s, t).rewrite(exp)) ==
+        Heaviside(t)*besseli(b, a*t))
+    assert (
+        ILT(a**b*(s + sqrt(s**2 + a**2))**(-b)/sqrt(s**2 + a**2),
+            s, t, simplify=True).rewrite(exp) ==
+        Heaviside(t)*besselj(b, a*t))
+    assert ILT(1/(s*sqrt(s + 1)), s, t) == Heaviside(t)*erf(sqrt(t))
+    # TODO can we make erf(t) work?
+    assert (ILT((s * eye(2) - Matrix([[1, 0], [0, 2]])).inv(), s, t) ==
+            Matrix([[exp(t)*Heaviside(t), 0], [0, exp(2*t)*Heaviside(t)]]))
+    # Test time_diff rule
+    assert (ILT(s**42*f(s), s, t) ==
+            Derivative(InverseLaplaceTransform(f(s), s, t, None), (t, 42)))
+    assert ILT(cos(s), s, t) == InverseLaplaceTransform(cos(s), s, t, None)
+    # Rules for testing different DiracDelta cases
+    assert (ILT(2*exp(3*s) - 5*exp(-7*s), s, t) ==
+            2*InverseLaplaceTransform(exp(3*s), s, t, None) -
+            5*DiracDelta(t - 7))
+    a = cos(sin(7)/2)
+    assert ILT(a*exp(-3*s), s, t) == a*DiracDelta(t - 3)
+    assert ILT(exp(2*s), s, t) == InverseLaplaceTransform(exp(2*s), s, t, None)
+    r = Symbol('r', real=True)
+    assert ILT(exp(r*s), s, t) == InverseLaplaceTransform(exp(r*s), s, t, None)
+    # Rules for testing whether Heaviside(t) is treated properly in diff rule
+    assert ILT(s**2/(a**2 + s**2), s, t) == (
+        -a*sin(a*t)*Heaviside(t) + DiracDelta(t))
+    assert ILT(s**2*(f(s) + 1/(a**2 + s**2)), s, t) == (
+        -a*sin(a*t)*Heaviside(t) + DiracDelta(t) +
+        Derivative(InverseLaplaceTransform(f(s), s, t, None), (t, 2)))
+    # Rules from the previous test_inverse_laplace_transform_delta_cond():
+    assert (ILT(exp(r*s), s, t, noconds=False) ==
+            (InverseLaplaceTransform(exp(r*s), s, t, None), True))
+    # inversion does not exist: verify it doesn't evaluate to DiracDelta
+    for z in (Symbol('z', extended_real=False),
+              Symbol('z', imaginary=True, zero=False)):
+        f = ILT(exp(z*s), s, t, noconds=False)
+        f = f[0] if isinstance(f, tuple) else f
+        assert f.func != DiracDelta
+
+
+@slow
+def test_expint():
+    x = Symbol('x')
+    a = Symbol('a')
+    u = Symbol('u', polar=True)
+
+    # TODO LT of Si, Shi, Chi is a mess ...
+    assert laplace_transform(Ci(x), x, s) == (-log(1 + s**2)/2/s, 0, True)
+    assert (laplace_transform(expint(a, x), x, s, simplify=True) ==
+            (lerchphi(s*exp_polar(I*pi), 1, a), 0, re(a) > S.Zero))
+    assert (laplace_transform(expint(1, x), x, s, simplify=True) ==
+            (log(s + 1)/s, 0, True))
+    assert (laplace_transform(expint(2, x), x, s, simplify=True) ==
+            ((s - log(s + 1))/s**2, 0, True))
+    assert (inverse_laplace_transform(-log(1 + s**2)/2/s, s, u).expand() ==
+            Heaviside(u)*Ci(u))
+    assert (
+        inverse_laplace_transform(log(s + 1)/s, s, x,
+                                  simplify=True).rewrite(expint) ==
+        Heaviside(x)*E1(x))
+    assert (
+        inverse_laplace_transform(
+            (s - log(s + 1))/s**2, s, x,
+            simplify=True).rewrite(expint).expand() ==
+        (expint(2, x)*Heaviside(x)).rewrite(Ei).rewrite(expint).expand())

rl/Lib/site-packages/sympy/integrals/tests/test_lineintegrals.py

@@ -0,0 +1,13 @@
+from sympy.core.numbers import E
+from sympy.core.symbol import symbols
+from sympy.functions.elementary.exponential import log
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.geometry.curve import Curve
+from sympy.integrals.integrals import line_integrate
+
+s, t, x, y, z = symbols('s,t,x,y,z')
+
+
+def test_lineintegral():
+    c = Curve([E**t + 1, E**t - 1], (t, 0, log(2)))
+    assert line_integrate(x + y, c, [x, y]) == 3*sqrt(2)

rl/Lib/site-packages/sympy/integrals/tests/test_manual.py

@@ -0,0 +1,714 @@
+from sympy.core.expr import Expr
+from sympy.core.mul import Mul
+from sympy.core.function import (Derivative, Function, diff, expand)
+from sympy.core.numbers import (I, Rational, pi)
+from sympy.core.relational import Ne
+from sympy.core.singleton import S
+from sympy.core.symbol import (Dummy, Symbol, symbols)
+from sympy.functions.elementary.exponential import (exp, log)
+from sympy.functions.elementary.hyperbolic import (asinh, csch, cosh, coth, sech, sinh, tanh)
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.piecewise import Piecewise, piecewise_fold
+from sympy.functions.elementary.trigonometric import (acos, acot, acsc, asec, asin, atan, cos, cot, csc, sec, sin, tan)
+from sympy.functions.special.delta_functions import Heaviside, DiracDelta
+from sympy.functions.special.elliptic_integrals import (elliptic_e, elliptic_f)
+from sympy.functions.special.error_functions import (Chi, Ci, Ei, Shi, Si, erf, erfi, fresnelc, fresnels, li)
+from sympy.functions.special.gamma_functions import uppergamma
+from sympy.functions.special.polynomials import (assoc_laguerre, chebyshevt, chebyshevu, gegenbauer, hermite, jacobi, laguerre, legendre)
+from sympy.functions.special.zeta_functions import polylog
+from sympy.integrals.integrals import (Integral, integrate)
+from sympy.logic.boolalg import And
+from sympy.integrals.manualintegrate import (manualintegrate, find_substitutions,
+    _parts_rule, integral_steps, manual_subs)
+from sympy.testing.pytest import raises, slow
+
+x, y, z, u, n, a, b, c, d, e = symbols('x y z u n a b c d e')
+f = Function('f')
+
+
+def assert_is_integral_of(f: Expr, F: Expr):
+    assert manualintegrate(f, x) == F
+    assert F.diff(x).equals(f)
+
+
+def test_find_substitutions():
+    assert find_substitutions((cot(x)**2 + 1)**2*csc(x)**2*cot(x)**2, x, u) == \
+        [(cot(x), 1, -u**6 - 2*u**4 - u**2)]
+    assert find_substitutions((sec(x)**2 + tan(x) * sec(x)) / (sec(x) + tan(x)),
+                              x, u) == [(sec(x) + tan(x), 1, 1/u)]
+    assert (-x**2, Rational(-1, 2), exp(u)) in find_substitutions(x * exp(-x**2), x, u)
+    assert not find_substitutions(Derivative(f(x), x)**2, x, u)
+
+
+def test_manualintegrate_polynomials():
+    assert manualintegrate(y, x) == x*y
+    assert manualintegrate(exp(2), x) == x * exp(2)
+    assert manualintegrate(x**2, x) == x**3 / 3
+    assert manualintegrate(3 * x**2 + 4 * x**3, x) == x**3 + x**4
+
+    assert manualintegrate((x + 2)**3, x) == (x + 2)**4 / 4
+    assert manualintegrate((3*x + 4)**2, x) == (3*x + 4)**3 / 9
+
+    assert manualintegrate((u + 2)**3, u) == (u + 2)**4 / 4
+    assert manualintegrate((3*u + 4)**2, u) == (3*u + 4)**3 / 9
+
+
+def test_manualintegrate_exponentials():
+    assert manualintegrate(exp(2*x), x) == exp(2*x) / 2
+    assert manualintegrate(2**x, x) == (2 ** x) / log(2)
+    assert_is_integral_of(1/sqrt(1-exp(2*x)),
+                          log(sqrt(1 - exp(2*x)) - 1)/2 - log(sqrt(1 - exp(2*x)) + 1)/2)
+
+    assert manualintegrate(1 / x, x) == log(x)
+    assert manualintegrate(1 / (2*x + 3), x) == log(2*x + 3) / 2
+    assert manualintegrate(log(x)**2 / x, x) == log(x)**3 / 3
+
+    assert_is_integral_of(x**x*(log(x)+1), x**x)
+
+
+def test_manualintegrate_parts():
+    assert manualintegrate(exp(x) * sin(x), x) == \
+        (exp(x) * sin(x)) / 2 - (exp(x) * cos(x)) / 2
+    assert manualintegrate(2*x*cos(x), x) == 2*x*sin(x) + 2*cos(x)
+    assert manualintegrate(x * log(x), x) == x**2*log(x)/2 - x**2/4
+    assert manualintegrate(log(x), x) == x * log(x) - x
+    assert manualintegrate((3*x**2 + 5) * exp(x), x) == \
+        3*x**2*exp(x) - 6*x*exp(x) + 11*exp(x)
+    assert manualintegrate(atan(x), x) == x*atan(x) - log(x**2 + 1)/2
+
+    # Make sure _parts_rule doesn't pick u = constant but can pick dv =
+    # constant if necessary, e.g. for integrate(atan(x))
+    assert _parts_rule(cos(x), x) == None
+    assert _parts_rule(exp(x), x) == None
+    assert _parts_rule(x**2, x) == None
+    result = _parts_rule(atan(x), x)
+    assert result[0] == atan(x) and result[1] == 1
+
+
+def test_manualintegrate_trigonometry():
+    assert manualintegrate(sin(x), x) == -cos(x)
+    assert manualintegrate(tan(x), x) == -log(cos(x))
+
+    assert manualintegrate(sec(x), x) == log(sec(x) + tan(x))
+    assert manualintegrate(csc(x), x) == -log(csc(x) + cot(x))
+
+    assert manualintegrate(sin(x) * cos(x), x) in [sin(x) ** 2 / 2, -cos(x)**2 / 2]
+    assert manualintegrate(-sec(x) * tan(x), x) == -sec(x)
+    assert manualintegrate(csc(x) * cot(x), x) == -csc(x)
+    assert manualintegrate(sec(x)**2, x) == tan(x)
+    assert manualintegrate(csc(x)**2, x) == -cot(x)
+
+    assert manualintegrate(x * sec(x**2), x) == log(tan(x**2) + sec(x**2))/2
+    assert manualintegrate(cos(x)*csc(sin(x)), x) == -log(cot(sin(x)) + csc(sin(x)))
+    assert manualintegrate(cos(3*x)*sec(x), x) == -x + sin(2*x)
+    assert manualintegrate(sin(3*x)*sec(x), x) == \
+        -3*log(cos(x)) + 2*log(cos(x)**2) - 2*cos(x)**2
+
+    assert_is_integral_of(sinh(2*x), cosh(2*x)/2)
+    assert_is_integral_of(x*cosh(x**2), sinh(x**2)/2)
+    assert_is_integral_of(tanh(x), log(cosh(x)))
+    assert_is_integral_of(coth(x), log(sinh(x)))
+    f, F = sech(x), 2*atan(tanh(x/2))
+    assert manualintegrate(f, x) == F
+    assert (F.diff(x) - f).rewrite(exp).simplify() == 0  # todo: equals returns None
+    f, F = csch(x), log(tanh(x/2))
+    assert manualintegrate(f, x) == F
+    assert (F.diff(x) - f).rewrite(exp).simplify() == 0
+
+
+@slow
+def test_manualintegrate_trigpowers():
+    assert manualintegrate(sin(x)**2 * cos(x), x) == sin(x)**3 / 3
+    assert manualintegrate(sin(x)**2 * cos(x) **2, x) == \
+        x / 8 - sin(4*x) / 32
+    assert manualintegrate(sin(x) * cos(x)**3, x) == -cos(x)**4 / 4
+    assert manualintegrate(sin(x)**3 * cos(x)**2, x) == \
+        cos(x)**5 / 5 - cos(x)**3 / 3
+
+    assert manualintegrate(tan(x)**3 * sec(x), x) == sec(x)**3/3 - sec(x)
+    assert manualintegrate(tan(x) * sec(x) **2, x) == sec(x)**2/2
+
+    assert manualintegrate(cot(x)**5 * csc(x), x) == \
+        -csc(x)**5/5 + 2*csc(x)**3/3 - csc(x)
+    assert manualintegrate(cot(x)**2 * csc(x)**6, x) == \
+        -cot(x)**7/7 - 2*cot(x)**5/5 - cot(x)**3/3
+
+
+@slow
+def test_manualintegrate_inversetrig():
+    # atan
+    assert manualintegrate(exp(x) / (1 + exp(2*x)), x) == atan(exp(x))
+    assert manualintegrate(1 / (4 + 9 * x**2), x) == atan(3 * x/2) / 6
+    assert manualintegrate(1 / (16 + 16 * x**2), x) == atan(x) / 16
+    assert manualintegrate(1 / (4 + x**2), x) == atan(x / 2) / 2
+    assert manualintegrate(1 / (1 + 4 * x**2), x) == atan(2*x) / 2
+    ra = Symbol('a', real=True)
+    rb = Symbol('b', real=True)
+    assert manualintegrate(1/(ra + rb*x**2), x) == \
+        Piecewise((atan(x/sqrt(ra/rb))/(rb*sqrt(ra/rb)), ra/rb > 0),
+                  ((log(x - sqrt(-ra/rb)) - log(x + sqrt(-ra/rb)))/(2*sqrt(rb)*sqrt(-ra)), True))
+    assert manualintegrate(1/(4 + rb*x**2), x) == \
+        Piecewise((atan(x/(2*sqrt(1/rb)))/(2*rb*sqrt(1/rb)), 1/rb > 0),
+                  (-I*(log(x - 2*sqrt(-1/rb)) - log(x + 2*sqrt(-1/rb)))/(4*sqrt(rb)), True))
+    assert manualintegrate(1/(ra + 4*x**2), x) == \
+        Piecewise((atan(2*x/sqrt(ra))/(2*sqrt(ra)), ra > 0),
+                  ((log(x - sqrt(-ra)/2) - log(x + sqrt(-ra)/2))/(4*sqrt(-ra)), True))
+    assert manualintegrate(1/(4 + 4*x**2), x) == atan(x) / 4
+
+    assert manualintegrate(1/(a + b*x**2), x) == Piecewise((atan(x/sqrt(a/b))/(b*sqrt(a/b)), Ne(a, 0)),
+                                                           (-1/(b*x), True))
+
+    # asin
+    assert manualintegrate(1/sqrt(1-x**2), x) == asin(x)
+    assert manualintegrate(1/sqrt(4-4*x**2), x) == asin(x)/2
+    assert manualintegrate(3/sqrt(1-9*x**2), x) == asin(3*x)
+    assert manualintegrate(1/sqrt(4-9*x**2), x) == asin(x*Rational(3, 2))/3
+
+    # asinh
+    assert manualintegrate(1/sqrt(x**2 + 1), x) == \
+        asinh(x)
+    assert manualintegrate(1/sqrt(x**2 + 4), x) == \
+        asinh(x/2)
+    assert manualintegrate(1/sqrt(4*x**2 + 4), x) == \
+        asinh(x)/2
+    assert manualintegrate(1/sqrt(4*x**2 + 1), x) == \
+        asinh(2*x)/2
+    assert manualintegrate(1/sqrt(ra*x**2 + 1), x) == \
+        Piecewise((asin(x*sqrt(-ra))/sqrt(-ra), ra < 0), (asinh(sqrt(ra)*x)/sqrt(ra), ra > 0), (x, True))
+    assert manualintegrate(1/sqrt(ra + x**2), x) == \
+        Piecewise((asinh(x*sqrt(1/ra)), ra > 0), (log(2*x + 2*sqrt(ra + x**2)), True))
+
+    # log
+    assert manualintegrate(1/sqrt(x**2 - 1), x) == log(2*x + 2*sqrt(x**2 - 1))
+    assert manualintegrate(1/sqrt(x**2 - 4), x) == log(2*x + 2*sqrt(x**2 - 4))
+    assert manualintegrate(1/sqrt(4*x**2 - 4), x) == log(8*x + 4*sqrt(4*x**2 - 4))/2
+    assert manualintegrate(1/sqrt(9*x**2 - 1), x) == log(18*x + 6*sqrt(9*x**2 - 1))/3
+    assert manualintegrate(1/sqrt(ra*x**2 - 4), x) == \
+           Piecewise((log(2*sqrt(ra)*sqrt(ra*x**2 - 4) + 2*ra*x)/sqrt(ra), Ne(ra, 0)), (-I*x/2, True))
+    assert manualintegrate(1/sqrt(-ra + 4*x**2), x) == \
+        Piecewise((asinh(2*x*sqrt(-1/ra))/2, ra < 0), (log(8*x + 4*sqrt(-ra + 4*x**2))/2, True))
+
+    # From https://www.wikiwand.com/en/List_of_integrals_of_inverse_trigonometric_functions
+    # asin
+    assert manualintegrate(asin(x), x) == x*asin(x) + sqrt(1 - x**2)
+    assert manualintegrate(asin(a*x), x) == Piecewise(((a*x*asin(a*x) + sqrt(-a**2*x**2 + 1))/a, Ne(a, 0)), (0, True))
+    assert manualintegrate(x*asin(a*x), x) == \
+           -a*Piecewise((-x*sqrt(-a**2*x**2 + 1)/(2*a**2) +
+                         log(-2*a**2*x + 2*sqrt(-a**2)*sqrt(-a**2*x**2 + 1))/(2*a**2*sqrt(-a**2)), Ne(a**2, 0)),
+                        (x**3/3, True))/2 + x**2*asin(a*x)/2
+    # acos
+    assert manualintegrate(acos(x), x) == x*acos(x) - sqrt(1 - x**2)
+    assert manualintegrate(acos(a*x), x) == Piecewise(((a*x*acos(a*x) - sqrt(-a**2*x**2 + 1))/a, Ne(a, 0)), (pi*x/2, True))
+    assert manualintegrate(x*acos(a*x), x) == \
+           a*Piecewise((-x*sqrt(-a**2*x**2 + 1)/(2*a**2) +
+                        log(-2*a**2*x + 2*sqrt(-a**2)*sqrt(-a**2*x**2 + 1))/(2*a**2*sqrt(-a**2)), Ne(a**2, 0)),
+                       (x**3/3, True))/2 + x**2*acos(a*x)/2
+    # atan
+    assert manualintegrate(atan(x), x) == x*atan(x) - log(x**2 + 1)/2
+    assert manualintegrate(atan(a*x), x) == Piecewise(((a*x*atan(a*x) - log(a**2*x**2 + 1)/2)/a, Ne(a, 0)), (0, True))
+    assert manualintegrate(x*atan(a*x), x) == -a*(x/a**2 - atan(x/sqrt(a**(-2)))/(a**4*sqrt(a**(-2))))/2 + x**2*atan(a*x)/2
+    # acsc
+    assert manualintegrate(acsc(x), x) == x*acsc(x) + Integral(1/(x*sqrt(1 - 1/x**2)), x)
+    assert manualintegrate(acsc(a*x), x) == x*acsc(a*x) + Integral(1/(x*sqrt(1 - 1/(a**2*x**2))), x)/a
+    assert manualintegrate(x*acsc(a*x), x) == x**2*acsc(a*x)/2 + Integral(1/sqrt(1 - 1/(a**2*x**2)), x)/(2*a)
+    # asec
+    assert manualintegrate(asec(x), x) == x*asec(x) - Integral(1/(x*sqrt(1 - 1/x**2)), x)
+    assert manualintegrate(asec(a*x), x) == x*asec(a*x) - Integral(1/(x*sqrt(1 - 1/(a**2*x**2))), x)/a
+    assert manualintegrate(x*asec(a*x), x) == x**2*asec(a*x)/2 - Integral(1/sqrt(1 - 1/(a**2*x**2)), x)/(2*a)
+    # acot
+    assert manualintegrate(acot(x), x) == x*acot(x) + log(x**2 + 1)/2
+    assert manualintegrate(acot(a*x), x) == Piecewise(((a*x*acot(a*x) + log(a**2*x**2 + 1)/2)/a, Ne(a, 0)), (pi*x/2, True))
+    assert manualintegrate(x*acot(a*x), x) == a*(x/a**2 - atan(x/sqrt(a**(-2)))/(a**4*sqrt(a**(-2))))/2 + x**2*acot(a*x)/2
+
+    # piecewise
+    assert manualintegrate(1/sqrt(ra-rb*x**2), x) == \
+        Piecewise((asin(x*sqrt(rb/ra))/sqrt(rb), And(-rb < 0, ra > 0)),
+                  (asinh(x*sqrt(-rb/ra))/sqrt(-rb), And(-rb > 0, ra > 0)),
+                  (log(-2*rb*x + 2*sqrt(-rb)*sqrt(ra - rb*x**2))/sqrt(-rb), Ne(rb, 0)),
+                  (x/sqrt(ra), True))
+    assert manualintegrate(1/sqrt(ra + rb*x**2), x) == \
+        Piecewise((asin(x*sqrt(-rb/ra))/sqrt(-rb), And(ra > 0, rb < 0)),
+                  (asinh(x*sqrt(rb/ra))/sqrt(rb), And(ra > 0, rb > 0)),
+                  (log(2*sqrt(rb)*sqrt(ra + rb*x**2) + 2*rb*x)/sqrt(rb), Ne(rb, 0)),
+                  (x/sqrt(ra), True))
+
+
+def test_manualintegrate_trig_substitution():
+    assert manualintegrate(sqrt(16*x**2 - 9)/x, x) == \
+        Piecewise((sqrt(16*x**2 - 9) - 3*acos(3/(4*x)),
+                   And(x < Rational(3, 4), x > Rational(-3, 4))))
+    assert manualintegrate(1/(x**4 * sqrt(25-x**2)), x) == \
+        Piecewise((-sqrt(-x**2/25 + 1)/(125*x) -
+                   (-x**2/25 + 1)**(3*S.Half)/(15*x**3), And(x < 5, x > -5)))
+    assert manualintegrate(x**7/(49*x**2 + 1)**(3 * S.Half), x) == \
+        ((49*x**2 + 1)**(5*S.Half)/28824005 -
+         (49*x**2 + 1)**(3*S.Half)/5764801 +
+         3*sqrt(49*x**2 + 1)/5764801 + 1/(5764801*sqrt(49*x**2 + 1)))
+
+def test_manualintegrate_trivial_substitution():
+    assert manualintegrate((exp(x) - exp(-x))/x, x) == -Ei(-x) + Ei(x)
+    f = Function('f')
+    assert manualintegrate((f(x) - f(-x))/x, x) == \
+        -Integral(f(-x)/x, x) + Integral(f(x)/x, x)
+
+
+def test_manualintegrate_rational():
+    assert manualintegrate(1/(4 - x**2), x) == -log(x - 2)/4 + log(x + 2)/4
+    assert manualintegrate(1/(-1 + x**2), x) == log(x - 1)/2 - log(x + 1)/2
+
+
+def test_manualintegrate_special():
+    f, F = 4*exp(-x**2/3), 2*sqrt(3)*sqrt(pi)*erf(sqrt(3)*x/3)
+    assert_is_integral_of(f, F)
+    f, F = 3*exp(4*x**2), 3*sqrt(pi)*erfi(2*x)/4
+    assert_is_integral_of(f, F)
+    f, F = x**Rational(1, 3)*exp(-x/8), -16*uppergamma(Rational(4, 3), x/8)
+    assert_is_integral_of(f, F)
+    f, F = exp(2*x)/x, Ei(2*x)
+    assert_is_integral_of(f, F)
+    f, F = exp(1 + 2*x - x**2), sqrt(pi)*exp(2)*erf(x - 1)/2
+    assert_is_integral_of(f, F)
+    f = sin(x**2 + 4*x + 1)
+    F = (sqrt(2)*sqrt(pi)*(-sin(3)*fresnelc(sqrt(2)*(2*x + 4)/(2*sqrt(pi))) +
+        cos(3)*fresnels(sqrt(2)*(2*x + 4)/(2*sqrt(pi))))/2)
+    assert_is_integral_of(f, F)
+    f, F = cos(4*x**2), sqrt(2)*sqrt(pi)*fresnelc(2*sqrt(2)*x/sqrt(pi))/4
+    assert_is_integral_of(f, F)
+    f, F = sin(3*x + 2)/x, sin(2)*Ci(3*x) + cos(2)*Si(3*x)
+    assert_is_integral_of(f, F)
+    f, F = sinh(3*x - 2)/x, -sinh(2)*Chi(3*x) + cosh(2)*Shi(3*x)
+    assert_is_integral_of(f, F)
+    f, F = 5*cos(2*x - 3)/x, 5*cos(3)*Ci(2*x) + 5*sin(3)*Si(2*x)
+    assert_is_integral_of(f, F)
+    f, F = cosh(x/2)/x, Chi(x/2)
+    assert_is_integral_of(f, F)
+    f, F = cos(x**2)/x, Ci(x**2)/2
+    assert_is_integral_of(f, F)
+    f, F = 1/log(2*x + 1), li(2*x + 1)/2
+    assert_is_integral_of(f, F)
+    f, F = polylog(2, 5*x)/x, polylog(3, 5*x)
+    assert_is_integral_of(f, F)
+    f, F = 5/sqrt(3 - 2*sin(x)**2), 5*sqrt(3)*elliptic_f(x, Rational(2, 3))/3
+    assert_is_integral_of(f, F)
+    f, F = sqrt(4 + 9*sin(x)**2), 2*elliptic_e(x, Rational(-9, 4))
+    assert_is_integral_of(f, F)
+
+
+def test_manualintegrate_derivative():
+    assert manualintegrate(pi * Derivative(x**2 + 2*x + 3), x) == \
+        pi * (x**2 + 2*x + 3)
+    assert manualintegrate(Derivative(x**2 + 2*x + 3, y), x) == \
+        Integral(Derivative(x**2 + 2*x + 3, y))
+    assert manualintegrate(Derivative(sin(x), x, x, x, y), x) == \
+        Derivative(sin(x), x, x, y)
+
+
+def test_manualintegrate_Heaviside():
+    assert_is_integral_of(DiracDelta(3*x+2), Heaviside(3*x+2)/3)
+    assert_is_integral_of(DiracDelta(3*x, 0), Heaviside(3*x)/3)
+    assert manualintegrate(DiracDelta(a+b*x, 1), x) == \
+        Piecewise((DiracDelta(a + b*x)/b, Ne(b, 0)), (x*DiracDelta(a, 1), True))
+    assert_is_integral_of(DiracDelta(x/3-1, 2), 3*DiracDelta(x/3-1, 1))
+    assert manualintegrate(Heaviside(x), x) == x*Heaviside(x)
+    assert manualintegrate(x*Heaviside(2), x) == x**2/2
+    assert manualintegrate(x*Heaviside(-2), x) == 0
+    assert manualintegrate(x*Heaviside( x), x) == x**2*Heaviside( x)/2
+    assert manualintegrate(x*Heaviside(-x), x) == x**2*Heaviside(-x)/2
+    assert manualintegrate(Heaviside(2*x + 4), x) == (x+2)*Heaviside(2*x + 4)
+    assert manualintegrate(x*Heaviside(x), x) == x**2*Heaviside(x)/2
+    assert manualintegrate(Heaviside(x + 1)*Heaviside(1 - x)*x**2, x) == \
+        ((x**3/3 + Rational(1, 3))*Heaviside(x + 1) - Rational(2, 3))*Heaviside(-x + 1)
+
+    y = Symbol('y')
+    assert manualintegrate(sin(7 + x)*Heaviside(3*x - 7), x) == \
+            (- cos(x + 7) + cos(Rational(28, 3)))*Heaviside(3*x - S(7))
+
+    assert manualintegrate(sin(y + x)*Heaviside(3*x - y), x) == \
+            (cos(y*Rational(4, 3)) - cos(x + y))*Heaviside(3*x - y)
+
+
+def test_manualintegrate_orthogonal_poly():
+    n = symbols('n')
+    a, b = 7, Rational(5, 3)
+    polys = [jacobi(n, a, b, x), gegenbauer(n, a, x), chebyshevt(n, x),
+        chebyshevu(n, x), legendre(n, x), hermite(n, x), laguerre(n, x),
+        assoc_laguerre(n, a, x)]
+    for p in polys:
+        integral = manualintegrate(p, x)
+        for deg in [-2, -1, 0, 1, 3, 5, 8]:
+            # some accept negative "degree", some do not
+            try:
+                p_subbed = p.subs(n, deg)
+            except ValueError:
+                continue
+            assert (integral.subs(n, deg).diff(x) - p_subbed).expand() == 0
+
+        # can also integrate simple expressions with these polynomials
+        q = x*p.subs(x, 2*x + 1)
+        integral = manualintegrate(q, x)
+        for deg in [2, 4, 7]:
+            assert (integral.subs(n, deg).diff(x) - q.subs(n, deg)).expand() == 0
+
+        # cannot integrate with respect to any other parameter
+        t = symbols('t')
+        for i in range(len(p.args) - 1):
+            new_args = list(p.args)
+            new_args[i] = t
+            assert isinstance(manualintegrate(p.func(*new_args), t), Integral)
+
+
+@slow
+def test_issue_6799():
+    r, x, phi = map(Symbol, 'r x phi'.split())
+    n = Symbol('n', integer=True, positive=True)
+
+    integrand = (cos(n*(x-phi))*cos(n*x))
+    limits = (x, -pi, pi)
+    assert manualintegrate(integrand, x) == \
+        ((n*x/2 + sin(2*n*x)/4)*cos(n*phi) - sin(n*phi)*cos(n*x)**2/2)/n
+    assert r * integrate(integrand, limits).trigsimp() / pi == r * cos(n * phi)
+    assert not integrate(integrand, limits).has(Dummy)
+
+
+def test_issue_12251():
+    assert manualintegrate(x**y, x) == Piecewise(
+        (x**(y + 1)/(y + 1), Ne(y, -1)), (log(x), True))
+
+
+def test_issue_3796():
+    assert manualintegrate(diff(exp(x + x**2)), x) == exp(x + x**2)
+    assert integrate(x * exp(x**4), x, risch=False) == -I*sqrt(pi)*erf(I*x**2)/4
+
+
+def test_manual_true():
+    assert integrate(exp(x) * sin(x), x, manual=True) == \
+        (exp(x) * sin(x)) / 2 - (exp(x) * cos(x)) / 2
+    assert integrate(sin(x) * cos(x), x, manual=True) in \
+        [sin(x) ** 2 / 2, -cos(x)**2 / 2]
+
+
+def test_issue_6746():
+    y = Symbol('y')
+    n = Symbol('n')
+    assert manualintegrate(y**x, x) == Piecewise(
+        (y**x/log(y), Ne(log(y), 0)), (x, True))
+    assert manualintegrate(y**(n*x), x) == Piecewise(
+        (Piecewise(
+            (y**(n*x)/log(y), Ne(log(y), 0)),
+            (n*x, True)
+        )/n, Ne(n, 0)),
+        (x, True))
+    assert manualintegrate(exp(n*x), x) == Piecewise(
+        (exp(n*x)/n, Ne(n, 0)), (x, True))
+
+    y = Symbol('y', positive=True)
+    assert manualintegrate((y + 1)**x, x) == (y + 1)**x/log(y + 1)
+    y = Symbol('y', zero=True)
+    assert manualintegrate((y + 1)**x, x) == x
+    y = Symbol('y')
+    n = Symbol('n', nonzero=True)
+    assert manualintegrate(y**(n*x), x) == Piecewise(
+        (y**(n*x)/log(y), Ne(log(y), 0)), (n*x, True))/n
+    y = Symbol('y', positive=True)
+    assert manualintegrate((y + 1)**(n*x), x) == \
+        (y + 1)**(n*x)/(n*log(y + 1))
+    a = Symbol('a', negative=True)
+    b = Symbol('b')
+    assert manualintegrate(1/(a + b*x**2), x) == atan(x/sqrt(a/b))/(b*sqrt(a/b))
+    b = Symbol('b', negative=True)
+    assert manualintegrate(1/(a + b*x**2), x) == \
+        atan(x/(sqrt(-a)*sqrt(-1/b)))/(b*sqrt(-a)*sqrt(-1/b))
+    assert manualintegrate(1/((x**a + y**b + 4)*sqrt(a*x**2 + 1)), x) == \
+        y**(-b)*Integral(x**(-a)/(y**(-b)*sqrt(a*x**2 + 1) +
+        x**(-a)*sqrt(a*x**2 + 1) + 4*x**(-a)*y**(-b)*sqrt(a*x**2 + 1)), x)
+    assert manualintegrate(1/((x**2 + 4)*sqrt(4*x**2 + 1)), x) == \
+        Integral(1/((x**2 + 4)*sqrt(4*x**2 + 1)), x)
+    assert manualintegrate(1/(x - a**x + x*b**2), x) == \
+        Integral(1/(-a**x + b**2*x + x), x)
+
+
+@slow
+def test_issue_2850():
+    assert manualintegrate(asin(x)*log(x), x) == -x*asin(x) - sqrt(-x**2 + 1) \
+            + (x*asin(x) + sqrt(-x**2 + 1))*log(x) - Integral(sqrt(-x**2 + 1)/x, x)
+    assert manualintegrate(acos(x)*log(x), x) == -x*acos(x) + sqrt(-x**2 + 1) + \
+        (x*acos(x) - sqrt(-x**2 + 1))*log(x) + Integral(sqrt(-x**2 + 1)/x, x)
+    assert manualintegrate(atan(x)*log(x), x) == -x*atan(x) + (x*atan(x) - \
+            log(x**2 + 1)/2)*log(x) + log(x**2 + 1)/2 + Integral(log(x**2 + 1)/x, x)/2
+
+
+def test_issue_9462():
+    assert manualintegrate(sin(2*x)*exp(x), x) == exp(x)*sin(2*x)/5 - 2*exp(x)*cos(2*x)/5
+    assert not integral_steps(sin(2*x)*exp(x), x).contains_dont_know()
+    assert manualintegrate((x - 3) / (x**2 - 2*x + 2)**2, x) == \
+                           Integral(x/(x**4 - 4*x**3 + 8*x**2 - 8*x + 4), x) \
+                           - 3*Integral(1/(x**4 - 4*x**3 + 8*x**2 - 8*x + 4), x)
+
+
+def test_cyclic_parts():
+    f = cos(x)*exp(x/4)
+    F = 16*exp(x/4)*sin(x)/17 + 4*exp(x/4)*cos(x)/17
+    assert manualintegrate(f, x) == F and F.diff(x) == f
+    f = x*cos(x)*exp(x/4)
+    F = (x*(16*exp(x/4)*sin(x)/17 + 4*exp(x/4)*cos(x)/17) -
+        128*exp(x/4)*sin(x)/289 + 240*exp(x/4)*cos(x)/289)
+    assert manualintegrate(f, x) == F and F.diff(x) == f
+
+
+@slow
+def test_issue_10847_slow():
+    assert manualintegrate((4*x**4 + 4*x**3 + 16*x**2 + 12*x + 8)
+                           / (x**6 + 2*x**5 + 3*x**4 + 4*x**3 + 3*x**2 + 2*x + 1), x) == \
+                           2*x/(x**2 + 1) + 3*atan(x) - 1/(x**2 + 1) - 3/(x + 1)
+
+
+@slow
+def test_issue_10847():
+
+    assert manualintegrate(x**2 / (x**2 - c), x) == \
+        c*Piecewise((atan(x/sqrt(-c))/sqrt(-c), Ne(c, 0)), (-1/x, True)) + x
+
+    rc = Symbol('c', real=True)
+    assert manualintegrate(x**2 / (x**2 - rc), x) == \
+        rc*Piecewise((atan(x/sqrt(-rc))/sqrt(-rc), rc < 0),
+                     ((log(-sqrt(rc) + x) - log(sqrt(rc) + x))/(2*sqrt(rc)), True)) + x
+
+    assert manualintegrate(sqrt(x - y) * log(z / x), x) == \
+        4*y**2*Piecewise((atan(sqrt(x - y)/sqrt(y))/sqrt(y), Ne(y, 0)),
+                         (-1/sqrt(x - y), True))/3 - 4*y*sqrt(x - y)/3 + \
+        2*(x - y)**Rational(3, 2)*log(z/x)/3 + 4*(x - y)**Rational(3, 2)/9
+    ry = Symbol('y', real=True)
+    rz = Symbol('z', real=True)
+    assert manualintegrate(sqrt(x - ry) * log(rz / x), x) == \
+        4*ry**2*Piecewise((atan(sqrt(x - ry)/sqrt(ry))/sqrt(ry), ry > 0),
+                         ((log(-sqrt(-ry) + sqrt(x - ry)) - log(sqrt(-ry) + sqrt(x - ry)))/(2*sqrt(-ry)), True))/3 \
+                         - 4*ry*sqrt(x - ry)/3 + 2*(x - ry)**Rational(3, 2)*log(rz/x)/3 \
+                         + 4*(x - ry)**Rational(3, 2)/9
+
+    assert manualintegrate(sqrt(x) * log(x), x) == 2*x**Rational(3, 2)*log(x)/3 - 4*x**Rational(3, 2)/9
+
+    result = manualintegrate(sqrt(a*x + b) / x, x)
+    assert result == Piecewise((-2*b*Piecewise(
+        (-atan(sqrt(a*x + b)/sqrt(-b))/sqrt(-b), Ne(b, 0)),
+        (1/sqrt(a*x + b), True)) + 2*sqrt(a*x + b), Ne(a, 0)),
+        (sqrt(b)*log(x), True))
+    assert piecewise_fold(result) == Piecewise(
+        (2*b*atan(sqrt(a*x + b)/sqrt(-b))/sqrt(-b) + 2*sqrt(a*x + b), Ne(a, 0) & Ne(b, 0)),
+        (-2*b/sqrt(a*x + b) + 2*sqrt(a*x + b), Ne(a, 0)),
+        (sqrt(b)*log(x), True))
+
+    ra = Symbol('a', real=True)
+    rb = Symbol('b', real=True)
+    assert manualintegrate(sqrt(ra*x + rb) / x, x) == \
+        Piecewise(
+            (-2*rb*Piecewise(
+                (-atan(sqrt(ra*x + rb)/sqrt(-rb))/sqrt(-rb), rb < 0),
+                (-I*(log(-sqrt(rb) + sqrt(ra*x + rb)) - log(sqrt(rb) + sqrt(ra*x + rb)))/(2*sqrt(-rb)), True)) +
+             2*sqrt(ra*x + rb), Ne(ra, 0)),
+            (sqrt(rb)*log(x), True))
+
+    assert expand(manualintegrate(sqrt(ra*x + rb) / (x + rc), x)) == \
+           Piecewise((-2*ra*rc*Piecewise((atan(sqrt(ra*x + rb)/sqrt(ra*rc - rb))/sqrt(ra*rc - rb), ra*rc - rb > 0),
+                                       (log(-sqrt(-ra*rc + rb) + sqrt(ra*x + rb))/(2*sqrt(-ra*rc + rb)) -
+                                        log(sqrt(-ra*rc + rb) + sqrt(ra*x + rb))/(2*sqrt(-ra*rc + rb)), True)) +
+                      2*rb*Piecewise((atan(sqrt(ra*x + rb)/sqrt(ra*rc - rb))/sqrt(ra*rc - rb), ra*rc - rb > 0),
+                                    (log(-sqrt(-ra*rc + rb) + sqrt(ra*x + rb))/(2*sqrt(-ra*rc + rb)) -
+                                     log(sqrt(-ra*rc + rb) + sqrt(ra*x + rb))/(2*sqrt(-ra*rc + rb)), True)) +
+                      2*sqrt(ra*x + rb), Ne(ra, 0)), (sqrt(rb)*log(rc + x), True))
+
+    assert manualintegrate(sqrt(2*x + 3) / (x + 1), x) == 2*sqrt(2*x + 3) - log(sqrt(2*x + 3) + 1) + log(sqrt(2*x + 3) - 1)
+    assert manualintegrate(sqrt(2*x + 3) / 2 * x, x) == (2*x + 3)**Rational(5, 2)/20 - (2*x + 3)**Rational(3, 2)/4
+    assert manualintegrate(x**Rational(3,2) * log(x), x) == 2*x**Rational(5,2)*log(x)/5 - 4*x**Rational(5,2)/25
+    assert manualintegrate(x**(-3) * log(x), x) == -log(x)/(2*x**2) - 1/(4*x**2)
+    assert manualintegrate(log(y)/(y**2*(1 - 1/y)), y) == \
+        log(y)*log(-1 + 1/y) - Integral(log(-1 + 1/y)/y, y)
+
+
+def test_issue_12899():
+    assert manualintegrate(f(x,y).diff(x),y) == Integral(Derivative(f(x,y),x),y)
+    assert manualintegrate(f(x,y).diff(y).diff(x),y) == Derivative(f(x,y),x)
+
+
+def test_constant_independent_of_symbol():
+    assert manualintegrate(Integral(y, (x, 1, 2)), x) == \
+        x*Integral(y, (x, 1, 2))
+
+
+def test_issue_12641():
+    assert manualintegrate(sin(2*x), x) == -cos(2*x)/2
+    assert manualintegrate(cos(x)*sin(2*x), x) == -2*cos(x)**3/3
+    assert manualintegrate((sin(2*x)*cos(x))/(1 + cos(x)), x) == \
+        -2*log(cos(x) + 1) - cos(x)**2 + 2*cos(x)
+
+
+@slow
+def test_issue_13297():
+    assert manualintegrate(sin(x) * cos(x)**5, x) == -cos(x)**6 / 6
+
+
+def test_issue_14470():
+    assert_is_integral_of(1/(x*sqrt(x + 1)), log(sqrt(x + 1) - 1) - log(sqrt(x + 1) + 1))
+
+
+@slow
+def test_issue_9858():
+    assert manualintegrate(exp(x)*cos(exp(x)), x) == sin(exp(x))
+    assert manualintegrate(exp(2*x)*cos(exp(x)), x) == \
+        exp(x)*sin(exp(x)) + cos(exp(x))
+    res = manualintegrate(exp(10*x)*sin(exp(x)), x)
+    assert not res.has(Integral)
+    assert res.diff(x) == exp(10*x)*sin(exp(x))
+    # an example with many similar integrations by parts
+    assert manualintegrate(sum(x*exp(k*x) for k in range(1, 8)), x) == (
+        x*exp(7*x)/7 + x*exp(6*x)/6 + x*exp(5*x)/5 + x*exp(4*x)/4 +
+        x*exp(3*x)/3 + x*exp(2*x)/2 + x*exp(x) - exp(7*x)/49 -exp(6*x)/36 -
+        exp(5*x)/25 - exp(4*x)/16 - exp(3*x)/9 - exp(2*x)/4 - exp(x))
+
+
+def test_issue_8520():
+    assert manualintegrate(x/(x**4 + 1), x) == atan(x**2)/2
+    assert manualintegrate(x**2/(x**6 + 25), x) == atan(x**3/5)/15
+    f = x/(9*x**4 + 4)**2
+    assert manualintegrate(f, x).diff(x).factor() == f
+
+
+def test_manual_subs():
+    x, y = symbols('x y')
+    expr = log(x) + exp(x)
+    # if log(x) is y, then exp(y) is x
+    assert manual_subs(expr, log(x), y) == y + exp(exp(y))
+    # if exp(x) is y, then log(y) need not be x
+    assert manual_subs(expr, exp(x), y) == log(x) + y
+
+    raises(ValueError, lambda: manual_subs(expr, x))
+    raises(ValueError, lambda: manual_subs(expr, exp(x), x, y))
+
+
+@slow
+def test_issue_15471():
+    f = log(x)*cos(log(x))/x**Rational(3, 4)
+    F = -128*x**Rational(1, 4)*sin(log(x))/289 + 240*x**Rational(1, 4)*cos(log(x))/289 + (16*x**Rational(1, 4)*sin(log(x))/17 + 4*x**Rational(1, 4)*cos(log(x))/17)*log(x)
+    assert_is_integral_of(f, F)
+
+
+def test_quadratic_denom():
+    f = (5*x + 2)/(3*x**2 - 2*x + 8)
+    assert manualintegrate(f, x) == 5*log(3*x**2 - 2*x + 8)/6 + 11*sqrt(23)*atan(3*sqrt(23)*(x - Rational(1, 3))/23)/69
+    g = 3/(2*x**2 + 3*x + 1)
+    assert manualintegrate(g, x) == 3*log(4*x + 2) - 3*log(4*x + 4)
+
+def test_issue_22757():
+    assert manualintegrate(sin(x), y) == y * sin(x)
+
+
+def test_issue_23348():
+    steps = integral_steps(tan(x), x)
+    constant_times_step = steps.substep.substep
+    assert constant_times_step.integrand == constant_times_step.constant * constant_times_step.other
+
+
+def test_issue_23566():
+    i = Integral(1/sqrt(x**2 - 1), (x, -2, -1)).doit(manual=True)
+    assert i == -log(4 - 2*sqrt(3)) + log(2)
+    assert str(i.n()) == '1.31695789692482'
+
+
+def test_issue_25093():
+    ap = Symbol('ap', positive=True)
+    an = Symbol('an', negative=True)
+    assert manualintegrate(exp(a*x**2 + b), x) == sqrt(pi)*exp(b)*erfi(sqrt(a)*x)/(2*sqrt(a))
+    assert manualintegrate(exp(ap*x**2 + b), x) == sqrt(pi)*exp(b)*erfi(sqrt(ap)*x)/(2*sqrt(ap))
+    assert manualintegrate(exp(an*x**2 + b), x) == -sqrt(pi)*exp(b)*erf(an*x/sqrt(-an))/(2*sqrt(-an))
+    assert manualintegrate(sin(a*x**2 + b), x) == (
+        sqrt(2)*sqrt(pi)*(sin(b)*fresnelc(sqrt(2)*sqrt(a)*x/sqrt(pi))
+        + cos(b)*fresnels(sqrt(2)*sqrt(a)*x/sqrt(pi)))/(2*sqrt(a)))
+    assert manualintegrate(cos(a*x**2 + b), x) == (
+        sqrt(2)*sqrt(pi)*(-sin(b)*fresnels(sqrt(2)*sqrt(a)*x/sqrt(pi))
+        + cos(b)*fresnelc(sqrt(2)*sqrt(a)*x/sqrt(pi)))/(2*sqrt(a)))
+
+
+def test_nested_pow():
+    assert_is_integral_of(sqrt(x**2), x*sqrt(x**2)/2)
+    assert_is_integral_of(sqrt(x**(S(5)/3)), 6*x*sqrt(x**(S(5)/3))/11)
+    assert_is_integral_of(1/sqrt(x**2), x*log(x)/sqrt(x**2))
+    assert_is_integral_of(x*sqrt(x**(-4)), x**2*sqrt(x**-4)*log(x))
+    f = (c*(a+b*x)**d)**e
+    F1 = (c*(a + b*x)**d)**e*(a/b + x)/(d*e + 1)
+    F2 = (c*(a + b*x)**d)**e*(a/b + x)*log(a/b + x)
+    assert manualintegrate(f, x) == \
+        Piecewise((Piecewise((F1, Ne(d*e, -1)), (F2, True)), Ne(b, 0)), (x*(a**d*c)**e, True))
+    assert F1.diff(x).equals(f)
+    assert F2.diff(x).subs(d*e, -1).equals(f)
+
+
+def test_manualintegrate_sqrt_linear():
+    assert_is_integral_of((5*x**3+4)/sqrt(2+3*x),
+                          10*(3*x + 2)**(S(7)/2)/567 - 4*(3*x + 2)**(S(5)/2)/27 +
+                          40*(3*x + 2)**(S(3)/2)/81 + 136*sqrt(3*x + 2)/81)
+    assert manualintegrate(x/sqrt(a+b*x)**3, x) == \
+        Piecewise((Mul(2, b**-2, a/sqrt(a + b*x) + sqrt(a + b*x)), Ne(b, 0)), (x**2/(2*a**(S(3)/2)), True))
+    assert_is_integral_of((sqrt(3*x+3)+1)/((2*x+2)**(1/S(3))+1),
+                          3*sqrt(6)*(2*x + 2)**(S(7)/6)/14 - 3*sqrt(6)*(2*x + 2)**(S(5)/6)/10 -
+                          3*sqrt(6)*(2*x + 2)**(S.One/6)/2 + 3*(2*x + 2)**(S(2)/3)/4 - 3*(2*x + 2)**(S.One/3)/2 +
+                          sqrt(6)*sqrt(2*x + 2)/2 + 3*log((2*x + 2)**(S.One/3) + 1)/2 +
+                          3*sqrt(6)*atan((2*x + 2)**(S.One/6))/2)
+    assert_is_integral_of(sqrt(x+sqrt(x)),
+                          2*sqrt(sqrt(x) + x)*(sqrt(x)/12 + x/3 - S(1)/8) + log(2*sqrt(x) + 2*sqrt(sqrt(x) + x) + 1)/8)
+    assert_is_integral_of(sqrt(2*x+3+sqrt(4*x+5))**3,
+                          sqrt(2*x + sqrt(4*x + 5) + 3) *
+                          (9*x/10 + 11*(4*x + 5)**(S(3)/2)/40 + sqrt(4*x + 5)/40 + (4*x + 5)**2/10 + S(11)/10)/2)
+
+
+def test_manualintegrate_sqrt_quadratic():
+    assert_is_integral_of(1/sqrt((x - I)**2-1), log(2*x + 2*sqrt(x**2 - 2*I*x - 2) - 2*I))
+    assert_is_integral_of(1/sqrt(3*x**2+4*x+5), sqrt(3)*asinh(3*sqrt(11)*(x + S(2)/3)/11)/3)
+    assert_is_integral_of(1/sqrt(-3*x**2+4*x+5), sqrt(3)*asin(3*sqrt(19)*(x - S(2)/3)/19)/3)
+    assert_is_integral_of(1/sqrt(3*x**2+4*x-5), sqrt(3)*log(6*x + 2*sqrt(3)*sqrt(3*x**2 + 4*x - 5) + 4)/3)
+    assert_is_integral_of(1/sqrt(4*x**2-4*x+1), (x - S.Half)*log(x - S.Half)/(2*sqrt((x - S.Half)**2)))
+    assert manualintegrate(1/sqrt(a+b*x+c*x**2), x) == \
+        Piecewise((log(b + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + 2*c*x)/sqrt(c), Ne(c, 0) & Ne(a - b**2/(4*c), 0)),
+                  ((b/(2*c) + x)*log(b/(2*c) + x)/sqrt(c*(b/(2*c) + x)**2), Ne(c, 0)),
+                  (2*sqrt(a + b*x)/b, Ne(b, 0)), (x/sqrt(a), True))
+
+    assert_is_integral_of((7*x+6)/sqrt(3*x**2+4*x+5),
+                          7*sqrt(3*x**2 + 4*x + 5)/3 + 4*sqrt(3)*asinh(3*sqrt(11)*(x + S(2)/3)/11)/9)
+    assert_is_integral_of((7*x+6)/sqrt(-3*x**2+4*x+5),
+                          -7*sqrt(-3*x**2 + 4*x + 5)/3 + 32*sqrt(3)*asin(3*sqrt(19)*(x - S(2)/3)/19)/9)
+    assert_is_integral_of((7*x+6)/sqrt(3*x**2+4*x-5),
+                          7*sqrt(3*x**2 + 4*x - 5)/3 + 4*sqrt(3)*log(6*x + 2*sqrt(3)*sqrt(3*x**2 + 4*x - 5) + 4)/9)
+    assert manualintegrate((d+e*x)/sqrt(a+b*x+c*x**2), x) == \
+        Piecewise(((-b*e/(2*c) + d) *
+                   Piecewise((log(b + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + 2*c*x)/sqrt(c), Ne(a - b**2/(4*c), 0)),
+                             ((b/(2*c) + x)*log(b/(2*c) + x)/sqrt(c*(b/(2*c) + x)**2), True)) +
+                   e*sqrt(a + b*x + c*x**2)/c, Ne(c, 0)),
+                  ((2*d*sqrt(a + b*x) + 2*e*(-a*sqrt(a + b*x) + (a + b*x)**(S(3)/2)/3)/b)/b, Ne(b, 0)),
+                  ((d*x + e*x**2/2)/sqrt(a), True))
+
+    assert manualintegrate((3*x**3-x**2+2*x-4)/sqrt(x**2-3*x+2), x) == \
+        sqrt(x**2 - 3*x + 2)*(x**2 + 13*x/4 + S(101)/8) + 135*log(2*x + 2*sqrt(x**2 - 3*x + 2) - 3)/16
+
+    assert_is_integral_of(sqrt(53225*x**2-66732*x+23013),
+                          (x/2 - S(16683)/53225)*sqrt(53225*x**2 - 66732*x + 23013) +
+                          111576969*sqrt(2129)*asinh(53225*x/10563 - S(11122)/3521)/1133160250)
+    assert manualintegrate(sqrt(a+c*x**2), x) == \
+        Piecewise((a*Piecewise((log(2*sqrt(c)*sqrt(a + c*x**2) + 2*c*x)/sqrt(c), Ne(a, 0)),
+                               (x*log(x)/sqrt(c*x**2), True))/2 + x*sqrt(a + c*x**2)/2, Ne(c, 0)),
+                  (sqrt(a)*x, True))
+    assert manualintegrate(sqrt(a+b*x+c*x**2), x) == \
+        Piecewise(((a/2 - b**2/(8*c)) *
+                   Piecewise((log(b + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + 2*c*x)/sqrt(c), Ne(a - b**2/(4*c), 0)),
+                             ((b/(2*c) + x)*log(b/(2*c) + x)/sqrt(c*(b/(2*c) + x)**2), True)) +
+                   (b/(4*c) + x/2)*sqrt(a + b*x + c*x**2), Ne(c, 0)),
+                  (2*(a + b*x)**(S(3)/2)/(3*b), Ne(b, 0)),
+                  (sqrt(a)*x, True))
+
+    assert_is_integral_of(x*sqrt(x**2+2*x+4),
+                          (x**2/3 + x/6 + S(5)/6)*sqrt(x**2 + 2*x + 4) - 3*asinh(sqrt(3)*(x + 1)/3)/2)
+
+
+def test_mul_pow_derivative():
+    assert_is_integral_of(x*sec(x)*tan(x), x*sec(x) - log(tan(x) + sec(x)))
+    assert_is_integral_of(x*sec(x)**2, x*tan(x) + log(cos(x)))
+    assert_is_integral_of(x**3*Derivative(f(x), (x, 4)),
+                          x**3*Derivative(f(x), (x, 3)) - 3*x**2*Derivative(f(x), (x, 2)) +
+                          6*x*Derivative(f(x), x) - 6*f(x))

rl/Lib/site-packages/sympy/integrals/tests/test_meijerint.py

@@ -0,0 +1,774 @@
+from sympy.core.function import expand_func
+from sympy.core.numbers import (I, Rational, oo, pi)
+from sympy.core.singleton import S
+from sympy.core.sorting import default_sort_key
+from sympy.functions.elementary.complexes import Abs, arg, re, unpolarify
+from sympy.functions.elementary.exponential import (exp, exp_polar, log)
+from sympy.functions.elementary.hyperbolic import cosh, acosh, sinh
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.piecewise import Piecewise, piecewise_fold
+from sympy.functions.elementary.trigonometric import (cos, sin, sinc, asin)
+from sympy.functions.special.error_functions import (erf, erfc)
+from sympy.functions.special.gamma_functions import (gamma, polygamma)
+from sympy.functions.special.hyper import (hyper, meijerg)
+from sympy.integrals.integrals import (Integral, integrate)
+from sympy.simplify.hyperexpand import hyperexpand
+from sympy.simplify.simplify import simplify
+from sympy.integrals.meijerint import (_rewrite_single, _rewrite1,
+    meijerint_indefinite, _inflate_g, _create_lookup_table,
+    meijerint_definite, meijerint_inversion)
+from sympy.testing.pytest import slow
+from sympy.core.random import (verify_numerically,
+        random_complex_number as randcplx)
+from sympy.abc import x, y, a, b, c, d, s, t, z
+
+
+def test_rewrite_single():
+    def t(expr, c, m):
+        e = _rewrite_single(meijerg([a], [b], [c], [d], expr), x)
+        assert e is not None
+        assert isinstance(e[0][0][2], meijerg)
+        assert e[0][0][2].argument.as_coeff_mul(x) == (c, (m,))
+
+    def tn(expr):
+        assert _rewrite_single(meijerg([a], [b], [c], [d], expr), x) is None
+
+    t(x, 1, x)
+    t(x**2, 1, x**2)
+    t(x**2 + y*x**2, y + 1, x**2)
+    tn(x**2 + x)
+    tn(x**y)
+
+    def u(expr, x):
+        from sympy.core.add import Add
+        r = _rewrite_single(expr, x)
+        e = Add(*[res[0]*res[2] for res in r[0]]).replace(
+            exp_polar, exp)  # XXX Hack?
+        assert verify_numerically(e, expr, x)
+
+    u(exp(-x)*sin(x), x)
+
+    # The following has stopped working because hyperexpand changed slightly.
+    # It is probably not worth fixing
+    #u(exp(-x)*sin(x)*cos(x), x)
+
+    # This one cannot be done numerically, since it comes out as a g-function
+    # of argument 4*pi
+    # NOTE This also tests a bug in inverse mellin transform (which used to
+    #      turn exp(4*pi*I*t) into a factor of exp(4*pi*I)**t instead of
+    #      exp_polar).
+    #u(exp(x)*sin(x), x)
+    assert _rewrite_single(exp(x)*sin(x), x) == \
+        ([(-sqrt(2)/(2*sqrt(pi)), 0,
+           meijerg(((Rational(-1, 2), 0, Rational(1, 4), S.Half, Rational(3, 4)), (1,)),
+                   ((), (Rational(-1, 2), 0)), 64*exp_polar(-4*I*pi)/x**4))], True)
+
+
+def test_rewrite1():
+    assert _rewrite1(x**3*meijerg([a], [b], [c], [d], x**2 + y*x**2)*5, x) == \
+        (5, x**3, [(1, 0, meijerg([a], [b], [c], [d], x**2*(y + 1)))], True)
+
+
+def test_meijerint_indefinite_numerically():
+    def t(fac, arg):
+        g = meijerg([a], [b], [c], [d], arg)*fac
+        subs = {a: randcplx()/10, b: randcplx()/10 + I,
+                c: randcplx(), d: randcplx()}
+        integral = meijerint_indefinite(g, x)
+        assert integral is not None
+        assert verify_numerically(g.subs(subs), integral.diff(x).subs(subs), x)
+    t(1, x)
+    t(2, x)
+    t(1, 2*x)
+    t(1, x**2)
+    t(5, x**S('3/2'))
+    t(x**3, x)
+    t(3*x**S('3/2'), 4*x**S('7/3'))
+
+
+def test_meijerint_definite():
+    v, b = meijerint_definite(x, x, 0, 0)
+    assert v.is_zero and b is True
+    v, b = meijerint_definite(x, x, oo, oo)
+    assert v.is_zero and b is True
+
+
+def test_inflate():
+    subs = {a: randcplx()/10, b: randcplx()/10 + I, c: randcplx(),
+            d: randcplx(), y: randcplx()/10}
+
+    def t(a, b, arg, n):
+        from sympy.core.mul import Mul
+        m1 = meijerg(a, b, arg)
+        m2 = Mul(*_inflate_g(m1, n))
+        # NOTE: (the random number)**9 must still be on the principal sheet.
+        # Thus make b&d small to create random numbers of small imaginary part.
+        return verify_numerically(m1.subs(subs), m2.subs(subs), x, b=0.1, d=-0.1)
+    assert t([[a], [b]], [[c], [d]], x, 3)
+    assert t([[a, y], [b]], [[c], [d]], x, 3)
+    assert t([[a], [b]], [[c, y], [d]], 2*x**3, 3)
+
+
+def test_recursive():
+    from sympy.core.symbol import symbols
+    a, b, c = symbols('a b c', positive=True)
+    r = exp(-(x - a)**2)*exp(-(x - b)**2)
+    e = integrate(r, (x, 0, oo), meijerg=True)
+    assert simplify(e.expand()) == (
+        sqrt(2)*sqrt(pi)*(
+        (erf(sqrt(2)*(a + b)/2) + 1)*exp(-a**2/2 + a*b - b**2/2))/4)
+    e = integrate(exp(-(x - a)**2)*exp(-(x - b)**2)*exp(c*x), (x, 0, oo), meijerg=True)
+    assert simplify(e) == (
+        sqrt(2)*sqrt(pi)*(erf(sqrt(2)*(2*a + 2*b + c)/4) + 1)*exp(-a**2 - b**2
+        + (2*a + 2*b + c)**2/8)/4)
+    assert simplify(integrate(exp(-(x - a - b - c)**2), (x, 0, oo), meijerg=True)) == \
+        sqrt(pi)/2*(1 + erf(a + b + c))
+    assert simplify(integrate(exp(-(x + a + b + c)**2), (x, 0, oo), meijerg=True)) == \
+        sqrt(pi)/2*(1 - erf(a + b + c))
+
+
+@slow
+def test_meijerint():
+    from sympy.core.function import expand
+    from sympy.core.symbol import symbols
+    s, t, mu = symbols('s t mu', real=True)
+    assert integrate(meijerg([], [], [0], [], s*t)
+                     *meijerg([], [], [mu/2], [-mu/2], t**2/4),
+                     (t, 0, oo)).is_Piecewise
+    s = symbols('s', positive=True)
+    assert integrate(x**s*meijerg([[], []], [[0], []], x), (x, 0, oo)) == \
+        gamma(s + 1)
+    assert integrate(x**s*meijerg([[], []], [[0], []], x), (x, 0, oo),
+                     meijerg=True) == gamma(s + 1)
+    assert isinstance(integrate(x**s*meijerg([[], []], [[0], []], x),
+                                (x, 0, oo), meijerg=False),
+                      Integral)
+
+    assert meijerint_indefinite(exp(x), x) == exp(x)
+
+    # TODO what simplifications should be done automatically?
+    # This tests "extra case" for antecedents_1.
+    a, b = symbols('a b', positive=True)
+    assert simplify(meijerint_definite(x**a, x, 0, b)[0]) == \
+        b**(a + 1)/(a + 1)
+
+    # This tests various conditions and expansions:
+    assert meijerint_definite((x + 1)**3*exp(-x), x, 0, oo) == (16, True)
+
+    # Again, how about simplifications?
+    sigma, mu = symbols('sigma mu', positive=True)
+    i, c = meijerint_definite(exp(-((x - mu)/(2*sigma))**2), x, 0, oo)
+    assert simplify(i) == sqrt(pi)*sigma*(2 - erfc(mu/(2*sigma)))
+    assert c == True
+
+    i, _ = meijerint_definite(exp(-mu*x)*exp(sigma*x), x, 0, oo)
+    # TODO it would be nice to test the condition
+    assert simplify(i) == 1/(mu - sigma)
+
+    # Test substitutions to change limits
+    assert meijerint_definite(exp(x), x, -oo, 2) == (exp(2), True)
+    # Note: causes a NaN in _check_antecedents
+    assert expand(meijerint_definite(exp(x), x, 0, I)[0]) == exp(I) - 1
+    assert expand(meijerint_definite(exp(-x), x, 0, x)[0]) == \
+        1 - exp(-exp(I*arg(x))*abs(x))
+
+    # Test -oo to oo
+    assert meijerint_definite(exp(-x**2), x, -oo, oo) == (sqrt(pi), True)
+    assert meijerint_definite(exp(-abs(x)), x, -oo, oo) == (2, True)
+    assert meijerint_definite(exp(-(2*x - 3)**2), x, -oo, oo) == \
+        (sqrt(pi)/2, True)
+    assert meijerint_definite(exp(-abs(2*x - 3)), x, -oo, oo) == (1, True)
+    assert meijerint_definite(exp(-((x - mu)/sigma)**2/2)/sqrt(2*pi*sigma**2),
+                              x, -oo, oo) == (1, True)
+    assert meijerint_definite(sinc(x)**2, x, -oo, oo) == (pi, True)
+
+    # Test one of the extra conditions for 2 g-functinos
+    assert meijerint_definite(exp(-x)*sin(x), x, 0, oo) == (S.Half, True)
+
+    # Test a bug
+    def res(n):
+        return (1/(1 + x**2)).diff(x, n).subs(x, 1)*(-1)**n
+    for n in range(6):
+        assert integrate(exp(-x)*sin(x)*x**n, (x, 0, oo), meijerg=True) == \
+            res(n)
+
+    # This used to test trigexpand... now it is done by linear substitution
+    assert simplify(integrate(exp(-x)*sin(x + a), (x, 0, oo), meijerg=True)
+                    ) == sqrt(2)*sin(a + pi/4)/2
+
+    # Test the condition 14 from prudnikov.
+    # (This is besselj*besselj in disguise, to stop the product from being
+    #  recognised in the tables.)
+    a, b, s = symbols('a b s')
+    assert meijerint_definite(meijerg([], [], [a/2], [-a/2], x/4)
+                  *meijerg([], [], [b/2], [-b/2], x/4)*x**(s - 1), x, 0, oo
+        ) == (
+        (4*2**(2*s - 2)*gamma(-2*s + 1)*gamma(a/2 + b/2 + s)
+         /(gamma(-a/2 + b/2 - s + 1)*gamma(a/2 - b/2 - s + 1)
+           *gamma(a/2 + b/2 - s + 1)),
+            (re(s) < 1) & (re(s) < S(1)/2) & (re(a)/2 + re(b)/2 + re(s) > 0)))
+
+    # test a bug
+    assert integrate(sin(x**a)*sin(x**b), (x, 0, oo), meijerg=True) == \
+        Integral(sin(x**a)*sin(x**b), (x, 0, oo))
+
+    # test better hyperexpand
+    assert integrate(exp(-x**2)*log(x), (x, 0, oo), meijerg=True) == \
+        (sqrt(pi)*polygamma(0, S.Half)/4).expand()
+
+    # Test hyperexpand bug.
+    from sympy.functions.special.gamma_functions import lowergamma
+    n = symbols('n', integer=True)
+    assert simplify(integrate(exp(-x)*x**n, x, meijerg=True)) == \
+        lowergamma(n + 1, x)
+
+    # Test a bug with argument 1/x
+    alpha = symbols('alpha', positive=True)
+    assert meijerint_definite((2 - x)**alpha*sin(alpha/x), x, 0, 2) == \
+        (sqrt(pi)*alpha*gamma(alpha + 1)*meijerg(((), (alpha/2 + S.Half,
+        alpha/2 + 1)), ((0, 0, S.Half), (Rational(-1, 2),)), alpha**2/16)/4, True)
+
+    # test a bug related to 3016
+    a, s = symbols('a s', positive=True)
+    assert simplify(integrate(x**s*exp(-a*x**2), (x, -oo, oo))) == \
+        a**(-s/2 - S.Half)*((-1)**s + 1)*gamma(s/2 + S.Half)/2
+
+
+def test_bessel():
+    from sympy.functions.special.bessel import (besseli, besselj)
+    assert simplify(integrate(besselj(a, z)*besselj(b, z)/z, (z, 0, oo),
+                     meijerg=True, conds='none')) == \
+        2*sin(pi*(a/2 - b/2))/(pi*(a - b)*(a + b))
+    assert simplify(integrate(besselj(a, z)*besselj(a, z)/z, (z, 0, oo),
+                     meijerg=True, conds='none')) == 1/(2*a)
+
+    # TODO more orthogonality integrals
+
+    assert simplify(integrate(sin(z*x)*(x**2 - 1)**(-(y + S.Half)),
+                              (x, 1, oo), meijerg=True, conds='none')
+                    *2/((z/2)**y*sqrt(pi)*gamma(S.Half - y))) == \
+        besselj(y, z)
+
+    # Werner Rosenheinrich
+    # SOME INDEFINITE INTEGRALS OF BESSEL FUNCTIONS
+
+    assert integrate(x*besselj(0, x), x, meijerg=True) == x*besselj(1, x)
+    assert integrate(x*besseli(0, x), x, meijerg=True) == x*besseli(1, x)
+    # TODO can do higher powers, but come out as high order ... should they be
+    #      reduced to order 0, 1?
+    assert integrate(besselj(1, x), x, meijerg=True) == -besselj(0, x)
+    assert integrate(besselj(1, x)**2/x, x, meijerg=True) == \
+        -(besselj(0, x)**2 + besselj(1, x)**2)/2
+    # TODO more besseli when tables are extended or recursive mellin works
+    assert integrate(besselj(0, x)**2/x**2, x, meijerg=True) == \
+        -2*x*besselj(0, x)**2 - 2*x*besselj(1, x)**2 \
+        + 2*besselj(0, x)*besselj(1, x) - besselj(0, x)**2/x
+    assert integrate(besselj(0, x)*besselj(1, x), x, meijerg=True) == \
+        -besselj(0, x)**2/2
+    assert integrate(x**2*besselj(0, x)*besselj(1, x), x, meijerg=True) == \
+        x**2*besselj(1, x)**2/2
+    assert integrate(besselj(0, x)*besselj(1, x)/x, x, meijerg=True) == \
+        (x*besselj(0, x)**2 + x*besselj(1, x)**2 -
+            besselj(0, x)*besselj(1, x))
+    # TODO how does besselj(0, a*x)*besselj(0, b*x) work?
+    # TODO how does besselj(0, x)**2*besselj(1, x)**2 work?
+    # TODO sin(x)*besselj(0, x) etc come out a mess
+    # TODO can x*log(x)*besselj(0, x) be done?
+    # TODO how does besselj(1, x)*besselj(0, x+a) work?
+    # TODO more indefinite integrals when struve functions etc are implemented
+
+    # test a substitution
+    assert integrate(besselj(1, x**2)*x, x, meijerg=True) == \
+        -besselj(0, x**2)/2
+
+
+def test_inversion():
+    from sympy.functions.special.bessel import besselj
+    from sympy.functions.special.delta_functions import Heaviside
+
+    def inv(f):
+        return piecewise_fold(meijerint_inversion(f, s, t))
+    assert inv(1/(s**2 + 1)) == sin(t)*Heaviside(t)
+    assert inv(s/(s**2 + 1)) == cos(t)*Heaviside(t)
+    assert inv(exp(-s)/s) == Heaviside(t - 1)
+    assert inv(1/sqrt(1 + s**2)) == besselj(0, t)*Heaviside(t)
+
+    # Test some antcedents checking.
+    assert meijerint_inversion(sqrt(s)/sqrt(1 + s**2), s, t) is None
+    assert inv(exp(s**2)) is None
+    assert meijerint_inversion(exp(-s**2), s, t) is None
+
+
+def test_inversion_conditional_output():
+    from sympy.core.symbol import Symbol
+    from sympy.integrals.transforms import InverseLaplaceTransform
+
+    a = Symbol('a', positive=True)
+    F = sqrt(pi/a)*exp(-2*sqrt(a)*sqrt(s))
+    f = meijerint_inversion(F, s, t)
+    assert not f.is_Piecewise
+
+    b = Symbol('b', real=True)
+    F = F.subs(a, b)
+    f2 = meijerint_inversion(F, s, t)
+    assert f2.is_Piecewise
+    # first piece is same as f
+    assert f2.args[0][0] == f.subs(a, b)
+    # last piece is an unevaluated transform
+    assert f2.args[-1][1]
+    ILT = InverseLaplaceTransform(F, s, t, None)
+    assert f2.args[-1][0] == ILT or f2.args[-1][0] == ILT.as_integral
+
+
+def test_inversion_exp_real_nonreal_shift():
+    from sympy.core.symbol import Symbol
+    from sympy.functions.special.delta_functions import DiracDelta
+    r = Symbol('r', real=True)
+    c = Symbol('c', extended_real=False)
+    a = 1 + 2*I
+    z = Symbol('z')
+    assert not meijerint_inversion(exp(r*s), s, t).is_Piecewise
+    assert meijerint_inversion(exp(a*s), s, t) is None
+    assert meijerint_inversion(exp(c*s), s, t) is None
+    f = meijerint_inversion(exp(z*s), s, t)
+    assert f.is_Piecewise
+    assert isinstance(f.args[0][0], DiracDelta)
+
+
+@slow
+def test_lookup_table():
+    from sympy.core.random import uniform, randrange
+    from sympy.core.add import Add
+    from sympy.integrals.meijerint import z as z_dummy
+    table = {}
+    _create_lookup_table(table)
+    for l in table.values():
+        for formula, terms, cond, hint in sorted(l, key=default_sort_key):
+            subs = {}
+            for ai in list(formula.free_symbols) + [z_dummy]:
+                if hasattr(ai, 'properties') and ai.properties:
+                    # these Wilds match positive integers
+                    subs[ai] = randrange(1, 10)
+                else:
+                    subs[ai] = uniform(1.5, 2.0)
+            if not isinstance(terms, list):
+                terms = terms(subs)
+
+            # First test that hyperexpand can do this.
+            expanded = [hyperexpand(g) for (_, g) in terms]
+            assert all(x.is_Piecewise or not x.has(meijerg) for x in expanded)
+
+            # Now test that the meijer g-function is indeed as advertised.
+            expanded = Add(*[f*x for (f, x) in terms])
+            a, b = formula.n(subs=subs), expanded.n(subs=subs)
+            r = min(abs(a), abs(b))
+            if r < 1:
+                assert abs(a - b).n() <= 1e-10
+            else:
+                assert (abs(a - b)/r).n() <= 1e-10
+
+
+def test_branch_bug():
+    from sympy.functions.special.gamma_functions import lowergamma
+    from sympy.simplify.powsimp import powdenest
+    # TODO gammasimp cannot prove that the factor is unity
+    assert powdenest(integrate(erf(x**3), x, meijerg=True).diff(x),
+           polar=True) == 2*erf(x**3)*gamma(Rational(2, 3))/3/gamma(Rational(5, 3))
+    assert integrate(erf(x**3), x, meijerg=True) == \
+        2*x*erf(x**3)*gamma(Rational(2, 3))/(3*gamma(Rational(5, 3))) \
+        - 2*gamma(Rational(2, 3))*lowergamma(Rational(2, 3), x**6)/(3*sqrt(pi)*gamma(Rational(5, 3)))
+
+
+def test_linear_subs():
+    from sympy.functions.special.bessel import besselj
+    assert integrate(sin(x - 1), x, meijerg=True) == -cos(1 - x)
+    assert integrate(besselj(1, x - 1), x, meijerg=True) == -besselj(0, 1 - x)
+
+
+@slow
+def test_probability():
+    # various integrals from probability theory
+    from sympy.core.function import expand_mul
+    from sympy.core.symbol import (Symbol, symbols)
+    from sympy.simplify.gammasimp import gammasimp
+    from sympy.simplify.powsimp import powsimp
+    mu1, mu2 = symbols('mu1 mu2', nonzero=True)
+    sigma1, sigma2 = symbols('sigma1 sigma2', positive=True)
+    rate = Symbol('lambda', positive=True)
+
+    def normal(x, mu, sigma):
+        return 1/sqrt(2*pi*sigma**2)*exp(-(x - mu)**2/2/sigma**2)
+
+    def exponential(x, rate):
+        return rate*exp(-rate*x)
+
+    assert integrate(normal(x, mu1, sigma1), (x, -oo, oo), meijerg=True) == 1
+    assert integrate(x*normal(x, mu1, sigma1), (x, -oo, oo), meijerg=True) == \
+        mu1
+    assert integrate(x**2*normal(x, mu1, sigma1), (x, -oo, oo), meijerg=True) \
+        == mu1**2 + sigma1**2
+    assert integrate(x**3*normal(x, mu1, sigma1), (x, -oo, oo), meijerg=True) \
+        == mu1**3 + 3*mu1*sigma1**2
+    assert integrate(normal(x, mu1, sigma1)*normal(y, mu2, sigma2),
+                     (x, -oo, oo), (y, -oo, oo), meijerg=True) == 1
+    assert integrate(x*normal(x, mu1, sigma1)*normal(y, mu2, sigma2),
+                     (x, -oo, oo), (y, -oo, oo), meijerg=True) == mu1
+    assert integrate(y*normal(x, mu1, sigma1)*normal(y, mu2, sigma2),
+                     (x, -oo, oo), (y, -oo, oo), meijerg=True) == mu2
+    assert integrate(x*y*normal(x, mu1, sigma1)*normal(y, mu2, sigma2),
+                     (x, -oo, oo), (y, -oo, oo), meijerg=True) == mu1*mu2
+    assert integrate((x + y + 1)*normal(x, mu1, sigma1)*normal(y, mu2, sigma2),
+                     (x, -oo, oo), (y, -oo, oo), meijerg=True) == 1 + mu1 + mu2
+    assert integrate((x + y - 1)*normal(x, mu1, sigma1)*normal(y, mu2, sigma2),
+                     (x, -oo, oo), (y, -oo, oo), meijerg=True) == \
+        -1 + mu1 + mu2
+
+    i = integrate(x**2*normal(x, mu1, sigma1)*normal(y, mu2, sigma2),
+                  (x, -oo, oo), (y, -oo, oo), meijerg=True)
+    assert not i.has(Abs)
+    assert simplify(i) == mu1**2 + sigma1**2
+    assert integrate(y**2*normal(x, mu1, sigma1)*normal(y, mu2, sigma2),
+                     (x, -oo, oo), (y, -oo, oo), meijerg=True) == \
+        sigma2**2 + mu2**2
+
+    assert integrate(exponential(x, rate), (x, 0, oo), meijerg=True) == 1
+    assert integrate(x*exponential(x, rate), (x, 0, oo), meijerg=True) == \
+        1/rate
+    assert integrate(x**2*exponential(x, rate), (x, 0, oo), meijerg=True) == \
+        2/rate**2
+
+    def E(expr):
+        res1 = integrate(expr*exponential(x, rate)*normal(y, mu1, sigma1),
+                         (x, 0, oo), (y, -oo, oo), meijerg=True)
+        res2 = integrate(expr*exponential(x, rate)*normal(y, mu1, sigma1),
+                        (y, -oo, oo), (x, 0, oo), meijerg=True)
+        assert expand_mul(res1) == expand_mul(res2)
+        return res1
+
+    assert E(1) == 1
+    assert E(x*y) == mu1/rate
+    assert E(x*y**2) == mu1**2/rate + sigma1**2/rate
+    ans = sigma1**2 + 1/rate**2
+    assert simplify(E((x + y + 1)**2) - E(x + y + 1)**2) == ans
+    assert simplify(E((x + y - 1)**2) - E(x + y - 1)**2) == ans
+    assert simplify(E((x + y)**2) - E(x + y)**2) == ans
+
+    # Beta' distribution
+    alpha, beta = symbols('alpha beta', positive=True)
+    betadist = x**(alpha - 1)*(1 + x)**(-alpha - beta)*gamma(alpha + beta) \
+        /gamma(alpha)/gamma(beta)
+    assert integrate(betadist, (x, 0, oo), meijerg=True) == 1
+    i = integrate(x*betadist, (x, 0, oo), meijerg=True, conds='separate')
+    assert (gammasimp(i[0]), i[1]) == (alpha/(beta - 1), 1 < beta)
+    j = integrate(x**2*betadist, (x, 0, oo), meijerg=True, conds='separate')
+    assert j[1] == (beta > 2)
+    assert gammasimp(j[0] - i[0]**2) == (alpha + beta - 1)*alpha \
+        /(beta - 2)/(beta - 1)**2
+
+    # Beta distribution
+    # NOTE: this is evaluated using antiderivatives. It also tests that
+    #       meijerint_indefinite returns the simplest possible answer.
+    a, b = symbols('a b', positive=True)
+    betadist = x**(a - 1)*(-x + 1)**(b - 1)*gamma(a + b)/(gamma(a)*gamma(b))
+    assert simplify(integrate(betadist, (x, 0, 1), meijerg=True)) == 1
+    assert simplify(integrate(x*betadist, (x, 0, 1), meijerg=True)) == \
+        a/(a + b)
+    assert simplify(integrate(x**2*betadist, (x, 0, 1), meijerg=True)) == \
+        a*(a + 1)/(a + b)/(a + b + 1)
+    assert simplify(integrate(x**y*betadist, (x, 0, 1), meijerg=True)) == \
+        gamma(a + b)*gamma(a + y)/gamma(a)/gamma(a + b + y)
+
+    # Chi distribution
+    k = Symbol('k', integer=True, positive=True)
+    chi = 2**(1 - k/2)*x**(k - 1)*exp(-x**2/2)/gamma(k/2)
+    assert powsimp(integrate(chi, (x, 0, oo), meijerg=True)) == 1
+    assert simplify(integrate(x*chi, (x, 0, oo), meijerg=True)) == \
+        sqrt(2)*gamma((k + 1)/2)/gamma(k/2)
+    assert simplify(integrate(x**2*chi, (x, 0, oo), meijerg=True)) == k
+
+    # Chi^2 distribution
+    chisquared = 2**(-k/2)/gamma(k/2)*x**(k/2 - 1)*exp(-x/2)
+    assert powsimp(integrate(chisquared, (x, 0, oo), meijerg=True)) == 1
+    assert simplify(integrate(x*chisquared, (x, 0, oo), meijerg=True)) == k
+    assert simplify(integrate(x**2*chisquared, (x, 0, oo), meijerg=True)) == \
+        k*(k + 2)
+    assert gammasimp(integrate(((x - k)/sqrt(2*k))**3*chisquared, (x, 0, oo),
+                    meijerg=True)) == 2*sqrt(2)/sqrt(k)
+
+    # Dagum distribution
+    a, b, p = symbols('a b p', positive=True)
+    # XXX (x/b)**a does not work
+    dagum = a*p/x*(x/b)**(a*p)/(1 + x**a/b**a)**(p + 1)
+    assert simplify(integrate(dagum, (x, 0, oo), meijerg=True)) == 1
+    # XXX conditions are a mess
+    arg = x*dagum
+    assert simplify(integrate(arg, (x, 0, oo), meijerg=True, conds='none')
+                    ) == a*b*gamma(1 - 1/a)*gamma(p + 1 + 1/a)/(
+                    (a*p + 1)*gamma(p))
+    assert simplify(integrate(x*arg, (x, 0, oo), meijerg=True, conds='none')
+                    ) == a*b**2*gamma(1 - 2/a)*gamma(p + 1 + 2/a)/(
+                    (a*p + 2)*gamma(p))
+
+    # F-distribution
+    d1, d2 = symbols('d1 d2', positive=True)
+    f = sqrt(((d1*x)**d1 * d2**d2)/(d1*x + d2)**(d1 + d2))/x \
+        /gamma(d1/2)/gamma(d2/2)*gamma((d1 + d2)/2)
+    assert simplify(integrate(f, (x, 0, oo), meijerg=True)) == 1
+    # TODO conditions are a mess
+    assert simplify(integrate(x*f, (x, 0, oo), meijerg=True, conds='none')
+                    ) == d2/(d2 - 2)
+    assert simplify(integrate(x**2*f, (x, 0, oo), meijerg=True, conds='none')
+                    ) == d2**2*(d1 + 2)/d1/(d2 - 4)/(d2 - 2)
+
+    # TODO gamma, rayleigh
+
+    # inverse gaussian
+    lamda, mu = symbols('lamda mu', positive=True)
+    dist = sqrt(lamda/2/pi)*x**(Rational(-3, 2))*exp(-lamda*(x - mu)**2/x/2/mu**2)
+    mysimp = lambda expr: simplify(expr.rewrite(exp))
+    assert mysimp(integrate(dist, (x, 0, oo))) == 1
+    assert mysimp(integrate(x*dist, (x, 0, oo))) == mu
+    assert mysimp(integrate((x - mu)**2*dist, (x, 0, oo))) == mu**3/lamda
+    assert mysimp(integrate((x - mu)**3*dist, (x, 0, oo))) == 3*mu**5/lamda**2
+
+    # Levi
+    c = Symbol('c', positive=True)
+    assert integrate(sqrt(c/2/pi)*exp(-c/2/(x - mu))/(x - mu)**S('3/2'),
+                    (x, mu, oo)) == 1
+    # higher moments oo
+
+    # log-logistic
+    alpha, beta = symbols('alpha beta', positive=True)
+    distn = (beta/alpha)*x**(beta - 1)/alpha**(beta - 1)/ \
+        (1 + x**beta/alpha**beta)**2
+    # FIXME: If alpha, beta are not declared as finite the line below hangs
+    # after the changes in:
+    #    https://github.com/sympy/sympy/pull/16603
+    assert simplify(integrate(distn, (x, 0, oo))) == 1
+    # NOTE the conditions are a mess, but correctly state beta > 1
+    assert simplify(integrate(x*distn, (x, 0, oo), conds='none')) == \
+        pi*alpha/beta/sin(pi/beta)
+    # (similar comment for conditions applies)
+    assert simplify(integrate(x**y*distn, (x, 0, oo), conds='none')) == \
+        pi*alpha**y*y/beta/sin(pi*y/beta)
+
+    # weibull
+    k = Symbol('k', positive=True)
+    n = Symbol('n', positive=True)
+    distn = k/lamda*(x/lamda)**(k - 1)*exp(-(x/lamda)**k)
+    assert simplify(integrate(distn, (x, 0, oo))) == 1
+    assert simplify(integrate(x**n*distn, (x, 0, oo))) == \
+        lamda**n*gamma(1 + n/k)
+
+    # rice distribution
+    from sympy.functions.special.bessel import besseli
+    nu, sigma = symbols('nu sigma', positive=True)
+    rice = x/sigma**2*exp(-(x**2 + nu**2)/2/sigma**2)*besseli(0, x*nu/sigma**2)
+    assert integrate(rice, (x, 0, oo), meijerg=True) == 1
+    # can someone verify higher moments?
+
+    # Laplace distribution
+    mu = Symbol('mu', real=True)
+    b = Symbol('b', positive=True)
+    laplace = exp(-abs(x - mu)/b)/2/b
+    assert integrate(laplace, (x, -oo, oo), meijerg=True) == 1
+    assert integrate(x*laplace, (x, -oo, oo), meijerg=True) == mu
+    assert integrate(x**2*laplace, (x, -oo, oo), meijerg=True) == \
+        2*b**2 + mu**2
+
+    # TODO are there other distributions supported on (-oo, oo) that we can do?
+
+    # misc tests
+    k = Symbol('k', positive=True)
+    assert gammasimp(expand_mul(integrate(log(x)*x**(k - 1)*exp(-x)/gamma(k),
+                              (x, 0, oo)))) == polygamma(0, k)
+
+
+@slow
+def test_expint():
+    """ Test various exponential integrals. """
+    from sympy.core.symbol import Symbol
+    from sympy.functions.elementary.hyperbolic import sinh
+    from sympy.functions.special.error_functions import (Chi, Ci, Ei, Shi, Si, expint)
+    assert simplify(unpolarify(integrate(exp(-z*x)/x**y, (x, 1, oo),
+                meijerg=True, conds='none'
+                ).rewrite(expint).expand(func=True))) == expint(y, z)
+
+    assert integrate(exp(-z*x)/x, (x, 1, oo), meijerg=True,
+                     conds='none').rewrite(expint).expand() == \
+        expint(1, z)
+    assert integrate(exp(-z*x)/x**2, (x, 1, oo), meijerg=True,
+                     conds='none').rewrite(expint).expand() == \
+        expint(2, z).rewrite(Ei).rewrite(expint)
+    assert integrate(exp(-z*x)/x**3, (x, 1, oo), meijerg=True,
+                     conds='none').rewrite(expint).expand() == \
+        expint(3, z).rewrite(Ei).rewrite(expint).expand()
+
+    t = Symbol('t', positive=True)
+    assert integrate(-cos(x)/x, (x, t, oo), meijerg=True).expand() == Ci(t)
+    assert integrate(-sin(x)/x, (x, t, oo), meijerg=True).expand() == \
+        Si(t) - pi/2
+    assert integrate(sin(x)/x, (x, 0, z), meijerg=True) == Si(z)
+    assert integrate(sinh(x)/x, (x, 0, z), meijerg=True) == Shi(z)
+    assert integrate(exp(-x)/x, x, meijerg=True).expand().rewrite(expint) == \
+        I*pi - expint(1, x)
+    assert integrate(exp(-x)/x**2, x, meijerg=True).rewrite(expint).expand() \
+        == expint(1, x) - exp(-x)/x - I*pi
+
+    u = Symbol('u', polar=True)
+    assert integrate(cos(u)/u, u, meijerg=True).expand().as_independent(u)[1] \
+        == Ci(u)
+    assert integrate(cosh(u)/u, u, meijerg=True).expand().as_independent(u)[1] \
+        == Chi(u)
+
+    assert integrate(expint(1, x), x, meijerg=True
+            ).rewrite(expint).expand() == x*expint(1, x) - exp(-x)
+    assert integrate(expint(2, x), x, meijerg=True
+            ).rewrite(expint).expand() == \
+        -x**2*expint(1, x)/2 + x*exp(-x)/2 - exp(-x)/2
+    assert simplify(unpolarify(integrate(expint(y, x), x,
+                 meijerg=True).rewrite(expint).expand(func=True))) == \
+        -expint(y + 1, x)
+
+    assert integrate(Si(x), x, meijerg=True) == x*Si(x) + cos(x)
+    assert integrate(Ci(u), u, meijerg=True).expand() == u*Ci(u) - sin(u)
+    assert integrate(Shi(x), x, meijerg=True) == x*Shi(x) - cosh(x)
+    assert integrate(Chi(u), u, meijerg=True).expand() == u*Chi(u) - sinh(u)
+
+    assert integrate(Si(x)*exp(-x), (x, 0, oo), meijerg=True) == pi/4
+    assert integrate(expint(1, x)*sin(x), (x, 0, oo), meijerg=True) == log(2)/2
+
+
+def test_messy():
+    from sympy.functions.elementary.hyperbolic import (acosh, acoth)
+    from sympy.functions.elementary.trigonometric import (asin, atan)
+    from sympy.functions.special.bessel import besselj
+    from sympy.functions.special.error_functions import (Chi, E1, Shi, Si)
+    from sympy.integrals.transforms import (fourier_transform, laplace_transform)
+    assert (laplace_transform(Si(x), x, s, simplify=True) ==
+            ((-atan(s) + pi/2)/s, 0, True))
+
+    assert laplace_transform(Shi(x), x, s, simplify=True) == (
+        acoth(s)/s, -oo, s**2 > 1)
+
+    # where should the logs be simplified?
+    assert laplace_transform(Chi(x), x, s, simplify=True) == (
+        (log(s**(-2)) - log(1 - 1/s**2))/(2*s), -oo, s**2 > 1)
+
+    # TODO maybe simplify the inequalities? when the simplification
+    # allows for generators instead of symbols this will work
+    assert laplace_transform(besselj(a, x), x, s)[1:] == \
+        (0, (re(a) > -2) & (re(a) > -1))
+
+    # NOTE s < 0 can be done, but argument reduction is not good enough yet
+    ans = fourier_transform(besselj(1, x)/x, x, s, noconds=False)
+    assert (ans[0].factor(deep=True).expand(), ans[1]) == \
+        (Piecewise((0, (s > 1/(2*pi)) | (s < -1/(2*pi))),
+                   (2*sqrt(-4*pi**2*s**2 + 1), True)), s > 0)
+    # TODO FT(besselj(0,x)) - conditions are messy (but for acceptable reasons)
+    #                       - folding could be better
+
+    assert integrate(E1(x)*besselj(0, x), (x, 0, oo), meijerg=True) == \
+        log(1 + sqrt(2))
+    assert integrate(E1(x)*besselj(1, x), (x, 0, oo), meijerg=True) == \
+        log(S.Half + sqrt(2)/2)
+
+    assert integrate(1/x/sqrt(1 - x**2), x, meijerg=True) == \
+        Piecewise((-acosh(1/x), abs(x**(-2)) > 1), (I*asin(1/x), True))
+
+
+def test_issue_6122():
+    assert integrate(exp(-I*x**2), (x, -oo, oo), meijerg=True) == \
+        -I*sqrt(pi)*exp(I*pi/4)
+
+
+def test_issue_6252():
+    expr = 1/x/(a + b*x)**Rational(1, 3)
+    anti = integrate(expr, x, meijerg=True)
+    assert not anti.has(hyper)
+    # XXX the expression is a mess, but actually upon differentiation and
+    # putting in numerical values seems to work...
+
+
+def test_issue_6348():
+    assert integrate(exp(I*x)/(1 + x**2), (x, -oo, oo)).simplify().rewrite(exp) \
+        == pi*exp(-1)
+
+
+def test_fresnel():
+    from sympy.functions.special.error_functions import (fresnelc, fresnels)
+
+    assert expand_func(integrate(sin(pi*x**2/2), x)) == fresnels(x)
+    assert expand_func(integrate(cos(pi*x**2/2), x)) == fresnelc(x)
+
+
+def test_issue_6860():
+    assert meijerint_indefinite(x**x**x, x) is None
+
+
+def test_issue_7337():
+    f = meijerint_indefinite(x*sqrt(2*x + 3), x).together()
+    assert f == sqrt(2*x + 3)*(2*x**2 + x - 3)/5
+    assert f._eval_interval(x, S.NegativeOne, S.One) == Rational(2, 5)
+
+
+def test_issue_8368():
+    assert meijerint_indefinite(cosh(x)*exp(-x*t), x) == (
+        (-t - 1)*exp(x) + (-t + 1)*exp(-x))*exp(-t*x)/2/(t**2 - 1)
+
+
+def test_issue_10211():
+    from sympy.abc import h, w
+    assert integrate((1/sqrt((y-x)**2 + h**2)**3), (x,0,w), (y,0,w)) == \
+        2*sqrt(1 + w**2/h**2)/h - 2/h
+
+
+def test_issue_11806():
+    from sympy.core.symbol import symbols
+    y, L = symbols('y L', positive=True)
+    assert integrate(1/sqrt(x**2 + y**2)**3, (x, -L, L)) == \
+        2*L/(y**2*sqrt(L**2 + y**2))
+
+def test_issue_10681():
+    from sympy.polys.domains.realfield import RR
+    from sympy.abc import R, r
+    f = integrate(r**2*(R**2-r**2)**0.5, r, meijerg=True)
+    g = (1.0/3)*R**1.0*r**3*hyper((-0.5, Rational(3, 2)), (Rational(5, 2),),
+                                  r**2*exp_polar(2*I*pi)/R**2)
+    assert RR.almosteq((f/g).n(), 1.0, 1e-12)
+
+def test_issue_13536():
+    from sympy.core.symbol import Symbol
+    a = Symbol('a', positive=True)
+    assert integrate(1/x**2, (x, oo, a)) == -1/a
+
+
+def test_issue_6462():
+    from sympy.core.symbol import Symbol
+    x = Symbol('x')
+    n = Symbol('n')
+    # Not the actual issue, still wrong answer for n = 1, but that there is no
+    # exception
+    assert integrate(cos(x**n)/x**n, x, meijerg=True).subs(n, 2).equals(
+            integrate(cos(x**2)/x**2, x, meijerg=True))
+
+
+def test_indefinite_1_bug():
+    assert integrate((b + t)**(-a), t, meijerg=True) == -b*(1 + t/b)**(1 - a)/(a*b**a - b**a)
+
+
+def test_pr_23583():
+    # This result is wrong. Check whether new result is correct when this test fail.
+    assert integrate(1/sqrt((x - I)**2-1), meijerg=True) == \
+           Piecewise((acosh(x - I), Abs((x - I)**2) > 1), (-I*asin(x - I), True))
+
+
+# 25786
+def test_integrate_function_of_square_over_negatives():
+    assert integrate(exp(-x**2), (x,-5,0), meijerg=True) == sqrt(pi)/2 * erf(5)
+
+
+def test_issue_25949():
+    from sympy.core.symbol import symbols
+    y = symbols("y", nonzero=True)
+    assert integrate(cosh(y*(x + 1)), (x, -1, -0.25), meijerg=True) == sinh(0.75*y)/y

rl/Lib/site-packages/sympy/integrals/tests/test_prde.py

@@ -0,0 +1,322 @@
+"""Most of these tests come from the examples in Bronstein's book."""
+from sympy.integrals.risch import DifferentialExtension, derivation
+from sympy.integrals.prde import (prde_normal_denom, prde_special_denom,
+    prde_linear_constraints, constant_system, prde_spde, prde_no_cancel_b_large,
+    prde_no_cancel_b_small, limited_integrate_reduce, limited_integrate,
+    is_deriv_k, is_log_deriv_k_t_radical, parametric_log_deriv_heu,
+    is_log_deriv_k_t_radical_in_field, param_poly_rischDE, param_rischDE,
+    prde_cancel_liouvillian)
+
+from sympy.polys.polymatrix import PolyMatrix as Matrix
+
+from sympy.core.numbers import Rational
+from sympy.core.singleton import S
+from sympy.core.symbol import symbols
+from sympy.polys.domains.rationalfield import QQ
+from sympy.polys.polytools import Poly
+from sympy.abc import x, t, n
+
+t0, t1, t2, t3, k = symbols('t:4 k')
+
+
+def test_prde_normal_denom():
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1 + t**2, t)]})
+    fa = Poly(1, t)
+    fd = Poly(x, t)
+    G = [(Poly(t, t), Poly(1 + t**2, t)), (Poly(1, t), Poly(x + x*t**2, t))]
+    assert prde_normal_denom(fa, fd, G, DE) == \
+        (Poly(x, t, domain='ZZ(x)'), (Poly(1, t, domain='ZZ(x)'), Poly(1, t,
+            domain='ZZ(x)')), [(Poly(x*t, t, domain='ZZ(x)'),
+         Poly(t**2 + 1, t, domain='ZZ(x)')), (Poly(1, t, domain='ZZ(x)'),
+             Poly(t**2 + 1, t, domain='ZZ(x)'))], Poly(1, t, domain='ZZ(x)'))
+    G = [(Poly(t, t), Poly(t**2 + 2*t + 1, t)), (Poly(x*t, t),
+        Poly(t**2 + 2*t + 1, t)), (Poly(x*t**2, t), Poly(t**2 + 2*t + 1, t))]
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]})
+    assert prde_normal_denom(Poly(x, t), Poly(1, t), G, DE) == \
+        (Poly(t + 1, t), (Poly((-1 + x)*t + x, t), Poly(1, t, domain='ZZ[x]')), [(Poly(t, t),
+        Poly(1, t)), (Poly(x*t, t), Poly(1, t, domain='ZZ[x]')), (Poly(x*t**2, t),
+        Poly(1, t, domain='ZZ[x]'))], Poly(t + 1, t))
+
+
+def test_prde_special_denom():
+    a = Poly(t + 1, t)
+    ba = Poly(t**2, t)
+    bd = Poly(1, t)
+    G = [(Poly(t, t), Poly(1, t)), (Poly(t**2, t), Poly(1, t)), (Poly(t**3, t), Poly(1, t))]
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]})
+    assert prde_special_denom(a, ba, bd, G, DE) == \
+        (Poly(t + 1, t), Poly(t**2, t), [(Poly(t, t), Poly(1, t)),
+        (Poly(t**2, t), Poly(1, t)), (Poly(t**3, t), Poly(1, t))], Poly(1, t))
+    G = [(Poly(t, t), Poly(1, t)), (Poly(1, t), Poly(t, t))]
+    assert prde_special_denom(Poly(1, t), Poly(t**2, t), Poly(1, t), G, DE) == \
+        (Poly(1, t), Poly(t**2 - 1, t), [(Poly(t**2, t), Poly(1, t)),
+        (Poly(1, t), Poly(1, t))], Poly(t, t))
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(-2*x*t0, t0)]})
+    DE.decrement_level()
+    G = [(Poly(t, t), Poly(t**2, t)), (Poly(2*t, t), Poly(t, t))]
+    assert prde_special_denom(Poly(5*x*t + 1, t), Poly(t**2 + 2*x**3*t, t), Poly(t**3 + 2, t), G, DE) == \
+        (Poly(5*x*t + 1, t), Poly(0, t, domain='ZZ[x]'), [(Poly(t, t), Poly(t**2, t)),
+        (Poly(2*t, t), Poly(t, t))], Poly(1, x))
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly((t**2 + 1)*2*x, t)]})
+    G = [(Poly(t + x, t), Poly(t*x, t)), (Poly(2*t, t), Poly(x**2, x))]
+    assert prde_special_denom(Poly(5*x*t + 1, t), Poly(t**2 + 2*x**3*t, t), Poly(t**3, t), G, DE) == \
+        (Poly(5*x*t + 1, t), Poly(0, t, domain='ZZ[x]'), [(Poly(t + x, t), Poly(x*t, t)),
+        (Poly(2*t, t, x), Poly(x**2, t, x))], Poly(1, t))
+    assert prde_special_denom(Poly(t + 1, t), Poly(t**2, t), Poly(t**3, t), G, DE) == \
+        (Poly(t + 1, t), Poly(0, t, domain='ZZ[x]'), [(Poly(t + x, t), Poly(x*t, t)), (Poly(2*t, t, x),
+        Poly(x**2, t, x))], Poly(1, t))
+
+
+def test_prde_linear_constraints():
+    DE = DifferentialExtension(extension={'D': [Poly(1, x)]})
+    G = [(Poly(2*x**3 + 3*x + 1, x), Poly(x**2 - 1, x)), (Poly(1, x), Poly(x - 1, x)),
+        (Poly(1, x), Poly(x + 1, x))]
+    assert prde_linear_constraints(Poly(1, x), Poly(0, x), G, DE) == \
+        ((Poly(2*x, x, domain='QQ'), Poly(0, x, domain='QQ'), Poly(0, x, domain='QQ')),
+            Matrix([[1, 1, -1], [5, 1, 1]], x))
+    G = [(Poly(t, t), Poly(1, t)), (Poly(t**2, t), Poly(1, t)), (Poly(t**3, t), Poly(1, t))]
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]})
+    assert prde_linear_constraints(Poly(t + 1, t), Poly(t**2, t), G, DE) == \
+        ((Poly(t, t, domain='QQ'), Poly(t**2, t, domain='QQ'), Poly(t**3, t, domain='QQ')),
+            Matrix(0, 3, [], t))
+    G = [(Poly(2*x, t), Poly(t, t)), (Poly(-x, t), Poly(t, t))]
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t)]})
+    assert prde_linear_constraints(Poly(1, t), Poly(0, t), G, DE) == \
+        ((Poly(0, t, domain='QQ[x]'), Poly(0, t, domain='QQ[x]')), Matrix([[2*x, -x]], t))
+
+
+def test_constant_system():
+    A = Matrix([[-(x + 3)/(x - 1), (x + 1)/(x - 1), 1],
+                [-x - 3, x + 1, x - 1],
+                [2*(x + 3)/(x - 1), 0, 0]], t)
+    u = Matrix([[(x + 1)/(x - 1)], [x + 1], [0]], t)
+    DE = DifferentialExtension(extension={'D': [Poly(1, x)]})
+    R = QQ.frac_field(x)[t]
+    assert constant_system(A, u, DE) == \
+        (Matrix([[1, 0, 0],
+                 [0, 1, 0],
+                 [0, 0, 0],
+                 [0, 0, 1]], ring=R), Matrix([0, 1, 0, 0], ring=R))
+
+
+def test_prde_spde():
+    D = [Poly(x, t), Poly(-x*t, t)]
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t)]})
+    # TODO: when bound_degree() can handle this, test degree bound from that too
+    assert prde_spde(Poly(t, t), Poly(-1/x, t), D, n, DE) == \
+        (Poly(t, t), Poly(0, t, domain='ZZ(x)'),
+        [Poly(2*x, t, domain='ZZ(x)'), Poly(-x, t, domain='ZZ(x)')],
+        [Poly(-x**2, t, domain='ZZ(x)'), Poly(0, t, domain='ZZ(x)')], n - 1)
+
+
+def test_prde_no_cancel():
+    # b large
+    DE = DifferentialExtension(extension={'D': [Poly(1, x)]})
+    assert prde_no_cancel_b_large(Poly(1, x), [Poly(x**2, x), Poly(1, x)], 2, DE) == \
+        ([Poly(x**2 - 2*x + 2, x), Poly(1, x)], Matrix([[1, 0, -1, 0],
+                                                        [0, 1, 0, -1]], x))
+    assert prde_no_cancel_b_large(Poly(1, x), [Poly(x**3, x), Poly(1, x)], 3, DE) == \
+        ([Poly(x**3 - 3*x**2 + 6*x - 6, x), Poly(1, x)], Matrix([[1, 0, -1, 0],
+                                                                 [0, 1, 0, -1]], x))
+    assert prde_no_cancel_b_large(Poly(x, x), [Poly(x**2, x), Poly(1, x)], 1, DE) == \
+        ([Poly(x, x, domain='ZZ'), Poly(0, x, domain='ZZ')], Matrix([[1, -1,  0,  0],
+                                                                    [1,  0, -1,  0],
+                                                                    [0,  1,  0, -1]], x))
+    # b small
+    # XXX: Is there a better example of a monomial with D.degree() > 2?
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**3 + 1, t)]})
+
+    # My original q was t**4 + t + 1, but this solution implies q == t**4
+    # (c1 = 4), with some of the ci for the original q equal to 0.
+    G = [Poly(t**6, t), Poly(x*t**5, t), Poly(t**3, t), Poly(x*t**2, t), Poly(1 + x, t)]
+    R = QQ.frac_field(x)[t]
+    assert prde_no_cancel_b_small(Poly(x*t, t), G, 4, DE) == \
+        ([Poly(t**4/4 - x/12*t**3 + x**2/24*t**2 + (Rational(-11, 12) - x**3/24)*t + x/24, t),
+        Poly(x/3*t**3 - x**2/6*t**2 + (Rational(-1, 3) + x**3/6)*t - x/6, t), Poly(t, t),
+        Poly(0, t), Poly(0, t)], Matrix([[1, 0,              -1, 0, 0,  0,  0,  0,  0,  0],
+                                         [0, 1, Rational(-1, 4), 0, 0,  0,  0,  0,  0,  0],
+                                         [0, 0,               0, 0, 0,  0,  0,  0,  0,  0],
+                                         [0, 0,               0, 1, 0,  0,  0,  0,  0,  0],
+                                         [0, 0,               0, 0, 1,  0,  0,  0,  0,  0],
+                                         [1, 0,               0, 0, 0, -1,  0,  0,  0,  0],
+                                         [0, 1,               0, 0, 0,  0, -1,  0,  0,  0],
+                                         [0, 0,               1, 0, 0,  0,  0, -1,  0,  0],
+                                         [0, 0,               0, 1, 0,  0,  0,  0, -1,  0],
+                                         [0, 0,               0, 0, 1,  0,  0,  0,  0, -1]], ring=R))
+
+    # TODO: Add test for deg(b) <= 0 with b small
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1 + t**2, t)]})
+    b = Poly(-1/x**2, t, field=True)  # deg(b) == 0
+    q = [Poly(x**i*t**j, t, field=True) for i in range(2) for j in range(3)]
+    h, A = prde_no_cancel_b_small(b, q, 3, DE)
+    V = A.nullspace()
+    R = QQ.frac_field(x)[t]
+    assert len(V) == 1
+    assert V[0] == Matrix([Rational(-1, 2), 0, 0, 1, 0, 0]*3, ring=R)
+    assert (Matrix([h])*V[0][6:, :])[0] == Poly(x**2/2, t, domain='QQ(x)')
+    assert (Matrix([q])*V[0][:6, :])[0] == Poly(x - S.Half, t, domain='QQ(x)')
+
+
+def test_prde_cancel_liouvillian():
+    ### 1. case == 'primitive'
+    # used when integrating f = log(x) - log(x - 1)
+    # Not taken from 'the' book
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t)]})
+    p0 = Poly(0, t, field=True)
+    p1 = Poly((x - 1)*t, t, domain='ZZ(x)')
+    p2 = Poly(x - 1, t, domain='ZZ(x)')
+    p3 = Poly(-x**2 + x, t, domain='ZZ(x)')
+    h, A = prde_cancel_liouvillian(Poly(-1/(x - 1), t), [Poly(-x + 1, t), Poly(1, t)], 1, DE)
+    V = A.nullspace()
+    assert h == [p0, p0, p1, p0, p0, p0, p0, p0, p0, p0, p2, p3, p0, p0, p0, p0]
+    assert A.rank() == 16
+    assert (Matrix([h])*V[0][:16, :]) == Matrix([[Poly(0, t, domain='QQ(x)')]])
+
+    ### 2. case == 'exp'
+    # used when integrating log(x/exp(x) + 1)
+    # Not taken from book
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(-t, t)]})
+    assert prde_cancel_liouvillian(Poly(0, t, domain='QQ[x]'), [Poly(1, t, domain='QQ(x)')], 0, DE) == \
+            ([Poly(1, t, domain='QQ'), Poly(x, t, domain='ZZ(x)')], Matrix([[-1, 0, 1]], DE.t))
+
+
+def test_param_poly_rischDE():
+    DE = DifferentialExtension(extension={'D': [Poly(1, x)]})
+    a = Poly(x**2 - x, x, field=True)
+    b = Poly(1, x, field=True)
+    q = [Poly(x, x, field=True), Poly(x**2, x, field=True)]
+    h, A = param_poly_rischDE(a, b, q, 3, DE)
+
+    assert A.nullspace() == [Matrix([0, 1, 1, 1], DE.t)]  # c1, c2, d1, d2
+    # Solution of a*Dp + b*p = c1*q1 + c2*q2 = q2 = x**2
+    # is d1*h1 + d2*h2 = h1 + h2 = x.
+    assert h[0] + h[1] == Poly(x, x, domain='QQ')
+    # a*Dp + b*p = q1 = x has no solution.
+
+    a = Poly(x**2 - x, x, field=True)
+    b = Poly(x**2 - 5*x + 3, x, field=True)
+    q = [Poly(1, x, field=True), Poly(x, x, field=True),
+         Poly(x**2, x, field=True)]
+    h, A = param_poly_rischDE(a, b, q, 3, DE)
+
+    assert A.nullspace() == [Matrix([3, -5, 1, -5, 1, 1], DE.t)]
+    p = -Poly(5, DE.t)*h[0] + h[1] + h[2]  # Poly(1, x)
+    assert a*derivation(p, DE) + b*p == Poly(x**2 - 5*x + 3, x, domain='QQ')
+
+
+def test_param_rischDE():
+    DE = DifferentialExtension(extension={'D': [Poly(1, x)]})
+    p1, px = Poly(1, x, field=True), Poly(x, x, field=True)
+    G = [(p1, px), (p1, p1), (px, p1)]  # [1/x, 1, x]
+    h, A = param_rischDE(-p1, Poly(x**2, x, field=True), G, DE)
+    assert len(h) == 3
+    p = [hi[0].as_expr()/hi[1].as_expr() for hi in h]
+    V = A.nullspace()
+    assert len(V) == 2
+    assert V[0] == Matrix([-1, 1, 0, -1, 1, 0], DE.t)
+    y = -p[0] + p[1] + 0*p[2]  # x
+    assert y.diff(x) - y/x**2 == 1 - 1/x  # Dy + f*y == -G0 + G1 + 0*G2
+
+    # the below test computation takes place while computing the integral
+    # of 'f = log(log(x + exp(x)))'
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]})
+    G = [(Poly(t + x, t, domain='ZZ(x)'), Poly(1, t, domain='QQ')), (Poly(0, t, domain='QQ'), Poly(1, t, domain='QQ'))]
+    h, A = param_rischDE(Poly(-t - 1, t, field=True), Poly(t + x, t, field=True), G, DE)
+    assert len(h) == 5
+    p = [hi[0].as_expr()/hi[1].as_expr() for hi in h]
+    V = A.nullspace()
+    assert len(V) == 3
+    assert V[0] == Matrix([0, 0, 0, 0, 1, 0, 0], DE.t)
+    y = 0*p[0] + 0*p[1] + 1*p[2] + 0*p[3] + 0*p[4]
+    assert y.diff(t) - y/(t + x) == 0   # Dy + f*y = 0*G0 + 0*G1
+
+
+def test_limited_integrate_reduce():
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t)]})
+    assert limited_integrate_reduce(Poly(x, t), Poly(t**2, t), [(Poly(x, t),
+    Poly(t, t))], DE) == \
+        (Poly(t, t), Poly(-1/x, t), Poly(t, t), 1, (Poly(x, t), Poly(1, t, domain='ZZ[x]')),
+        [(Poly(-x*t, t), Poly(1, t, domain='ZZ[x]'))])
+
+
+def test_limited_integrate():
+    DE = DifferentialExtension(extension={'D': [Poly(1, x)]})
+    G = [(Poly(x, x), Poly(x + 1, x))]
+    assert limited_integrate(Poly(-(1 + x + 5*x**2 - 3*x**3), x),
+    Poly(1 - x - x**2 + x**3, x), G, DE) == \
+        ((Poly(x**2 - x + 2, x), Poly(x - 1, x, domain='QQ')), [2])
+    G = [(Poly(1, x), Poly(x, x))]
+    assert limited_integrate(Poly(5*x**2, x), Poly(3, x), G, DE) == \
+        ((Poly(5*x**3/9, x), Poly(1, x, domain='QQ')), [0])
+
+
+def test_is_log_deriv_k_t_radical():
+    DE = DifferentialExtension(extension={'D': [Poly(1, x)], 'exts': [None],
+        'extargs': [None]})
+    assert is_log_deriv_k_t_radical(Poly(2*x, x), Poly(1, x), DE) is None
+
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(2*t1, t1), Poly(1/x, t2)],
+        'exts': [None, 'exp', 'log'], 'extargs': [None, 2*x, x]})
+    assert is_log_deriv_k_t_radical(Poly(x + t2/2, t2), Poly(1, t2), DE) == \
+        ([(t1, 1), (x, 1)], t1*x, 2, 0)
+    # TODO: Add more tests
+
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t0, t0), Poly(1/x, t)],
+        'exts': [None, 'exp', 'log'], 'extargs': [None, x, x]})
+    assert is_log_deriv_k_t_radical(Poly(x + t/2 + 3, t), Poly(1, t), DE) == \
+        ([(t0, 2), (x, 1)], x*t0**2, 2, 3)
+
+
+def test_is_deriv_k():
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t1), Poly(1/(x + 1), t2)],
+        'exts': [None, 'log', 'log'], 'extargs': [None, x, x + 1]})
+    assert is_deriv_k(Poly(2*x**2 + 2*x, t2), Poly(1, t2), DE) == \
+        ([(t1, 1), (t2, 1)], t1 + t2, 2)
+
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t1), Poly(t2, t2)],
+        'exts': [None, 'log', 'exp'], 'extargs': [None, x, x]})
+    assert is_deriv_k(Poly(x**2*t2**3, t2), Poly(1, t2), DE) == \
+        ([(x, 3), (t1, 2)], 2*t1 + 3*x, 1)
+    # TODO: Add more tests, including ones with exponentials
+
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(2/x, t1)],
+        'exts': [None, 'log'], 'extargs': [None, x**2]})
+    assert is_deriv_k(Poly(x, t1), Poly(1, t1), DE) == \
+        ([(t1, S.Half)], t1/2, 1)
+
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(2/(1 + x), t0)],
+        'exts': [None, 'log'], 'extargs': [None, x**2 + 2*x + 1]})
+    assert is_deriv_k(Poly(1 + x, t0), Poly(1, t0), DE) == \
+        ([(t0, S.Half)], t0/2, 1)
+
+    # Issue 10798
+    # DE = DifferentialExtension(log(1/x), x)
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(-1/x, t)],
+        'exts': [None, 'log'], 'extargs': [None, 1/x]})
+    assert is_deriv_k(Poly(1, t), Poly(x, t), DE) == ([(t, 1)], t, 1)
+
+
+def test_is_log_deriv_k_t_radical_in_field():
+    # NOTE: any potential constant factor in the second element of the result
+    # doesn't matter, because it cancels in Da/a.
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t)]})
+    assert is_log_deriv_k_t_radical_in_field(Poly(5*t + 1, t), Poly(2*t*x, t), DE) == \
+        (2, t*x**5)
+    assert is_log_deriv_k_t_radical_in_field(Poly(2 + 3*t, t), Poly(5*x*t, t), DE) == \
+        (5, x**3*t**2)
+
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(-t/x**2, t)]})
+    assert is_log_deriv_k_t_radical_in_field(Poly(-(1 + 2*t), t),
+    Poly(2*x**2 + 2*x**2*t, t), DE) == \
+        (2, t + t**2)
+    assert is_log_deriv_k_t_radical_in_field(Poly(-1, t), Poly(x**2, t), DE) == \
+        (1, t)
+    assert is_log_deriv_k_t_radical_in_field(Poly(1, t), Poly(2*x**2, t), DE) == \
+        (2, 1/t)
+
+
+def test_parametric_log_deriv():
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t)]})
+    assert parametric_log_deriv_heu(Poly(5*t**2 + t - 6, t), Poly(2*x*t**2, t),
+    Poly(-1, t), Poly(x*t**2, t), DE) == \
+        (2, 6, t*x**5)

rl/Lib/site-packages/sympy/integrals/tests/test_quadrature.py

@@ -0,0 +1,601 @@
+from sympy.core import S, Rational
+from sympy.integrals.quadrature import (gauss_legendre, gauss_laguerre,
+                                        gauss_hermite, gauss_gen_laguerre,
+                                        gauss_chebyshev_t, gauss_chebyshev_u,
+                                        gauss_jacobi, gauss_lobatto)
+
+
+def test_legendre():
+    x, w = gauss_legendre(1, 17)
+    assert [str(r) for r in x] == ['0']
+    assert [str(r) for r in w] == ['2.0000000000000000']
+
+    x, w = gauss_legendre(2, 17)
+    assert [str(r) for r in x] == [
+            '-0.57735026918962576',
+            '0.57735026918962576']
+    assert [str(r) for r in w] == [
+            '1.0000000000000000',
+            '1.0000000000000000']
+
+    x, w = gauss_legendre(3, 17)
+    assert [str(r) for r in x] == [
+            '-0.77459666924148338',
+            '0',
+            '0.77459666924148338']
+    assert [str(r) for r in w] == [
+            '0.55555555555555556',
+            '0.88888888888888889',
+            '0.55555555555555556']
+
+    x, w = gauss_legendre(4, 17)
+    assert [str(r) for r in x] == [
+            '-0.86113631159405258',
+            '-0.33998104358485626',
+            '0.33998104358485626',
+            '0.86113631159405258']
+    assert [str(r) for r in w] == [
+            '0.34785484513745386',
+            '0.65214515486254614',
+            '0.65214515486254614',
+            '0.34785484513745386']
+
+
+def test_legendre_precise():
+    x, w = gauss_legendre(3, 40)
+    assert [str(r) for r in x] == [
+            '-0.7745966692414833770358530799564799221666',
+            '0',
+            '0.7745966692414833770358530799564799221666']
+    assert [str(r) for r in w] == [
+            '0.5555555555555555555555555555555555555556',
+            '0.8888888888888888888888888888888888888889',
+            '0.5555555555555555555555555555555555555556']
+
+
+def test_laguerre():
+    x, w = gauss_laguerre(1, 17)
+    assert [str(r) for r in x] == ['1.0000000000000000']
+    assert [str(r) for r in w] == ['1.0000000000000000']
+
+    x, w = gauss_laguerre(2, 17)
+    assert [str(r) for r in x] == [
+            '0.58578643762690495',
+            '3.4142135623730950']
+    assert [str(r) for r in w] == [
+            '0.85355339059327376',
+            '0.14644660940672624']
+
+    x, w = gauss_laguerre(3, 17)
+    assert [str(r) for r in x] == [
+            '0.41577455678347908',
+            '2.2942803602790417',
+            '6.2899450829374792',
+            ]
+    assert [str(r) for r in w] == [
+            '0.71109300992917302',
+            '0.27851773356924085',
+            '0.010389256501586136',
+            ]
+
+    x, w = gauss_laguerre(4, 17)
+    assert [str(r) for r in x] == [
+            '0.32254768961939231',
+            '1.7457611011583466',
+            '4.5366202969211280',
+            '9.3950709123011331']
+    assert [str(r) for r in w] == [
+            '0.60315410434163360',
+            '0.35741869243779969',
+            '0.038887908515005384',
+            '0.00053929470556132745']
+
+    x, w = gauss_laguerre(5, 17)
+    assert [str(r) for r in x] == [
+            '0.26356031971814091',
+            '1.4134030591065168',
+            '3.5964257710407221',
+            '7.0858100058588376',
+            '12.640800844275783']
+    assert [str(r) for r in w] == [
+            '0.52175561058280865',
+            '0.39866681108317593',
+            '0.075942449681707595',
+            '0.0036117586799220485',
+            '2.3369972385776228e-5']
+
+
+def test_laguerre_precise():
+    x, w = gauss_laguerre(3, 40)
+    assert [str(r) for r in x] == [
+            '0.4157745567834790833115338731282744735466',
+            '2.294280360279041719822050361359593868960',
+            '6.289945082937479196866415765512131657493']
+    assert [str(r) for r in w] == [
+            '0.7110930099291730154495901911425944313094',
+            '0.2785177335692408488014448884567264810349',
+            '0.01038925650158613574896492040067908765572']
+
+
+def test_hermite():
+    x, w = gauss_hermite(1, 17)
+    assert [str(r) for r in x] == ['0']
+    assert [str(r) for r in w] == ['1.7724538509055160']
+
+    x, w = gauss_hermite(2, 17)
+    assert [str(r) for r in x] == [
+            '-0.70710678118654752',
+            '0.70710678118654752']
+    assert [str(r) for r in w] == [
+            '0.88622692545275801',
+            '0.88622692545275801']
+
+    x, w = gauss_hermite(3, 17)
+    assert [str(r) for r in x] == [
+            '-1.2247448713915890',
+            '0',
+            '1.2247448713915890']
+    assert [str(r) for r in w] == [
+            '0.29540897515091934',
+            '1.1816359006036774',
+            '0.29540897515091934']
+
+    x, w = gauss_hermite(4, 17)
+    assert [str(r) for r in x] == [
+            '-1.6506801238857846',
+            '-0.52464762327529032',
+            '0.52464762327529032',
+            '1.6506801238857846']
+    assert [str(r) for r in w] == [
+            '0.081312835447245177',
+            '0.80491409000551284',
+            '0.80491409000551284',
+            '0.081312835447245177']
+
+    x, w = gauss_hermite(5, 17)
+    assert [str(r) for r in x] == [
+            '-2.0201828704560856',
+            '-0.95857246461381851',
+            '0',
+            '0.95857246461381851',
+            '2.0201828704560856']
+    assert [str(r) for r in w] == [
+            '0.019953242059045913',
+            '0.39361932315224116',
+            '0.94530872048294188',
+            '0.39361932315224116',
+            '0.019953242059045913']
+
+
+def test_hermite_precise():
+    x, w = gauss_hermite(3, 40)
+    assert [str(r) for r in x] == [
+        '-1.224744871391589049098642037352945695983',
+        '0',
+        '1.224744871391589049098642037352945695983']
+    assert [str(r) for r in w] == [
+        '0.2954089751509193378830279138901908637996',
+        '1.181635900603677351532111655560763455198',
+        '0.2954089751509193378830279138901908637996']
+
+
+def test_gen_laguerre():
+    x, w = gauss_gen_laguerre(1, Rational(-1, 2), 17)
+    assert [str(r) for r in x] == ['0.50000000000000000']
+    assert [str(r) for r in w] == ['1.7724538509055160']
+
+    x, w = gauss_gen_laguerre(2, Rational(-1, 2), 17)
+    assert [str(r) for r in x] == [
+            '0.27525512860841095',
+            '2.7247448713915890']
+    assert [str(r) for r in w] == [
+            '1.6098281800110257',
+            '0.16262567089449035']
+
+    x, w = gauss_gen_laguerre(3, Rational(-1, 2), 17)
+    assert [str(r) for r in x] == [
+            '0.19016350919348813',
+            '1.7844927485432516',
+            '5.5253437422632603']
+    assert [str(r) for r in w] == [
+            '1.4492591904487850',
+            '0.31413464064571329',
+            '0.0090600198110176913']
+
+    x, w = gauss_gen_laguerre(4, Rational(-1, 2), 17)
+    assert [str(r) for r in x] == [
+            '0.14530352150331709',
+            '1.3390972881263614',
+            '3.9269635013582872',
+            '8.5886356890120343']
+    assert [str(r) for r in w] == [
+            '1.3222940251164826',
+            '0.41560465162978376',
+            '0.034155966014826951',
+            '0.00039920814442273524']
+
+    x, w = gauss_gen_laguerre(5, Rational(-1, 2), 17)
+    assert [str(r) for r in x] == [
+            '0.11758132021177814',
+            '1.0745620124369040',
+            '3.0859374437175500',
+            '6.4147297336620305',
+            '11.807189489971737']
+    assert [str(r) for r in w] == [
+            '1.2217252674706516',
+            '0.48027722216462937',
+            '0.067748788910962126',
+            '0.0026872914935624654',
+            '1.5280865710465241e-5']
+
+    x, w = gauss_gen_laguerre(1, 2, 17)
+    assert [str(r) for r in x] == ['3.0000000000000000']
+    assert [str(r) for r in w] == ['2.0000000000000000']
+
+    x, w = gauss_gen_laguerre(2, 2, 17)
+    assert [str(r) for r in x] == [
+            '2.0000000000000000',
+            '6.0000000000000000']
+    assert [str(r) for r in w] == [
+            '1.5000000000000000',
+            '0.50000000000000000']
+
+    x, w = gauss_gen_laguerre(3, 2, 17)
+    assert [str(r) for r in x] == [
+            '1.5173870806774125',
+            '4.3115831337195203',
+            '9.1710297856030672']
+    assert [str(r) for r in w] == [
+            '1.0374949614904253',
+            '0.90575000470306537',
+            '0.056755033806509347']
+
+    x, w = gauss_gen_laguerre(4, 2, 17)
+    assert [str(r) for r in x] == [
+            '1.2267632635003021',
+            '3.4125073586969460',
+            '6.9026926058516134',
+            '12.458036771951139']
+    assert [str(r) for r in w] == [
+            '0.72552499769865438',
+            '1.0634242919791946',
+            '0.20669613102835355',
+            '0.0043545792937974889']
+
+    x, w = gauss_gen_laguerre(5, 2, 17)
+    assert [str(r) for r in x] == [
+            '1.0311091440933816',
+            '2.8372128239538217',
+            '5.6202942725987079',
+            '9.6829098376640271',
+            '15.828473921690062']
+    assert [str(r) for r in w] == [
+            '0.52091739683509184',
+            '1.0667059331592211',
+            '0.38354972366693113',
+            '0.028564233532974658',
+            '0.00026271280578124935']
+
+
+def test_gen_laguerre_precise():
+    x, w = gauss_gen_laguerre(3, Rational(-1, 2), 40)
+    assert [str(r) for r in x] == [
+            '0.1901635091934881328718554276203028970878',
+            '1.784492748543251591186722461957367638500',
+            '5.525343742263260275941422110422329464413']
+    assert [str(r) for r in w] == [
+            '1.449259190448785048183829411195134343108',
+            '0.3141346406457132878326231270167565378246',
+            '0.009060019811017691281714945129254301865020']
+
+    x, w = gauss_gen_laguerre(3, 2, 40)
+    assert [str(r) for r in x] == [
+            '1.517387080677412495020323111016672547482',
+            '4.311583133719520302881184669723530562299',
+            '9.171029785603067202098492219259796890218']
+    assert [str(r) for r in w] == [
+            '1.037494961490425285817554606541269153041',
+            '0.9057500047030653669269785048806009945254',
+            '0.05675503380650934725546688857812985243312']
+
+
+def test_chebyshev_t():
+    x, w = gauss_chebyshev_t(1, 17)
+    assert [str(r) for r in x] == ['0']
+    assert [str(r) for r in w] == ['3.1415926535897932']
+
+    x, w = gauss_chebyshev_t(2, 17)
+    assert [str(r) for r in x] == [
+            '0.70710678118654752',
+            '-0.70710678118654752']
+    assert [str(r) for r in w] == [
+            '1.5707963267948966',
+            '1.5707963267948966']
+
+    x, w = gauss_chebyshev_t(3, 17)
+    assert [str(r) for r in x] == [
+            '0.86602540378443865',
+            '0',
+            '-0.86602540378443865']
+    assert [str(r) for r in w] == [
+            '1.0471975511965977',
+            '1.0471975511965977',
+            '1.0471975511965977']
+
+    x, w = gauss_chebyshev_t(4, 17)
+    assert [str(r) for r in x] == [
+            '0.92387953251128676',
+            '0.38268343236508977',
+            '-0.38268343236508977',
+            '-0.92387953251128676']
+    assert [str(r) for r in w] == [
+            '0.78539816339744831',
+            '0.78539816339744831',
+            '0.78539816339744831',
+            '0.78539816339744831']
+
+    x, w = gauss_chebyshev_t(5, 17)
+    assert [str(r) for r in x] == [
+            '0.95105651629515357',
+            '0.58778525229247313',
+            '0',
+            '-0.58778525229247313',
+            '-0.95105651629515357']
+    assert [str(r) for r in w] == [
+            '0.62831853071795865',
+            '0.62831853071795865',
+            '0.62831853071795865',
+            '0.62831853071795865',
+            '0.62831853071795865']
+
+
+def test_chebyshev_t_precise():
+    x, w = gauss_chebyshev_t(3, 40)
+    assert [str(r) for r in x] == [
+            '0.8660254037844386467637231707529361834714',
+            '0',
+            '-0.8660254037844386467637231707529361834714']
+    assert [str(r) for r in w] == [
+            '1.047197551196597746154214461093167628066',
+            '1.047197551196597746154214461093167628066',
+            '1.047197551196597746154214461093167628066']
+
+
+def test_chebyshev_u():
+    x, w = gauss_chebyshev_u(1, 17)
+    assert [str(r) for r in x] == ['0']
+    assert [str(r) for r in w] == ['1.5707963267948966']
+
+    x, w = gauss_chebyshev_u(2, 17)
+    assert [str(r) for r in x] == [
+            '0.50000000000000000',
+            '-0.50000000000000000']
+    assert [str(r) for r in w] == [
+            '0.78539816339744831',
+            '0.78539816339744831']
+
+    x, w = gauss_chebyshev_u(3, 17)
+    assert [str(r) for r in x] == [
+            '0.70710678118654752',
+            '0',
+            '-0.70710678118654752']
+    assert [str(r) for r in w] == [
+            '0.39269908169872415',
+            '0.78539816339744831',
+            '0.39269908169872415']
+
+    x, w = gauss_chebyshev_u(4, 17)
+    assert [str(r) for r in x] == [
+            '0.80901699437494742',
+            '0.30901699437494742',
+            '-0.30901699437494742',
+            '-0.80901699437494742']
+    assert [str(r) for r in w] == [
+            '0.21707871342270599',
+            '0.56831944997474231',
+            '0.56831944997474231',
+            '0.21707871342270599']
+
+    x, w = gauss_chebyshev_u(5, 17)
+    assert [str(r) for r in x] == [
+            '0.86602540378443865',
+            '0.50000000000000000',
+            '0',
+            '-0.50000000000000000',
+            '-0.86602540378443865']
+    assert [str(r) for r in w] == [
+            '0.13089969389957472',
+            '0.39269908169872415',
+            '0.52359877559829887',
+            '0.39269908169872415',
+            '0.13089969389957472']
+
+
+def test_chebyshev_u_precise():
+    x, w = gauss_chebyshev_u(3, 40)
+    assert [str(r) for r in x] == [
+            '0.7071067811865475244008443621048490392848',
+            '0',
+            '-0.7071067811865475244008443621048490392848']
+    assert [str(r) for r in w] == [
+            '0.3926990816987241548078304229099378605246',
+            '0.7853981633974483096156608458198757210493',
+            '0.3926990816987241548078304229099378605246']
+
+
+def test_jacobi():
+    x, w = gauss_jacobi(1, Rational(-1, 2), S.Half, 17)
+    assert [str(r) for r in x] == ['0.50000000000000000']
+    assert [str(r) for r in w] == ['3.1415926535897932']
+
+    x, w = gauss_jacobi(2, Rational(-1, 2), S.Half, 17)
+    assert [str(r) for r in x] == [
+            '-0.30901699437494742',
+            '0.80901699437494742']
+    assert [str(r) for r in w] == [
+            '0.86831485369082398',
+            '2.2732777998989693']
+
+    x, w = gauss_jacobi(3, Rational(-1, 2), S.Half, 17)
+    assert [str(r) for r in x] == [
+            '-0.62348980185873353',
+            '0.22252093395631440',
+            '0.90096886790241913']
+    assert [str(r) for r in w] == [
+            '0.33795476356635433',
+            '1.0973322242791115',
+            '1.7063056657443274']
+
+    x, w = gauss_jacobi(4, Rational(-1, 2), S.Half, 17)
+    assert [str(r) for r in x] == [
+            '-0.76604444311897804',
+            '-0.17364817766693035',
+            '0.50000000000000000',
+            '0.93969262078590838']
+    assert [str(r) for r in w] == [
+            '0.16333179083642836',
+            '0.57690240318269103',
+            '1.0471975511965977',
+            '1.3541609083740761']
+
+    x, w = gauss_jacobi(5, Rational(-1, 2), S.Half, 17)
+    assert [str(r) for r in x] == [
+            '-0.84125353283118117',
+            '-0.41541501300188643',
+            '0.14231483827328514',
+            '0.65486073394528506',
+            '0.95949297361449739']
+    assert [str(r) for r in w] == [
+            '0.090675770007435372',
+            '0.33391416373675607',
+            '0.65248870981926643',
+            '0.94525424081394926',
+            '1.1192597692123861']
+
+    x, w = gauss_jacobi(1, 2, 3, 17)
+    assert [str(r) for r in x] == ['0.14285714285714286']
+    assert [str(r) for r in w] == ['1.0666666666666667']
+
+    x, w = gauss_jacobi(2, 2, 3, 17)
+    assert [str(r) for r in x] == [
+            '-0.24025307335204215',
+            '0.46247529557426437']
+    assert [str(r) for r in w] == [
+            '0.48514624517838660',
+            '0.58152042148828007']
+
+    x, w = gauss_jacobi(3, 2, 3, 17)
+    assert [str(r) for r in x] == [
+            '-0.46115870378089762',
+            '0.10438533038323902',
+            '0.62950064612493132']
+    assert [str(r) for r in w] == [
+            '0.17937613502213266',
+            '0.61595640991147154',
+            '0.27133412173306246']
+
+    x, w = gauss_jacobi(4, 2, 3, 17)
+    assert [str(r) for r in x] == [
+            '-0.59903470850824782',
+            '-0.14761105199952565',
+            '0.32554377081188859',
+            '0.72879429738819258']
+    assert [str(r) for r in w] == [
+            '0.067809641836772187',
+            '0.38956404952032481',
+            '0.47995970868024150',
+            '0.12933326662932816']
+
+    x, w = gauss_jacobi(5, 2, 3, 17)
+    assert [str(r) for r in x] == [
+            '-0.69045775012676106',
+            '-0.32651993134900065',
+            '0.082337849552034905',
+            '0.47517887061283164',
+            '0.79279429464422850']
+    assert [str(r) for r in w] == [
+            '0.027410178066337099',
+            '0.21291786060364828',
+            '0.43908437944395081',
+            '0.32220656547221822',
+            '0.065047683080512268']
+
+
+def test_jacobi_precise():
+    x, w = gauss_jacobi(3, Rational(-1, 2), S.Half, 40)
+    assert [str(r) for r in x] == [
+            '-0.6234898018587335305250048840042398106323',
+            '0.2225209339563144042889025644967947594664',
+            '0.9009688679024191262361023195074450511659']
+    assert [str(r) for r in w] == [
+            '0.3379547635663543330553835737094171534907',
+            '1.097332224279111467485302294320899710461',
+            '1.706305665744327437921957515249186020246']
+
+    x, w = gauss_jacobi(3, 2, 3, 40)
+    assert [str(r) for r in x] == [
+            '-0.4611587037808976179121958105554375981274',
+            '0.1043853303832390210914918407615869143233',
+            '0.6295006461249313240934312425211234110769']
+    assert [str(r) for r in w] == [
+            '0.1793761350221326596137764371503859752628',
+            '0.6159564099114715430909548532229749439714',
+            '0.2713341217330624639619353762933057474325']
+
+
+def test_lobatto():
+    x, w = gauss_lobatto(2, 17)
+    assert [str(r) for r in x] == [
+            '-1',
+            '1']
+    assert [str(r) for r in w] == [
+            '1.0000000000000000',
+            '1.0000000000000000']
+
+    x, w = gauss_lobatto(3, 17)
+    assert [str(r) for r in x] == [
+            '-1',
+            '0',
+            '1']
+    assert [str(r) for r in w] == [
+            '0.33333333333333333',
+            '1.3333333333333333',
+            '0.33333333333333333']
+
+    x, w = gauss_lobatto(4, 17)
+    assert [str(r) for r in x] == [
+            '-1',
+            '-0.44721359549995794',
+            '0.44721359549995794',
+            '1']
+    assert [str(r) for r in w] == [
+            '0.16666666666666667',
+            '0.83333333333333333',
+            '0.83333333333333333',
+            '0.16666666666666667']
+
+    x, w = gauss_lobatto(5, 17)
+    assert [str(r) for r in x] == [
+            '-1',
+            '-0.65465367070797714',
+            '0',
+            '0.65465367070797714',
+            '1']
+    assert [str(r) for r in w] == [
+            '0.10000000000000000',
+            '0.54444444444444444',
+            '0.71111111111111111',
+            '0.54444444444444444',
+            '0.10000000000000000']
+
+
+def test_lobatto_precise():
+    x, w = gauss_lobatto(3, 40)
+    assert [str(r) for r in x] == [
+            '-1',
+            '0',
+            '1']
+    assert [str(r) for r in w] == [
+            '0.3333333333333333333333333333333333333333',
+            '1.333333333333333333333333333333333333333',
+            '0.3333333333333333333333333333333333333333']

rl/Lib/site-packages/sympy/integrals/tests/test_rationaltools.py

@@ -0,0 +1,183 @@
+from sympy.core.numbers import (I, Rational)
+from sympy.core.singleton import S
+from sympy.core.symbol import (Dummy, symbols)
+from sympy.functions.elementary.exponential import log
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.trigonometric import atan
+from sympy.integrals.integrals import integrate
+from sympy.polys.polytools import Poly
+from sympy.simplify.simplify import simplify
+
+from sympy.integrals.rationaltools import ratint, ratint_logpart, log_to_atan
+
+from sympy.abc import a, b, x, t
+
+half = S.Half
+
+
+def test_ratint():
+    assert ratint(S.Zero, x) == 0
+    assert ratint(S(7), x) == 7*x
+
+    assert ratint(x, x) == x**2/2
+    assert ratint(2*x, x) == x**2
+    assert ratint(-2*x, x) == -x**2
+
+    assert ratint(8*x**7 + 2*x + 1, x) == x**8 + x**2 + x
+
+    f = S.One
+    g = x + 1
+
+    assert ratint(f / g, x) == log(x + 1)
+    assert ratint((f, g), x) == log(x + 1)
+
+    f = x**3 - x
+    g = x - 1
+
+    assert ratint(f/g, x) == x**3/3 + x**2/2
+
+    f = x
+    g = (x - a)*(x + a)
+
+    assert ratint(f/g, x) == log(x**2 - a**2)/2
+
+    f = S.One
+    g = x**2 + 1
+
+    assert ratint(f/g, x, real=None) == atan(x)
+    assert ratint(f/g, x, real=True) == atan(x)
+
+    assert ratint(f/g, x, real=False) == I*log(x + I)/2 - I*log(x - I)/2
+
+    f = S(36)
+    g = x**5 - 2*x**4 - 2*x**3 + 4*x**2 + x - 2
+
+    assert ratint(f/g, x) == \
+        -4*log(x + 1) + 4*log(x - 2) + (12*x + 6)/(x**2 - 1)
+
+    f = x**4 - 3*x**2 + 6
+    g = x**6 - 5*x**4 + 5*x**2 + 4
+
+    assert ratint(f/g, x) == \
+        atan(x) + atan(x**3) + atan(x/2 - Rational(3, 2)*x**3 + S.Half*x**5)
+
+    f = x**7 - 24*x**4 - 4*x**2 + 8*x - 8
+    g = x**8 + 6*x**6 + 12*x**4 + 8*x**2
+
+    assert ratint(f/g, x) == \
+        (4 + 6*x + 8*x**2 + 3*x**3)/(4*x + 4*x**3 + x**5) + log(x)
+
+    assert ratint((x**3*f)/(x*g), x) == \
+        -(12 - 16*x + 6*x**2 - 14*x**3)/(4 + 4*x**2 + x**4) - \
+        5*sqrt(2)*atan(x*sqrt(2)/2) + S.Half*x**2 - 3*log(2 + x**2)
+
+    f = x**5 - x**4 + 4*x**3 + x**2 - x + 5
+    g = x**4 - 2*x**3 + 5*x**2 - 4*x + 4
+
+    assert ratint(f/g, x) == \
+        x + S.Half*x**2 + S.Half*log(2 - x + x**2) + (9 - 4*x)/(7*x**2 - 7*x + 14) + \
+        13*sqrt(7)*atan(Rational(-1, 7)*sqrt(7) + 2*x*sqrt(7)/7)/49
+
+    assert ratint(1/(x**2 + x + 1), x) == \
+        2*sqrt(3)*atan(sqrt(3)/3 + 2*x*sqrt(3)/3)/3
+
+    assert ratint(1/(x**3 + 1), x) == \
+        -log(1 - x + x**2)/6 + log(1 + x)/3 + sqrt(3)*atan(-sqrt(3)
+             /3 + 2*x*sqrt(3)/3)/3
+
+    assert ratint(1/(x**2 + x + 1), x, real=False) == \
+        -I*3**half*log(half + x - half*I*3**half)/3 + \
+        I*3**half*log(half + x + half*I*3**half)/3
+
+    assert ratint(1/(x**3 + 1), x, real=False) == log(1 + x)/3 + \
+        (Rational(-1, 6) + I*3**half/6)*log(-half + x + I*3**half/2) + \
+        (Rational(-1, 6) - I*3**half/6)*log(-half + x - I*3**half/2)
+
+    # issue 4991
+    assert ratint(1/(x*(a + b*x)**3), x) == \
+        (3*a + 2*b*x)/(2*a**4 + 4*a**3*b*x + 2*a**2*b**2*x**2) + (
+            log(x) - log(a/b + x))/a**3
+
+    assert ratint(x/(1 - x**2), x) == -log(x**2 - 1)/2
+    assert ratint(-x/(1 - x**2), x) == log(x**2 - 1)/2
+
+    assert ratint((x/4 - 4/(1 - x)).diff(x), x) == x/4 + 4/(x - 1)
+
+    ans = atan(x)
+    assert ratint(1/(x**2 + 1), x, symbol=x) == ans
+    assert ratint(1/(x**2 + 1), x, symbol='x') == ans
+    assert ratint(1/(x**2 + 1), x, symbol=a) == ans
+    # this asserts that as_dummy must return a unique symbol
+    # even if the symbol is already a Dummy
+    d = Dummy()
+    assert ratint(1/(d**2 + 1), d, symbol=d) == atan(d)
+
+
+def test_ratint_logpart():
+    assert ratint_logpart(x, x**2 - 9, x, t) == \
+        [(Poly(x**2 - 9, x), Poly(-2*t + 1, t))]
+    assert ratint_logpart(x**2, x**3 - 5, x, t) == \
+        [(Poly(x**3 - 5, x), Poly(-3*t + 1, t))]
+
+
+def test_issue_5414():
+    assert ratint(1/(x**2 + 16), x) == atan(x/4)/4
+
+
+def test_issue_5249():
+    assert ratint(
+        1/(x**2 + a**2), x) == (-I*log(-I*a + x)/2 + I*log(I*a + x)/2)/a
+
+
+def test_issue_5817():
+    a, b, c = symbols('a,b,c', positive=True)
+
+    assert simplify(ratint(a/(b*c*x**2 + a**2 + b*a), x)) == \
+        sqrt(a)*atan(sqrt(
+            b)*sqrt(c)*x/(sqrt(a)*sqrt(a + b)))/(sqrt(b)*sqrt(c)*sqrt(a + b))
+
+
+def test_issue_5981():
+    u = symbols('u')
+    assert integrate(1/(u**2 + 1)) == atan(u)
+
+def test_issue_10488():
+    a,b,c,x = symbols('a b c x', positive=True)
+    assert integrate(x/(a*x+b),x) == x/a - b*log(a*x + b)/a**2
+
+
+def test_issues_8246_12050_13501_14080():
+    a = symbols('a', nonzero=True)
+    assert integrate(a/(x**2 + a**2), x) == atan(x/a)
+    assert integrate(1/(x**2 + a**2), x) == atan(x/a)/a
+    assert integrate(1/(1 + a**2*x**2), x) == atan(a*x)/a
+
+
+def test_issue_6308():
+    k, a0 = symbols('k a0', real=True)
+    assert integrate((x**2 + 1 - k**2)/(x**2 + 1 + a0**2), x) == \
+        x - (a0**2 + k**2)*atan(x/sqrt(a0**2 + 1))/sqrt(a0**2 + 1)
+
+
+def test_issue_5907():
+    a = symbols('a', nonzero=True)
+    assert integrate(1/(x**2 + a**2)**2, x) == \
+         x/(2*a**4 + 2*a**2*x**2) + atan(x/a)/(2*a**3)
+
+
+def test_log_to_atan():
+    f, g = (Poly(x + S.Half, x, domain='QQ'), Poly(sqrt(3)/2, x, domain='EX'))
+    fg_ans = 2*atan(2*sqrt(3)*x/3 + sqrt(3)/3)
+    assert log_to_atan(f, g) == fg_ans
+    assert log_to_atan(g, f) == -fg_ans
+
+
+def test_issue_25896():
+    # for both tests, C = 0 in log_to_real
+    # but this only has a log result
+    e = (2*x + 1)/(x**2 + x + 1) + 1/x
+    assert ratint(e, x) == log(x**3 + x**2 + x)
+    # while this has more
+    assert ratint((4*x + 7)/(x**2 + 4*x + 6) + 2/x, x) == (
+        2*log(x) + 2*log(x**2 + 4*x + 6) - sqrt(2)*atan(
+        sqrt(2)*x/2 + sqrt(2))/2)

rl/Lib/site-packages/sympy/integrals/tests/test_rde.py

@@ -0,0 +1,202 @@
+"""Most of these tests come from the examples in Bronstein's book."""
+from sympy.core.numbers import (I, Rational, oo)
+from sympy.core.symbol import symbols
+from sympy.polys.polytools import Poly
+from sympy.integrals.risch import (DifferentialExtension,
+    NonElementaryIntegralException)
+from sympy.integrals.rde import (order_at, order_at_oo, weak_normalizer,
+    normal_denom, special_denom, bound_degree, spde, solve_poly_rde,
+    no_cancel_equal, cancel_primitive, cancel_exp, rischDE)
+
+from sympy.testing.pytest import raises
+from sympy.abc import x, t, z, n
+
+t0, t1, t2, k = symbols('t:3 k')
+
+
+def test_order_at():
+    a = Poly(t**4, t)
+    b = Poly((t**2 + 1)**3*t, t)
+    c = Poly((t**2 + 1)**6*t, t)
+    d = Poly((t**2 + 1)**10*t**10, t)
+    e = Poly((t**2 + 1)**100*t**37, t)
+    p1 = Poly(t, t)
+    p2 = Poly(1 + t**2, t)
+    assert order_at(a, p1, t) == 4
+    assert order_at(b, p1, t) == 1
+    assert order_at(c, p1, t) == 1
+    assert order_at(d, p1, t) == 10
+    assert order_at(e, p1, t) == 37
+    assert order_at(a, p2, t) == 0
+    assert order_at(b, p2, t) == 3
+    assert order_at(c, p2, t) == 6
+    assert order_at(d, p1, t) == 10
+    assert order_at(e, p2, t) == 100
+    assert order_at(Poly(0, t), Poly(t, t), t) is oo
+    assert order_at_oo(Poly(t**2 - 1, t), Poly(t + 1), t) == \
+        order_at_oo(Poly(t - 1, t), Poly(1, t), t) == -1
+    assert order_at_oo(Poly(0, t), Poly(1, t), t) is oo
+
+def test_weak_normalizer():
+    a = Poly((1 + x)*t**5 + 4*t**4 + (-1 - 3*x)*t**3 - 4*t**2 + (-2 + 2*x)*t, t)
+    d = Poly(t**4 - 3*t**2 + 2, t)
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]})
+    r = weak_normalizer(a, d, DE, z)
+    assert r == (Poly(t**5 - t**4 - 4*t**3 + 4*t**2 + 4*t - 4, t, domain='ZZ[x]'),
+        (Poly((1 + x)*t**2 + x*t, t, domain='ZZ[x]'),
+         Poly(t + 1, t, domain='ZZ[x]')))
+    assert weak_normalizer(r[1][0], r[1][1], DE) == (Poly(1, t), r[1])
+    r = weak_normalizer(Poly(1 + t**2), Poly(t**2 - 1, t), DE, z)
+    assert r == (Poly(t**4 - 2*t**2 + 1, t), (Poly(-3*t**2 + 1, t), Poly(t**2 - 1, t)))
+    assert weak_normalizer(r[1][0], r[1][1], DE, z) == (Poly(1, t), r[1])
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1 + t**2)]})
+    r = weak_normalizer(Poly(1 + t**2), Poly(t, t), DE, z)
+    assert r == (Poly(t, t), (Poly(0, t), Poly(1, t)))
+    assert weak_normalizer(r[1][0], r[1][1], DE, z) == (Poly(1, t), r[1])
+
+
+def test_normal_denom():
+    DE = DifferentialExtension(extension={'D': [Poly(1, x)]})
+    raises(NonElementaryIntegralException, lambda: normal_denom(Poly(1, x), Poly(1, x),
+    Poly(1, x), Poly(x, x), DE))
+    fa, fd = Poly(t**2 + 1, t), Poly(1, t)
+    ga, gd = Poly(1, t), Poly(t**2, t)
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]})
+    assert normal_denom(fa, fd, ga, gd, DE) == \
+        (Poly(t, t), (Poly(t**3 - t**2 + t - 1, t), Poly(1, t)), (Poly(1, t),
+        Poly(1, t)), Poly(t, t))
+
+
+def test_special_denom():
+    # TODO: add more tests here
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]})
+    assert special_denom(Poly(1, t), Poly(t**2, t), Poly(1, t), Poly(t**2 - 1, t),
+    Poly(t, t), DE) == \
+        (Poly(1, t), Poly(t**2 - 1, t), Poly(t**2 - 1, t), Poly(t, t))
+#    assert special_denom(Poly(1, t), Poly(2*x, t), Poly((1 + 2*x)*t, t), DE) == 1
+
+    # issue 3940
+    # Note, this isn't a very good test, because the denominator is just 1,
+    # but at least it tests the exp cancellation case
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(-2*x*t0, t0),
+        Poly(I*k*t1, t1)]})
+    DE.decrement_level()
+    assert special_denom(Poly(1, t0), Poly(I*k, t0), Poly(1, t0), Poly(t0, t0),
+    Poly(1, t0), DE) == \
+        (Poly(1, t0, domain='ZZ'), Poly(I*k, t0, domain='ZZ_I[k,x]'),
+                Poly(t0, t0, domain='ZZ'), Poly(1, t0, domain='ZZ'))
+
+
+    assert special_denom(Poly(1, t), Poly(t**2, t), Poly(1, t), Poly(t**2 - 1, t),
+    Poly(t, t), DE, case='tan') == \
+           (Poly(1, t, t0, domain='ZZ'), Poly(t**2, t0, t, domain='ZZ[x]'),
+            Poly(t, t, t0, domain='ZZ'), Poly(1, t0, domain='ZZ'))
+
+    raises(ValueError, lambda: special_denom(Poly(1, t), Poly(t**2, t), Poly(1, t), Poly(t**2 - 1, t),
+    Poly(t, t), DE, case='unrecognized_case'))
+
+
+def test_bound_degree_fail():
+    # Primitive
+    DE = DifferentialExtension(extension={'D': [Poly(1, x),
+        Poly(t0/x**2, t0), Poly(1/x, t)]})
+    assert bound_degree(Poly(t**2, t), Poly(-(1/x**2*t**2 + 1/x), t),
+        Poly((2*x - 1)*t**4 + (t0 + x)/x*t**3 - (t0 + 4*x**2)/2*x*t**2 + x*t,
+        t), DE) == 3
+
+
+def test_bound_degree():
+    # Base
+    DE = DifferentialExtension(extension={'D': [Poly(1, x)]})
+    assert bound_degree(Poly(1, x), Poly(-2*x, x), Poly(1, x), DE) == 0
+
+    # Primitive (see above test_bound_degree_fail)
+    # TODO: Add test for when the degree bound becomes larger after limited_integrate
+    # TODO: Add test for db == da - 1 case
+
+    # Exp
+    # TODO: Add tests
+    # TODO: Add test for when the degree becomes larger after parametric_log_deriv()
+
+    # Nonlinear
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]})
+    assert bound_degree(Poly(t, t), Poly((t - 1)*(t**2 + 1), t), Poly(1, t), DE) == 0
+
+
+def test_spde():
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]})
+    raises(NonElementaryIntegralException, lambda: spde(Poly(t, t), Poly((t - 1)*(t**2 + 1), t), Poly(1, t), 0, DE))
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]})
+    assert spde(Poly(t**2 + x*t*2 + x**2, t), Poly(t**2/x**2 + (2/x - 1)*t, t),
+        Poly(t**2/x**2 + (2/x - 1)*t, t), 0, DE) == \
+        (Poly(0, t), Poly(0, t), 0, Poly(0, t), Poly(1, t, domain='ZZ(x)'))
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t0/x**2, t0), Poly(1/x, t)]})
+    assert spde(Poly(t**2, t), Poly(-t**2/x**2 - 1/x, t),
+    Poly((2*x - 1)*t**4 + (t0 + x)/x*t**3 - (t0 + 4*x**2)/(2*x)*t**2 + x*t, t), 3, DE) == \
+        (Poly(0, t), Poly(0, t), 0, Poly(0, t),
+        Poly(t0*t**2/2 + x**2*t**2 - x**2*t, t, domain='ZZ(x,t0)'))
+    DE = DifferentialExtension(extension={'D': [Poly(1, x)]})
+    assert spde(Poly(x**2 + x + 1, x), Poly(-2*x - 1, x), Poly(x**5/2 +
+    3*x**4/4 + x**3 - x**2 + 1, x), 4, DE) == \
+        (Poly(0, x, domain='QQ'), Poly(x/2 - Rational(1, 4), x), 2, Poly(x**2 + x + 1, x), Poly(x*Rational(5, 4), x))
+    assert spde(Poly(x**2 + x + 1, x), Poly(-2*x - 1, x), Poly(x**5/2 +
+    3*x**4/4 + x**3 - x**2 + 1, x), n, DE) == \
+        (Poly(0, x, domain='QQ'), Poly(x/2 - Rational(1, 4), x), -2 + n, Poly(x**2 + x + 1, x), Poly(x*Rational(5, 4), x))
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1, t)]})
+    raises(NonElementaryIntegralException, lambda: spde(Poly((t - 1)*(t**2 + 1)**2, t), Poly((t - 1)*(t**2 + 1), t), Poly(1, t), 0, DE))
+    DE = DifferentialExtension(extension={'D': [Poly(1, x)]})
+    assert spde(Poly(x**2 - x, x), Poly(1, x), Poly(9*x**4 - 10*x**3 + 2*x**2, x), 4, DE) == \
+        (Poly(0, x, domain='ZZ'), Poly(0, x), 0, Poly(0, x), Poly(3*x**3 - 2*x**2, x, domain='QQ'))
+    assert spde(Poly(x**2 - x, x), Poly(x**2 - 5*x + 3, x), Poly(x**7 - x**6 - 2*x**4 + 3*x**3 - x**2, x), 5, DE) == \
+        (Poly(1, x, domain='QQ'), Poly(x + 1, x, domain='QQ'), 1, Poly(x**4 - x**3, x), Poly(x**3 - x**2, x, domain='QQ'))
+
+def test_solve_poly_rde_no_cancel():
+    # deg(b) large
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1 + t**2, t)]})
+    assert solve_poly_rde(Poly(t**2 + 1, t), Poly(t**3 + (x + 1)*t**2 + t + x + 2, t),
+    oo, DE) == Poly(t + x, t)
+    # deg(b) small
+    DE = DifferentialExtension(extension={'D': [Poly(1, x)]})
+    assert solve_poly_rde(Poly(0, x), Poly(x/2 - Rational(1, 4), x), oo, DE) == \
+        Poly(x**2/4 - x/4, x)
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]})
+    assert solve_poly_rde(Poly(2, t), Poly(t**2 + 2*t + 3, t), 1, DE) == \
+        Poly(t + 1, t, x)
+    # deg(b) == deg(D) - 1
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]})
+    assert no_cancel_equal(Poly(1 - t, t),
+    Poly(t**3 + t**2 - 2*x*t - 2*x, t), oo, DE) == \
+        (Poly(t**2, t), 1, Poly((-2 - 2*x)*t - 2*x, t))
+
+
+def test_solve_poly_rde_cancel():
+    # exp
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]})
+    assert cancel_exp(Poly(2*x, t), Poly(2*x, t), 0, DE) == \
+        Poly(1, t)
+    assert cancel_exp(Poly(2*x, t), Poly((1 + 2*x)*t, t), 1, DE) == \
+        Poly(t, t)
+    # TODO: Add more exp tests, including tests that require is_deriv_in_field()
+
+    # primitive
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t)]})
+
+    # If the DecrementLevel context manager is working correctly, this shouldn't
+    # cause any problems with the further tests.
+    raises(NonElementaryIntegralException, lambda: cancel_primitive(Poly(1, t), Poly(t, t), oo, DE))
+
+    assert cancel_primitive(Poly(1, t), Poly(t + 1/x, t), 2, DE) == \
+        Poly(t, t)
+    assert cancel_primitive(Poly(4*x, t), Poly(4*x*t**2 + 2*t/x, t), 3, DE) == \
+        Poly(t**2, t)
+
+    # TODO: Add more primitive tests, including tests that require is_deriv_in_field()
+
+
+def test_rischDE():
+    # TODO: Add more tests for rischDE, including ones from the text
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]})
+    DE.decrement_level()
+    assert rischDE(Poly(-2*x, x), Poly(1, x), Poly(1 - 2*x - 2*x**2, x),
+    Poly(1, x), DE) == \
+        (Poly(x + 1, x), Poly(1, x))

rl/Lib/site-packages/sympy/integrals/tests/test_risch.py

@@ -0,0 +1,763 @@
+"""Most of these tests come from the examples in Bronstein's book."""
+from sympy.core.function import (Function, Lambda, diff, expand_log)
+from sympy.core.numbers import (I, Rational, pi)
+from sympy.core.relational import Ne
+from sympy.core.singleton import S
+from sympy.core.symbol import (Symbol, symbols)
+from sympy.functions.elementary.exponential import (exp, log)
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.piecewise import Piecewise
+from sympy.functions.elementary.trigonometric import (atan, sin, tan)
+from sympy.polys.polytools import (Poly, cancel, factor)
+from sympy.integrals.risch import (gcdex_diophantine, frac_in, as_poly_1t,
+    derivation, splitfactor, splitfactor_sqf, canonical_representation,
+    hermite_reduce, polynomial_reduce, residue_reduce, residue_reduce_to_basic,
+    integrate_primitive, integrate_hyperexponential_polynomial,
+    integrate_hyperexponential, integrate_hypertangent_polynomial,
+    integrate_nonlinear_no_specials, integer_powers, DifferentialExtension,
+    risch_integrate, DecrementLevel, NonElementaryIntegral, recognize_log_derivative,
+    recognize_derivative, laurent_series)
+from sympy.testing.pytest import raises
+
+from sympy.abc import x, t, nu, z, a, y
+t0, t1, t2 = symbols('t:3')
+i = Symbol('i')
+
+def test_gcdex_diophantine():
+    assert gcdex_diophantine(Poly(x**4 - 2*x**3 - 6*x**2 + 12*x + 15),
+    Poly(x**3 + x**2 - 4*x - 4), Poly(x**2 - 1)) == \
+        (Poly((-x**2 + 4*x - 3)/5), Poly((x**3 - 7*x**2 + 16*x - 10)/5))
+    assert gcdex_diophantine(Poly(x**3 + 6*x + 7), Poly(x**2 + 3*x + 2), Poly(x + 1)) == \
+        (Poly(1/13, x, domain='QQ'), Poly(-1/13*x + 3/13, x, domain='QQ'))
+
+
+def test_frac_in():
+    assert frac_in(Poly((x + 1)/x*t, t), x) == \
+        (Poly(t*x + t, x), Poly(x, x))
+    assert frac_in((x + 1)/x*t, x) == \
+        (Poly(t*x + t, x), Poly(x, x))
+    assert frac_in((Poly((x + 1)/x*t, t), Poly(t + 1, t)), x) == \
+        (Poly(t*x + t, x), Poly((1 + t)*x, x))
+    raises(ValueError, lambda: frac_in((x + 1)/log(x)*t, x))
+    assert frac_in(Poly((2 + 2*x + x*(1 + x))/(1 + x)**2, t), x, cancel=True) == \
+        (Poly(x + 2, x), Poly(x + 1, x))
+
+
+def test_as_poly_1t():
+    assert as_poly_1t(2/t + t, t, z) in [
+        Poly(t + 2*z, t, z), Poly(t + 2*z, z, t)]
+    assert as_poly_1t(2/t + 3/t**2, t, z) in [
+        Poly(2*z + 3*z**2, t, z), Poly(2*z + 3*z**2, z, t)]
+    assert as_poly_1t(2/((exp(2) + 1)*t), t, z) in [
+        Poly(2/(exp(2) + 1)*z, t, z), Poly(2/(exp(2) + 1)*z, z, t)]
+    assert as_poly_1t(2/((exp(2) + 1)*t) + t, t, z) in [
+        Poly(t + 2/(exp(2) + 1)*z, t, z), Poly(t + 2/(exp(2) + 1)*z, z, t)]
+    assert as_poly_1t(S.Zero, t, z) == Poly(0, t, z)
+
+
+def test_derivation():
+    p = Poly(4*x**4*t**5 + (-4*x**3 - 4*x**4)*t**4 + (-3*x**2 + 2*x**3)*t**3 +
+        (2*x + 7*x**2 + 2*x**3)*t**2 + (1 - 4*x - 4*x**2)*t - 1 + 2*x, t)
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(-t**2 - 3/(2*x)*t + 1/(2*x), t)]})
+    assert derivation(p, DE) == Poly(-20*x**4*t**6 + (2*x**3 + 16*x**4)*t**5 +
+        (21*x**2 + 12*x**3)*t**4 + (x*Rational(7, 2) - 25*x**2 - 12*x**3)*t**3 +
+        (-5 - x*Rational(15, 2) + 7*x**2)*t**2 - (3 - 8*x - 10*x**2 - 4*x**3)/(2*x)*t +
+        (1 - 4*x**2)/(2*x), t)
+    assert derivation(Poly(1, t), DE) == Poly(0, t)
+    assert derivation(Poly(t, t), DE) == DE.d
+    assert derivation(Poly(t**2 + 1/x*t + (1 - 2*x)/(4*x**2), t), DE) == \
+        Poly(-2*t**3 - 4/x*t**2 - (5 - 2*x)/(2*x**2)*t - (1 - 2*x)/(2*x**3), t, domain='ZZ(x)')
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t1), Poly(t, t)]})
+    assert derivation(Poly(x*t*t1, t), DE) == Poly(t*t1 + x*t*t1 + t, t)
+    assert derivation(Poly(x*t*t1, t), DE, coefficientD=True) == \
+        Poly((1 + t1)*t, t)
+    DE = DifferentialExtension(extension={'D': [Poly(1, x)]})
+    assert derivation(Poly(x, x), DE) == Poly(1, x)
+    # Test basic option
+    assert derivation((x + 1)/(x - 1), DE, basic=True) == -2/(1 - 2*x + x**2)
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]})
+    assert derivation((t + 1)/(t - 1), DE, basic=True) == -2*t/(1 - 2*t + t**2)
+    assert derivation(t + 1, DE, basic=True) == t
+
+
+def test_splitfactor():
+    p = Poly(4*x**4*t**5 + (-4*x**3 - 4*x**4)*t**4 + (-3*x**2 + 2*x**3)*t**3 +
+        (2*x + 7*x**2 + 2*x**3)*t**2 + (1 - 4*x - 4*x**2)*t - 1 + 2*x, t, field=True)
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(-t**2 - 3/(2*x)*t + 1/(2*x), t)]})
+    assert splitfactor(p, DE) == (Poly(4*x**4*t**3 + (-8*x**3 - 4*x**4)*t**2 +
+        (4*x**2 + 8*x**3)*t - 4*x**2, t, domain='ZZ(x)'),
+        Poly(t**2 + 1/x*t + (1 - 2*x)/(4*x**2), t, domain='ZZ(x)'))
+    assert splitfactor(Poly(x, t), DE) == (Poly(x, t), Poly(1, t))
+    r = Poly(-4*x**4*z**2 + 4*x**6*z**2 - z*x**3 - 4*x**5*z**3 + 4*x**3*z**3 + x**4 + z*x**5 - x**6, t)
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t)]})
+    assert splitfactor(r, DE, coefficientD=True) == \
+        (Poly(x*z - x**2 - z*x**3 + x**4, t), Poly(-x**2 + 4*x**2*z**2, t))
+    assert splitfactor_sqf(r, DE, coefficientD=True) == \
+        (((Poly(x*z - x**2 - z*x**3 + x**4, t), 1),), ((Poly(-x**2 + 4*x**2*z**2, t), 1),))
+    assert splitfactor(Poly(0, t), DE) == (Poly(0, t), Poly(1, t))
+    assert splitfactor_sqf(Poly(0, t), DE) == (((Poly(0, t), 1),), ())
+
+
+def test_canonical_representation():
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1 + t**2, t)]})
+    assert canonical_representation(Poly(x - t, t), Poly(t**2, t), DE) == \
+        (Poly(0, t, domain='ZZ[x]'), (Poly(0, t, domain='QQ[x]'),
+        Poly(1, t, domain='ZZ')), (Poly(-t + x, t, domain='QQ[x]'),
+        Poly(t**2, t)))
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]})
+    assert canonical_representation(Poly(t**5 + t**3 + x**2*t + 1, t),
+    Poly((t**2 + 1)**3, t), DE) == \
+        (Poly(0, t, domain='ZZ[x]'), (Poly(t**5 + t**3 + x**2*t + 1, t, domain='QQ[x]'),
+         Poly(t**6 + 3*t**4 + 3*t**2 + 1, t, domain='QQ')),
+        (Poly(0, t, domain='QQ[x]'), Poly(1, t, domain='QQ')))
+
+
+def test_hermite_reduce():
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]})
+
+    assert hermite_reduce(Poly(x - t, t), Poly(t**2, t), DE) == \
+        ((Poly(-x, t, domain='QQ[x]'), Poly(t, t, domain='QQ[x]')),
+         (Poly(0, t, domain='QQ[x]'), Poly(1, t, domain='QQ[x]')),
+         (Poly(-x, t, domain='QQ[x]'), Poly(1, t, domain='QQ[x]')))
+
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(-t**2 - t/x - (1 - nu**2/x**2), t)]})
+
+    assert hermite_reduce(
+            Poly(x**2*t**5 + x*t**4 - nu**2*t**3 - x*(x**2 + 1)*t**2 - (x**2 - nu**2)*t - x**5/4, t),
+            Poly(x**2*t**4 + x**2*(x**2 + 2)*t**2 + x**2 + x**4 + x**6/4, t), DE) == \
+        ((Poly(-x**2 - 4, t, domain='ZZ(x,nu)'), Poly(4*t**2 + 2*x**2 + 4, t, domain='ZZ(x,nu)')),
+         (Poly((-2*nu**2 - x**4)*t - (2*x**3 + 2*x), t, domain='ZZ(x,nu)'),
+          Poly(2*x**2*t**2 + x**4 + 2*x**2, t, domain='ZZ(x,nu)')),
+         (Poly(x*t + 1, t, domain='ZZ(x,nu)'), Poly(x, t, domain='ZZ(x,nu)')))
+
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t)]})
+
+    a = Poly((-2 + 3*x)*t**3 + (-1 + x)*t**2 + (-4*x + 2*x**2)*t + x**2, t)
+    d = Poly(x*t**6 - 4*x**2*t**5 + 6*x**3*t**4 - 4*x**4*t**3 + x**5*t**2, t)
+
+    assert hermite_reduce(a, d, DE) == \
+        ((Poly(3*t**2 + t + 3*x, t, domain='ZZ(x)'),
+          Poly(3*t**4 - 9*x*t**3 + 9*x**2*t**2 - 3*x**3*t, t, domain='ZZ(x)')),
+         (Poly(0, t, domain='ZZ(x)'), Poly(1, t, domain='ZZ(x)')),
+         (Poly(0, t, domain='ZZ(x)'), Poly(1, t, domain='ZZ(x)')))
+
+    assert hermite_reduce(
+            Poly(-t**2 + 2*t + 2, t, domain='ZZ(x)'),
+            Poly(-x*t**2 + 2*x*t - x, t, domain='ZZ(x)'), DE) == \
+        ((Poly(3, t, domain='ZZ(x)'), Poly(t - 1, t, domain='ZZ(x)')),
+         (Poly(0, t, domain='ZZ(x)'), Poly(1, t, domain='ZZ(x)')),
+         (Poly(1, t, domain='ZZ(x)'), Poly(x, t, domain='ZZ(x)')))
+
+    assert hermite_reduce(
+            Poly(-x**2*t**6 + (-1 - 2*x**3 + x**4)*t**3 + (-3 - 3*x**4)*t**2 -
+                2*x*t - x - 3*x**2, t, domain='ZZ(x)'),
+            Poly(x**4*t**6 - 2*x**2*t**3 + 1, t, domain='ZZ(x)'), DE) == \
+        ((Poly(x**3*t + x**4 + 1, t, domain='ZZ(x)'), Poly(x**3*t**3 - x, t, domain='ZZ(x)')),
+         (Poly(0, t, domain='ZZ(x)'), Poly(1, t, domain='ZZ(x)')),
+         (Poly(-1, t, domain='ZZ(x)'), Poly(x**2, t, domain='ZZ(x)')))
+
+    assert hermite_reduce(
+            Poly((-2 + 3*x)*t**3 + (-1 + x)*t**2 + (-4*x + 2*x**2)*t + x**2, t),
+            Poly(x*t**6 - 4*x**2*t**5 + 6*x**3*t**4 - 4*x**4*t**3 + x**5*t**2, t), DE) == \
+        ((Poly(3*t**2 + t + 3*x, t, domain='ZZ(x)'),
+          Poly(3*t**4 - 9*x*t**3 + 9*x**2*t**2 - 3*x**3*t, t, domain='ZZ(x)')),
+         (Poly(0, t, domain='ZZ(x)'), Poly(1, t, domain='ZZ(x)')),
+         (Poly(0, t, domain='ZZ(x)'), Poly(1, t, domain='ZZ(x)')))
+
+
+def test_polynomial_reduce():
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1 + t**2, t)]})
+    assert polynomial_reduce(Poly(1 + x*t + t**2, t), DE) == \
+        (Poly(t, t), Poly(x*t, t))
+    assert polynomial_reduce(Poly(0, t), DE) == \
+        (Poly(0, t), Poly(0, t))
+
+
+def test_laurent_series():
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1, t)]})
+    a = Poly(36, t)
+    d = Poly((t - 2)*(t**2 - 1)**2, t)
+    F = Poly(t**2 - 1, t)
+    n = 2
+    assert laurent_series(a, d, F, n, DE) == \
+        (Poly(-3*t**3 + 3*t**2 - 6*t - 8, t), Poly(t**5 + t**4 - 2*t**3 - 2*t**2 + t + 1, t),
+        [Poly(-3*t**3 - 6*t**2, t, domain='QQ'), Poly(2*t**6 + 6*t**5 - 8*t**3, t, domain='QQ')])
+
+
+def test_recognize_derivative():
+    DE = DifferentialExtension(extension={'D': [Poly(1, t)]})
+    a = Poly(36, t)
+    d = Poly((t - 2)*(t**2 - 1)**2, t)
+    assert recognize_derivative(a, d, DE) == False
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t)]})
+    a = Poly(2, t)
+    d = Poly(t**2 - 1, t)
+    assert recognize_derivative(a, d, DE) == False
+    assert recognize_derivative(Poly(x*t, t), Poly(1, t), DE) == True
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]})
+    assert recognize_derivative(Poly(t, t), Poly(1, t), DE) == True
+
+
+def test_recognize_log_derivative():
+
+    a = Poly(2*x**2 + 4*x*t - 2*t - x**2*t, t)
+    d = Poly((2*x + t)*(t + x**2), t)
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]})
+    assert recognize_log_derivative(a, d, DE, z) == True
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t)]})
+    assert recognize_log_derivative(Poly(t + 1, t), Poly(t + x, t), DE) == True
+    assert recognize_log_derivative(Poly(2, t), Poly(t**2 - 1, t), DE) == True
+    DE = DifferentialExtension(extension={'D': [Poly(1, x)]})
+    assert recognize_log_derivative(Poly(1, x), Poly(x**2 - 2, x), DE) == False
+    assert recognize_log_derivative(Poly(1, x), Poly(x**2 + x, x), DE) == True
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]})
+    assert recognize_log_derivative(Poly(1, t), Poly(t**2 - 2, t), DE) == False
+    assert recognize_log_derivative(Poly(1, t), Poly(t**2 + t, t), DE) == False
+
+
+def test_residue_reduce():
+    a = Poly(2*t**2 - t - x**2, t)
+    d = Poly(t**3 - x**2*t, t)
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t)], 'Tfuncs': [log]})
+    assert residue_reduce(a, d, DE, z, invert=False) == \
+        ([(Poly(z**2 - Rational(1, 4), z, domain='ZZ(x)'),
+          Poly((1 + 3*x*z - 6*z**2 - 2*x**2 + 4*x**2*z**2)*t - x*z + x**2 +
+              2*x**2*z**2 - 2*z*x**3, t, domain='ZZ(z, x)'))], False)
+    assert residue_reduce(a, d, DE, z, invert=True) == \
+        ([(Poly(z**2 - Rational(1, 4), z, domain='ZZ(x)'), Poly(t + 2*x*z, t))], False)
+    assert residue_reduce(Poly(-2/x, t), Poly(t**2 - 1, t,), DE, z, invert=False) == \
+        ([(Poly(z**2 - 1, z, domain='QQ'), Poly(-2*z*t/x - 2/x, t, domain='ZZ(z,x)'))], True)
+    ans = residue_reduce(Poly(-2/x, t), Poly(t**2 - 1, t), DE, z, invert=True)
+    assert ans == ([(Poly(z**2 - 1, z, domain='QQ'), Poly(t + z, t))], True)
+    assert residue_reduce_to_basic(ans[0], DE, z) == -log(-1 + log(x)) + log(1 + log(x))
+
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(-t**2 - t/x - (1 - nu**2/x**2), t)]})
+    # TODO: Skip or make faster
+    assert residue_reduce(Poly((-2*nu**2 - x**4)/(2*x**2)*t - (1 + x**2)/x, t),
+    Poly(t**2 + 1 + x**2/2, t), DE, z) == \
+        ([(Poly(z + S.Half, z, domain='QQ'), Poly(t**2 + 1 + x**2/2, t,
+            domain='ZZ(x,nu)'))], True)
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1 + t**2, t)]})
+    assert residue_reduce(Poly(-2*x*t + 1 - x**2, t),
+    Poly(t**2 + 2*x*t + 1 + x**2, t), DE, z) == \
+        ([(Poly(z**2 + Rational(1, 4), z), Poly(t + x + 2*z, t))], True)
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]})
+    assert residue_reduce(Poly(t, t), Poly(t + sqrt(2), t), DE, z) == \
+        ([(Poly(z - 1, z, domain='QQ'), Poly(t + sqrt(2), t))], True)
+
+
+def test_integrate_hyperexponential():
+    # TODO: Add tests for integrate_hyperexponential() from the book
+    a = Poly((1 + 2*t1 + t1**2 + 2*t1**3)*t**2 + (1 + t1**2)*t + 1 + t1**2, t)
+    d = Poly(1, t)
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1 + t1**2, t1),
+        Poly(t*(1 + t1**2), t)], 'Tfuncs': [tan, Lambda(i, exp(tan(i)))]})
+    assert integrate_hyperexponential(a, d, DE) == \
+        (exp(2*tan(x))*tan(x) + exp(tan(x)), 1 + t1**2, True)
+    a = Poly((t1**3 + (x + 1)*t1**2 + t1 + x + 2)*t, t)
+    assert integrate_hyperexponential(a, d, DE) == \
+        ((x + tan(x))*exp(tan(x)), 0, True)
+
+    a = Poly(t, t)
+    d = Poly(1, t)
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(2*x*t, t)],
+        'Tfuncs': [Lambda(i, exp(x**2))]})
+
+    assert integrate_hyperexponential(a, d, DE) == \
+        (0, NonElementaryIntegral(exp(x**2), x), False)
+
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)], 'Tfuncs': [exp]})
+    assert integrate_hyperexponential(a, d, DE) == (exp(x), 0, True)
+
+    a = Poly(25*t**6 - 10*t**5 + 7*t**4 - 8*t**3 + 13*t**2 + 2*t - 1, t)
+    d = Poly(25*t**6 + 35*t**4 + 11*t**2 + 1, t)
+    assert integrate_hyperexponential(a, d, DE) == \
+        (-(11 - 10*exp(x))/(5 + 25*exp(2*x)) + log(1 + exp(2*x)), -1, True)
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t0, t0), Poly(t0*t, t)],
+        'Tfuncs': [exp, Lambda(i, exp(exp(i)))]})
+    assert integrate_hyperexponential(Poly(2*t0*t**2, t), Poly(1, t), DE) == (exp(2*exp(x)), 0, True)
+
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t0, t0), Poly(-t0*t, t)],
+        'Tfuncs': [exp, Lambda(i, exp(-exp(i)))]})
+    assert integrate_hyperexponential(Poly(-27*exp(9) - 162*t0*exp(9) +
+    27*x*t0*exp(9), t), Poly((36*exp(18) + x**2*exp(18) - 12*x*exp(18))*t, t), DE) == \
+        (27*exp(exp(x))/(-6*exp(9) + x*exp(9)), 0, True)
+
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)], 'Tfuncs': [exp]})
+    assert integrate_hyperexponential(Poly(x**2/2*t, t), Poly(1, t), DE) == \
+        ((2 - 2*x + x**2)*exp(x)/2, 0, True)
+    assert integrate_hyperexponential(Poly(1 + t, t), Poly(t, t), DE) == \
+        (-exp(-x), 1, True)  # x - exp(-x)
+    assert integrate_hyperexponential(Poly(x, t), Poly(t + 1, t), DE) == \
+        (0, NonElementaryIntegral(x/(1 + exp(x)), x), False)
+
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t0), Poly(2*x*t1, t1)],
+        'Tfuncs': [log, Lambda(i, exp(i**2))]})
+
+    elem, nonelem, b = integrate_hyperexponential(Poly((8*x**7 - 12*x**5 + 6*x**3 - x)*t1**4 +
+        (8*t0*x**7 - 8*t0*x**6 - 4*t0*x**5 + 2*t0*x**3 + 2*t0*x**2 - t0*x +
+        24*x**8 - 36*x**6 - 4*x**5 + 22*x**4 + 4*x**3 - 7*x**2 - x + 1)*t1**3
+        + (8*t0*x**8 - 4*t0*x**6 - 16*t0*x**5 - 2*t0*x**4 + 12*t0*x**3 +
+        t0*x**2 - 2*t0*x + 24*x**9 - 36*x**7 - 8*x**6 + 22*x**5 + 12*x**4 -
+        7*x**3 - 6*x**2 + x + 1)*t1**2 + (8*t0*x**8 - 8*t0*x**6 - 16*t0*x**5 +
+        6*t0*x**4 + 10*t0*x**3 - 2*t0*x**2 - t0*x + 8*x**10 - 12*x**8 - 4*x**7
+        + 2*x**6 + 12*x**5 + 3*x**4 - 9*x**3 - x**2 + 2*x)*t1 + 8*t0*x**7 -
+        12*t0*x**6 - 4*t0*x**5 + 8*t0*x**4 - t0*x**2 - 4*x**7 + 4*x**6 +
+        4*x**5 - 4*x**4 - x**3 + x**2, t1), Poly((8*x**7 - 12*x**5 + 6*x**3 -
+        x)*t1**4 + (24*x**8 + 8*x**7 - 36*x**6 - 12*x**5 + 18*x**4 + 6*x**3 -
+        3*x**2 - x)*t1**3 + (24*x**9 + 24*x**8 - 36*x**7 - 36*x**6 + 18*x**5 +
+        18*x**4 - 3*x**3 - 3*x**2)*t1**2 + (8*x**10 + 24*x**9 - 12*x**8 -
+        36*x**7 + 6*x**6 + 18*x**5 - x**4 - 3*x**3)*t1 + 8*x**10 - 12*x**8 +
+        6*x**6 - x**4, t1), DE)
+
+    assert factor(elem) == -((x - 1)*log(x)/((x + exp(x**2))*(2*x**2 - 1)))
+    assert (nonelem, b) == (NonElementaryIntegral(exp(x**2)/(exp(x**2) + 1), x), False)
+
+def test_integrate_hyperexponential_polynomial():
+    # Without proper cancellation within integrate_hyperexponential_polynomial(),
+    # this will take a long time to complete, and will return a complicated
+    # expression
+    p = Poly((-28*x**11*t0 - 6*x**8*t0 + 6*x**9*t0 - 15*x**8*t0**2 +
+        15*x**7*t0**2 + 84*x**10*t0**2 - 140*x**9*t0**3 - 20*x**6*t0**3 +
+        20*x**7*t0**3 - 15*x**6*t0**4 + 15*x**5*t0**4 + 140*x**8*t0**4 -
+        84*x**7*t0**5 - 6*x**4*t0**5 + 6*x**5*t0**5 + x**3*t0**6 - x**4*t0**6 +
+        28*x**6*t0**6 - 4*x**5*t0**7 + x**9 - x**10 + 4*x**12)/(-8*x**11*t0 +
+        28*x**10*t0**2 - 56*x**9*t0**3 + 70*x**8*t0**4 - 56*x**7*t0**5 +
+        28*x**6*t0**6 - 8*x**5*t0**7 + x**4*t0**8 + x**12)*t1**2 +
+        (-28*x**11*t0 - 12*x**8*t0 + 12*x**9*t0 - 30*x**8*t0**2 +
+        30*x**7*t0**2 + 84*x**10*t0**2 - 140*x**9*t0**3 - 40*x**6*t0**3 +
+        40*x**7*t0**3 - 30*x**6*t0**4 + 30*x**5*t0**4 + 140*x**8*t0**4 -
+        84*x**7*t0**5 - 12*x**4*t0**5 + 12*x**5*t0**5 - 2*x**4*t0**6 +
+        2*x**3*t0**6 + 28*x**6*t0**6 - 4*x**5*t0**7 + 2*x**9 - 2*x**10 +
+        4*x**12)/(-8*x**11*t0 + 28*x**10*t0**2 - 56*x**9*t0**3 +
+        70*x**8*t0**4 - 56*x**7*t0**5 + 28*x**6*t0**6 - 8*x**5*t0**7 +
+        x**4*t0**8 + x**12)*t1 + (-2*x**2*t0 + 2*x**3*t0 + x*t0**2 -
+        x**2*t0**2 + x**3 - x**4)/(-4*x**5*t0 + 6*x**4*t0**2 - 4*x**3*t0**3 +
+        x**2*t0**4 + x**6), t1, z, expand=False)
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t0), Poly(2*x*t1, t1)]})
+    assert integrate_hyperexponential_polynomial(p, DE, z) == (
+        Poly((x - t0)*t1**2 + (-2*t0 + 2*x)*t1, t1), Poly(-2*x*t0 + x**2 +
+        t0**2, t1), True)
+
+    DE = DifferentialExtension(extension={'D':[Poly(1, x), Poly(t0, t0)]})
+    assert integrate_hyperexponential_polynomial(Poly(0, t0), DE, z) == (
+        Poly(0, t0), Poly(1, t0), True)
+
+
+def test_integrate_hyperexponential_returns_piecewise():
+    a, b = symbols('a b')
+    DE = DifferentialExtension(a**x, x)
+    assert integrate_hyperexponential(DE.fa, DE.fd, DE) == (Piecewise(
+        (exp(x*log(a))/log(a), Ne(log(a), 0)), (x, True)), 0, True)
+    DE = DifferentialExtension(a**(b*x), x)
+    assert integrate_hyperexponential(DE.fa, DE.fd, DE) == (Piecewise(
+        (exp(b*x*log(a))/(b*log(a)), Ne(b*log(a), 0)), (x, True)), 0, True)
+    DE = DifferentialExtension(exp(a*x), x)
+    assert integrate_hyperexponential(DE.fa, DE.fd, DE) == (Piecewise(
+        (exp(a*x)/a, Ne(a, 0)), (x, True)), 0, True)
+    DE = DifferentialExtension(x*exp(a*x), x)
+    assert integrate_hyperexponential(DE.fa, DE.fd, DE) == (Piecewise(
+        ((a*x - 1)*exp(a*x)/a**2, Ne(a**2, 0)), (x**2/2, True)), 0, True)
+    DE = DifferentialExtension(x**2*exp(a*x), x)
+    assert integrate_hyperexponential(DE.fa, DE.fd, DE) == (Piecewise(
+        ((x**2*a**2 - 2*a*x + 2)*exp(a*x)/a**3, Ne(a**3, 0)),
+        (x**3/3, True)), 0, True)
+    DE = DifferentialExtension(x**y + z, y)
+    assert integrate_hyperexponential(DE.fa, DE.fd, DE) == (Piecewise(
+        (exp(log(x)*y)/log(x), Ne(log(x), 0)), (y, True)), z, True)
+    DE = DifferentialExtension(x**y + z + x**(2*y), y)
+    assert integrate_hyperexponential(DE.fa, DE.fd, DE) == (Piecewise(
+        ((exp(2*log(x)*y)*log(x) +
+            2*exp(log(x)*y)*log(x))/(2*log(x)**2), Ne(2*log(x)**2, 0)),
+            (2*y, True),
+        ), z, True)
+    # TODO: Add a test where two different parts of the extension use a
+    # Piecewise, like y**x + z**x.
+
+
+def test_issue_13947():
+    a, t, s = symbols('a t s')
+    assert risch_integrate(2**(-pi)/(2**t + 1), t) == \
+        2**(-pi)*t - 2**(-pi)*log(2**t + 1)/log(2)
+    assert risch_integrate(a**(t - s)/(a**t + 1), t) == \
+        exp(-s*log(a))*log(a**t + 1)/log(a)
+
+
+def test_integrate_primitive():
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t)],
+        'Tfuncs': [log]})
+    assert integrate_primitive(Poly(t, t), Poly(1, t), DE) == (x*log(x), -1, True)
+    assert integrate_primitive(Poly(x, t), Poly(t, t), DE) == (0, NonElementaryIntegral(x/log(x), x), False)
+
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t1), Poly(1/(x + 1), t2)],
+        'Tfuncs': [log, Lambda(i, log(i + 1))]})
+    assert integrate_primitive(Poly(t1, t2), Poly(t2, t2), DE) == \
+        (0, NonElementaryIntegral(log(x)/log(1 + x), x), False)
+
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t1), Poly(1/(x*t1), t2)],
+        'Tfuncs': [log, Lambda(i, log(log(i)))]})
+    assert integrate_primitive(Poly(t2, t2), Poly(t1, t2), DE) == \
+        (0, NonElementaryIntegral(log(log(x))/log(x), x), False)
+
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(1/x, t0)],
+        'Tfuncs': [log]})
+    assert integrate_primitive(Poly(x**2*t0**3 + (3*x**2 + x)*t0**2 + (3*x**2
+    + 2*x)*t0 + x**2 + x, t0), Poly(x**2*t0**4 + 4*x**2*t0**3 + 6*x**2*t0**2 +
+    4*x**2*t0 + x**2, t0), DE) == \
+        (-1/(log(x) + 1), NonElementaryIntegral(1/(log(x) + 1), x), False)
+
+def test_integrate_hypertangent_polynomial():
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t**2 + 1, t)]})
+    assert integrate_hypertangent_polynomial(Poly(t**2 + x*t + 1, t), DE) == \
+        (Poly(t, t), Poly(x/2, t))
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(a*(t**2 + 1), t)]})
+    assert integrate_hypertangent_polynomial(Poly(t**5, t), DE) == \
+        (Poly(1/(4*a)*t**4 - 1/(2*a)*t**2, t), Poly(1/(2*a), t))
+
+
+def test_integrate_nonlinear_no_specials():
+    a, d, = Poly(x**2*t**5 + x*t**4 - nu**2*t**3 - x*(x**2 + 1)*t**2 - (x**2 -
+    nu**2)*t - x**5/4, t), Poly(x**2*t**4 + x**2*(x**2 + 2)*t**2 + x**2 + x**4 + x**6/4, t)
+    # f(x) == phi_nu(x), the logarithmic derivative of J_v, the Bessel function,
+    # which has no specials (see Chapter 5, note 4 of Bronstein's book).
+    f = Function('phi_nu')
+    DE = DifferentialExtension(extension={'D': [Poly(1, x),
+        Poly(-t**2 - t/x - (1 - nu**2/x**2), t)], 'Tfuncs': [f]})
+    assert integrate_nonlinear_no_specials(a, d, DE) == \
+        (-log(1 + f(x)**2 + x**2/2)/2 + (- 4 - x**2)/(4 + 2*x**2 + 4*f(x)**2), True)
+    assert integrate_nonlinear_no_specials(Poly(t, t), Poly(1, t), DE) == \
+        (0, False)
+
+
+def test_integer_powers():
+    assert integer_powers([x, x/2, x**2 + 1, x*Rational(2, 3)]) == [
+            (x/6, [(x, 6), (x/2, 3), (x*Rational(2, 3), 4)]),
+            (1 + x**2, [(1 + x**2, 1)])]
+
+
+def test_DifferentialExtension_exp():
+    assert DifferentialExtension(exp(x) + exp(x**2), x)._important_attrs == \
+        (Poly(t1 + t0, t1), Poly(1, t1), [Poly(1, x,), Poly(t0, t0),
+        Poly(2*x*t1, t1)], [x, t0, t1], [Lambda(i, exp(i)),
+        Lambda(i, exp(i**2))], [], [None, 'exp', 'exp'], [None, x, x**2])
+    assert DifferentialExtension(exp(x) + exp(2*x), x)._important_attrs == \
+        (Poly(t0**2 + t0, t0), Poly(1, t0), [Poly(1, x), Poly(t0, t0)], [x, t0],
+        [Lambda(i, exp(i))], [], [None, 'exp'], [None, x])
+    assert DifferentialExtension(exp(x) + exp(x/2), x)._important_attrs == \
+        (Poly(t0**2 + t0, t0), Poly(1, t0), [Poly(1, x), Poly(t0/2, t0)],
+        [x, t0], [Lambda(i, exp(i/2))], [], [None, 'exp'], [None, x/2])
+    assert DifferentialExtension(exp(x) + exp(x**2) + exp(x + x**2), x)._important_attrs == \
+        (Poly((1 + t0)*t1 + t0, t1), Poly(1, t1), [Poly(1, x), Poly(t0, t0),
+        Poly(2*x*t1, t1)], [x, t0, t1], [Lambda(i, exp(i)),
+        Lambda(i, exp(i**2))], [], [None, 'exp', 'exp'], [None, x, x**2])
+    assert DifferentialExtension(exp(x) + exp(x**2) + exp(x + x**2 + 1), x)._important_attrs == \
+        (Poly((1 + S.Exp1*t0)*t1 + t0, t1), Poly(1, t1), [Poly(1, x),
+        Poly(t0, t0), Poly(2*x*t1, t1)], [x, t0, t1], [Lambda(i, exp(i)),
+        Lambda(i, exp(i**2))], [], [None, 'exp', 'exp'], [None, x, x**2])
+    assert DifferentialExtension(exp(x) + exp(x**2) + exp(x/2 + x**2), x)._important_attrs == \
+        (Poly((t0 + 1)*t1 + t0**2, t1), Poly(1, t1), [Poly(1, x),
+        Poly(t0/2, t0), Poly(2*x*t1, t1)], [x, t0, t1],
+        [Lambda(i, exp(i/2)), Lambda(i, exp(i**2))],
+        [(exp(x/2), sqrt(exp(x)))], [None, 'exp', 'exp'], [None, x/2, x**2])
+    assert DifferentialExtension(exp(x) + exp(x**2) + exp(x/2 + x**2 + 3), x)._important_attrs == \
+        (Poly((t0*exp(3) + 1)*t1 + t0**2, t1), Poly(1, t1), [Poly(1, x),
+        Poly(t0/2, t0), Poly(2*x*t1, t1)], [x, t0, t1], [Lambda(i, exp(i/2)),
+        Lambda(i, exp(i**2))], [(exp(x/2), sqrt(exp(x)))], [None, 'exp', 'exp'],
+        [None, x/2, x**2])
+    assert DifferentialExtension(sqrt(exp(x)), x)._important_attrs == \
+        (Poly(t0, t0), Poly(1, t0), [Poly(1, x), Poly(t0/2, t0)], [x, t0],
+        [Lambda(i, exp(i/2))], [(exp(x/2), sqrt(exp(x)))], [None, 'exp'], [None, x/2])
+
+    assert DifferentialExtension(exp(x/2), x)._important_attrs == \
+        (Poly(t0, t0), Poly(1, t0), [Poly(1, x), Poly(t0/2, t0)], [x, t0],
+        [Lambda(i, exp(i/2))], [], [None, 'exp'], [None, x/2])
+
+
+def test_DifferentialExtension_log():
+    assert DifferentialExtension(log(x)*log(x + 1)*log(2*x**2 + 2*x), x)._important_attrs == \
+        (Poly(t0*t1**2 + (t0*log(2) + t0**2)*t1, t1), Poly(1, t1),
+        [Poly(1, x), Poly(1/x, t0),
+        Poly(1/(x + 1), t1, expand=False)], [x, t0, t1],
+        [Lambda(i, log(i)), Lambda(i, log(i + 1))], [], [None, 'log', 'log'],
+        [None, x, x + 1])
+    assert DifferentialExtension(x**x*log(x), x)._important_attrs == \
+        (Poly(t0*t1, t1), Poly(1, t1), [Poly(1, x), Poly(1/x, t0),
+        Poly((1 + t0)*t1, t1)], [x, t0, t1], [Lambda(i, log(i)),
+        Lambda(i, exp(t0*i))], [(exp(x*log(x)), x**x)], [None, 'log', 'exp'],
+        [None, x, t0*x])
+
+
+def test_DifferentialExtension_symlog():
+    # See comment on test_risch_integrate below
+    assert DifferentialExtension(log(x**x), x)._important_attrs == \
+        (Poly(t0*x, t1), Poly(1, t1), [Poly(1, x), Poly(1/x, t0), Poly((t0 +
+            1)*t1, t1)], [x, t0, t1], [Lambda(i, log(i)), Lambda(i, exp(i*t0))],
+            [(exp(x*log(x)), x**x)], [None, 'log', 'exp'], [None, x, t0*x])
+    assert DifferentialExtension(log(x**y), x)._important_attrs == \
+        (Poly(y*t0, t0), Poly(1, t0), [Poly(1, x), Poly(1/x, t0)], [x, t0],
+        [Lambda(i, log(i))], [(y*log(x), log(x**y))], [None, 'log'],
+        [None, x])
+    assert DifferentialExtension(log(sqrt(x)), x)._important_attrs == \
+        (Poly(t0, t0), Poly(2, t0), [Poly(1, x), Poly(1/x, t0)], [x, t0],
+        [Lambda(i, log(i))], [(log(x)/2, log(sqrt(x)))], [None, 'log'],
+        [None, x])
+
+
+def test_DifferentialExtension_handle_first():
+    assert DifferentialExtension(exp(x)*log(x), x, handle_first='log')._important_attrs == \
+        (Poly(t0*t1, t1), Poly(1, t1), [Poly(1, x), Poly(1/x, t0),
+        Poly(t1, t1)], [x, t0, t1], [Lambda(i, log(i)), Lambda(i, exp(i))],
+        [], [None, 'log', 'exp'], [None, x, x])
+    assert DifferentialExtension(exp(x)*log(x), x, handle_first='exp')._important_attrs == \
+        (Poly(t0*t1, t1), Poly(1, t1), [Poly(1, x), Poly(t0, t0),
+        Poly(1/x, t1)], [x, t0, t1], [Lambda(i, exp(i)), Lambda(i, log(i))],
+        [], [None, 'exp', 'log'], [None, x, x])
+
+    # This one must have the log first, regardless of what we set it to
+    # (because the log is inside of the exponential: x**x == exp(x*log(x)))
+    assert DifferentialExtension(-x**x*log(x)**2 + x**x - x**x/x, x,
+    handle_first='exp')._important_attrs == \
+        DifferentialExtension(-x**x*log(x)**2 + x**x - x**x/x, x,
+        handle_first='log')._important_attrs == \
+        (Poly((-1 + x - x*t0**2)*t1, t1), Poly(x, t1),
+            [Poly(1, x), Poly(1/x, t0), Poly((1 + t0)*t1, t1)], [x, t0, t1],
+            [Lambda(i, log(i)), Lambda(i, exp(t0*i))], [(exp(x*log(x)), x**x)],
+            [None, 'log', 'exp'], [None, x, t0*x])
+
+
+def test_DifferentialExtension_all_attrs():
+    # Test 'unimportant' attributes
+    DE = DifferentialExtension(exp(x)*log(x), x, handle_first='exp')
+    assert DE.f == exp(x)*log(x)
+    assert DE.newf == t0*t1
+    assert DE.x == x
+    assert DE.cases == ['base', 'exp', 'primitive']
+    assert DE.case == 'primitive'
+
+    assert DE.level == -1
+    assert DE.t == t1 == DE.T[DE.level]
+    assert DE.d == Poly(1/x, t1) == DE.D[DE.level]
+    raises(ValueError, lambda: DE.increment_level())
+    DE.decrement_level()
+    assert DE.level == -2
+    assert DE.t == t0 == DE.T[DE.level]
+    assert DE.d == Poly(t0, t0) == DE.D[DE.level]
+    assert DE.case == 'exp'
+    DE.decrement_level()
+    assert DE.level == -3
+    assert DE.t == x == DE.T[DE.level] == DE.x
+    assert DE.d == Poly(1, x) == DE.D[DE.level]
+    assert DE.case == 'base'
+    raises(ValueError, lambda: DE.decrement_level())
+    DE.increment_level()
+    DE.increment_level()
+    assert DE.level == -1
+    assert DE.t == t1 == DE.T[DE.level]
+    assert DE.d == Poly(1/x, t1) == DE.D[DE.level]
+    assert DE.case == 'primitive'
+
+    # Test methods
+    assert DE.indices('log') == [2]
+    assert DE.indices('exp') == [1]
+
+
+def test_DifferentialExtension_extension_flag():
+    raises(ValueError, lambda: DifferentialExtension(extension={'T': [x, t]}))
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]})
+    assert DE._important_attrs == (None, None, [Poly(1, x), Poly(t, t)], [x, t],
+        None, None, None, None)
+    assert DE.d == Poly(t, t)
+    assert DE.t == t
+    assert DE.level == -1
+    assert DE.cases == ['base', 'exp']
+    assert DE.x == x
+    assert DE.case == 'exp'
+
+    DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)],
+        'exts': [None, 'exp'], 'extargs': [None, x]})
+    assert DE._important_attrs == (None, None, [Poly(1, x), Poly(t, t)], [x, t],
+        None, None, [None, 'exp'], [None, x])
+    raises(ValueError, lambda: DifferentialExtension())
+
+
+def test_DifferentialExtension_misc():
+    # Odd ends
+    assert DifferentialExtension(sin(y)*exp(x), x)._important_attrs == \
+        (Poly(sin(y)*t0, t0, domain='ZZ[sin(y)]'), Poly(1, t0, domain='ZZ'),
+        [Poly(1, x, domain='ZZ'), Poly(t0, t0, domain='ZZ')], [x, t0],
+        [Lambda(i, exp(i))], [], [None, 'exp'], [None, x])
+    raises(NotImplementedError, lambda: DifferentialExtension(sin(x), x))
+    assert DifferentialExtension(10**x, x)._important_attrs == \
+        (Poly(t0, t0), Poly(1, t0), [Poly(1, x), Poly(log(10)*t0, t0)], [x, t0],
+        [Lambda(i, exp(i*log(10)))], [(exp(x*log(10)), 10**x)], [None, 'exp'],
+        [None, x*log(10)])
+    assert DifferentialExtension(log(x) + log(x**2), x)._important_attrs in [
+        (Poly(3*t0, t0), Poly(2, t0), [Poly(1, x), Poly(2/x, t0)], [x, t0],
+        [Lambda(i, log(i**2))], [], [None, ], [], [1], [x**2]),
+        (Poly(3*t0, t0), Poly(1, t0), [Poly(1, x), Poly(1/x, t0)], [x, t0],
+        [Lambda(i, log(i))], [], [None, 'log'], [None, x])]
+    assert DifferentialExtension(S.Zero, x)._important_attrs == \
+        (Poly(0, x), Poly(1, x), [Poly(1, x)], [x], [], [], [None], [None])
+    assert DifferentialExtension(tan(atan(x).rewrite(log)), x)._important_attrs == \
+        (Poly(x, x), Poly(1, x), [Poly(1, x)], [x], [], [], [None], [None])
+
+
+def test_DifferentialExtension_Rothstein():
+    # Rothstein's integral
+    f = (2581284541*exp(x) + 1757211400)/(39916800*exp(3*x) +
+    119750400*exp(x)**2 + 119750400*exp(x) + 39916800)*exp(1/(exp(x) + 1) - 10*x)
+    assert DifferentialExtension(f, x)._important_attrs == \
+        (Poly((1757211400 + 2581284541*t0)*t1, t1), Poly(39916800 +
+        119750400*t0 + 119750400*t0**2 + 39916800*t0**3, t1),
+        [Poly(1, x), Poly(t0, t0), Poly(-(10 + 21*t0 + 10*t0**2)/(1 + 2*t0 +
+        t0**2)*t1, t1, domain='ZZ(t0)')], [x, t0, t1],
+        [Lambda(i, exp(i)), Lambda(i, exp(1/(t0 + 1) - 10*i))], [],
+        [None, 'exp', 'exp'], [None, x, 1/(t0 + 1) - 10*x])
+
+
+class _TestingException(Exception):
+    """Dummy Exception class for testing."""
+    pass
+
+
+def test_DecrementLevel():
+    DE = DifferentialExtension(x*log(exp(x) + 1), x)
+    assert DE.level == -1
+    assert DE.t == t1
+    assert DE.d == Poly(t0/(t0 + 1), t1)
+    assert DE.case == 'primitive'
+
+    with DecrementLevel(DE):
+        assert DE.level == -2
+        assert DE.t == t0
+        assert DE.d == Poly(t0, t0)
+        assert DE.case == 'exp'
+
+        with DecrementLevel(DE):
+            assert DE.level == -3
+            assert DE.t == x
+            assert DE.d == Poly(1, x)
+            assert DE.case == 'base'
+
+        assert DE.level == -2
+        assert DE.t == t0
+        assert DE.d == Poly(t0, t0)
+        assert DE.case == 'exp'
+
+    assert DE.level == -1
+    assert DE.t == t1
+    assert DE.d == Poly(t0/(t0 + 1), t1)
+    assert DE.case == 'primitive'
+
+    # Test that __exit__ is called after an exception correctly
+    try:
+        with DecrementLevel(DE):
+            raise _TestingException
+    except _TestingException:
+        pass
+    else:
+        raise AssertionError("Did not raise.")
+
+    assert DE.level == -1
+    assert DE.t == t1
+    assert DE.d == Poly(t0/(t0 + 1), t1)
+    assert DE.case == 'primitive'
+
+
+def test_risch_integrate():
+    assert risch_integrate(t0*exp(x), x) == t0*exp(x)
+    assert risch_integrate(sin(x), x, rewrite_complex=True) == -exp(I*x)/2 - exp(-I*x)/2
+
+    # From my GSoC writeup
+    assert risch_integrate((1 + 2*x**2 + x**4 + 2*x**3*exp(2*x**2))/
+    (x**4*exp(x**2) + 2*x**2*exp(x**2) + exp(x**2)), x) == \
+        NonElementaryIntegral(exp(-x**2), x) + exp(x**2)/(1 + x**2)
+
+
+    assert risch_integrate(0, x) == 0
+
+    # also tests prde_cancel()
+    e1 = log(x/exp(x) + 1)
+    ans1 = risch_integrate(e1, x)
+    assert ans1 == (x*log(x*exp(-x) + 1) + NonElementaryIntegral((x**2 - x)/(x + exp(x)), x))
+    assert cancel(diff(ans1, x) - e1) == 0
+
+    # also tests issue #10798
+    e2 = (log(-1/y)/2 - log(1/y)/2)/y - (log(1 - 1/y)/2 - log(1 + 1/y)/2)/y
+    ans2 = risch_integrate(e2, y)
+    assert ans2 == log(1/y)*log(1 - 1/y)/2 - log(1/y)*log(1 + 1/y)/2 + \
+            NonElementaryIntegral((I*pi*y**2 - 2*y*log(1/y) - I*pi)/(2*y**3 - 2*y), y)
+    assert expand_log(cancel(diff(ans2, y) - e2), force=True) == 0
+
+    # These are tested here in addition to in test_DifferentialExtension above
+    # (symlogs) to test that backsubs works correctly.  The integrals should be
+    # written in terms of the original logarithms in the integrands.
+
+    # XXX: Unfortunately, making backsubs work on this one is a little
+    # trickier, because x**x is converted to exp(x*log(x)), and so log(x**x)
+    # is converted to x*log(x). (x**2*log(x)).subs(x*log(x), log(x**x)) is
+    # smart enough, the issue is that these splits happen at different places
+    # in the algorithm.  Maybe a heuristic is in order
+    assert risch_integrate(log(x**x), x) == x**2*log(x)/2 - x**2/4
+
+    assert risch_integrate(log(x**y), x) == x*log(x**y) - x*y
+    assert risch_integrate(log(sqrt(x)), x) == x*log(sqrt(x)) - x/2
+
+
+def test_risch_integrate_float():
+    assert risch_integrate((-60*exp(x) - 19.2*exp(4*x))*exp(4*x), x) == -2.4*exp(8*x) - 12.0*exp(5*x)
+
+
+def test_NonElementaryIntegral():
+    assert isinstance(risch_integrate(exp(x**2), x), NonElementaryIntegral)
+    assert isinstance(risch_integrate(x**x*log(x), x), NonElementaryIntegral)
+    # Make sure methods of Integral still give back a NonElementaryIntegral
+    assert isinstance(NonElementaryIntegral(x**x*t0, x).subs(t0, log(x)), NonElementaryIntegral)
+
+
+def test_xtothex():
+    a = risch_integrate(x**x, x)
+    assert a == NonElementaryIntegral(x**x, x)
+    assert isinstance(a, NonElementaryIntegral)
+
+
+def test_DifferentialExtension_equality():
+    DE1 = DE2 = DifferentialExtension(log(x), x)
+    assert DE1 == DE2
+
+
+def test_DifferentialExtension_printing():
+    DE = DifferentialExtension(exp(2*x**2) + log(exp(x**2) + 1), x)
+    assert repr(DE) == ("DifferentialExtension(dict([('f', exp(2*x**2) + log(exp(x**2) + 1)), "
+        "('x', x), ('T', [x, t0, t1]), ('D', [Poly(1, x, domain='ZZ'), Poly(2*x*t0, t0, domain='ZZ[x]'), "
+        "Poly(2*t0*x/(t0 + 1), t1, domain='ZZ(x,t0)')]), ('fa', Poly(t1 + t0**2, t1, domain='ZZ[t0]')), "
+        "('fd', Poly(1, t1, domain='ZZ')), ('Tfuncs', [Lambda(i, exp(i**2)), Lambda(i, log(t0 + 1))]), "
+        "('backsubs', []), ('exts', [None, 'exp', 'log']), ('extargs', [None, x**2, t0 + 1]), "
+        "('cases', ['base', 'exp', 'primitive']), ('case', 'primitive'), ('t', t1), "
+        "('d', Poly(2*t0*x/(t0 + 1), t1, domain='ZZ(x,t0)')), ('newf', t0**2 + t1), ('level', -1), "
+        "('dummy', False)]))")
+
+    assert str(DE) == ("DifferentialExtension({fa=Poly(t1 + t0**2, t1, domain='ZZ[t0]'), "
+        "fd=Poly(1, t1, domain='ZZ'), D=[Poly(1, x, domain='ZZ'), Poly(2*x*t0, t0, domain='ZZ[x]'), "
+        "Poly(2*t0*x/(t0 + 1), t1, domain='ZZ(x,t0)')]})")
+
+
+def test_issue_23948():
+    f = (
+        ( (-2*x**5 + 28*x**4 - 144*x**3 + 324*x**2 - 270*x)*log(x)**2
+         +(-4*x**6 + 56*x**5 - 288*x**4 + 648*x**3 - 540*x**2)*log(x)
+         +(2*x**5 - 28*x**4 + 144*x**3 - 324*x**2 + 270*x)*exp(x)
+         +(2*x**5 - 28*x**4 + 144*x**3 - 324*x**2 + 270*x)*log(5)
+         -2*x**7 + 26*x**6 - 116*x**5 + 180*x**4 + 54*x**3 - 270*x**2
+        )*log(-log(x)**2 - 2*x*log(x) + exp(x) + log(5) - x**2 - x)**2
+       +( (4*x**5 - 44*x**4 + 168*x**3 - 216*x**2 - 108*x + 324)*log(x)
+         +(-2*x**5 + 24*x**4 - 108*x**3 + 216*x**2 - 162*x)*exp(x)
+         +4*x**6 - 42*x**5 + 144*x**4 - 108*x**3 - 324*x**2 + 486*x
+        )*log(-log(x)**2 - 2*x*log(x) + exp(x) + log(5) - x**2 - x)
+    )/(x*exp(x)**2*log(x)**2 + 2*x**2*exp(x)**2*log(x) - x*exp(x)**3
+       +(-x*log(5) + x**3 + x**2)*exp(x)**2)
+
+    F = ((x**4 - 12*x**3 + 54*x**2 - 108*x + 81)*exp(-2*x)
+        *log(-x**2 - 2*x*log(x) - x + exp(x) - log(x)**2 + log(5))**2)
+
+    assert risch_integrate(f, x) == F

rl/Lib/site-packages/sympy/integrals/tests/test_singularityfunctions.py

@@ -0,0 +1,22 @@
+from sympy.integrals.singularityfunctions import singularityintegrate
+from sympy.core.function import Function
+from sympy.core.symbol import symbols
+from sympy.functions.special.singularity_functions import SingularityFunction
+
+x, a, n, y = symbols('x a n y')
+f = Function('f')
+
+
+def test_singularityintegrate():
+    assert singularityintegrate(x, x) is None
+    assert singularityintegrate(x + SingularityFunction(x, 9, 1), x) is None
+
+    assert 4*singularityintegrate(SingularityFunction(x, a, 3), x) == 4*SingularityFunction(x, a, 4)/4
+    assert singularityintegrate(5*SingularityFunction(x, 5, -2), x) == 5*SingularityFunction(x, 5, -1)
+    assert singularityintegrate(6*SingularityFunction(x, 5, -1), x) == 6*SingularityFunction(x, 5, 0)
+    assert singularityintegrate(x*SingularityFunction(x, 0, -1), x) == 0
+    assert singularityintegrate((x - 5)*SingularityFunction(x, 5, -1), x) == 0
+    assert singularityintegrate(SingularityFunction(x, 0, -1) * f(x), x) == f(0) * SingularityFunction(x, 0, 0)
+    assert singularityintegrate(SingularityFunction(x, 1, -1) * f(x), x) == f(1) * SingularityFunction(x, 1, 0)
+    assert singularityintegrate(y*SingularityFunction(x, 0, -1)**2, x) == \
+        y*SingularityFunction(0, 0, -1)*SingularityFunction(x, 0, 0)

rl/Lib/site-packages/sympy/integrals/tests/test_transforms.py

@@ -0,0 +1,636 @@
+from sympy.integrals.transforms import (
+    mellin_transform, inverse_mellin_transform,
+    fourier_transform, inverse_fourier_transform,
+    sine_transform, inverse_sine_transform,
+    cosine_transform, inverse_cosine_transform,
+    hankel_transform, inverse_hankel_transform,
+    FourierTransform, SineTransform, CosineTransform, InverseFourierTransform,
+    InverseSineTransform, InverseCosineTransform, IntegralTransformError)
+from sympy.integrals.laplace import (
+    laplace_transform, inverse_laplace_transform)
+from sympy.core.function import Function, expand_mul
+from sympy.core import EulerGamma
+from sympy.core.numbers import I, Rational, oo, pi
+from sympy.core.singleton import S
+from sympy.core.symbol import Symbol, symbols
+from sympy.functions.combinatorial.factorials import factorial
+from sympy.functions.elementary.complexes import re, unpolarify
+from sympy.functions.elementary.exponential import exp, exp_polar, log
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.trigonometric import atan, cos, sin, tan
+from sympy.functions.special.bessel import besseli, besselj, besselk, bessely
+from sympy.functions.special.delta_functions import Heaviside
+from sympy.functions.special.error_functions import erf, expint
+from sympy.functions.special.gamma_functions import gamma
+from sympy.functions.special.hyper import meijerg
+from sympy.simplify.gammasimp import gammasimp
+from sympy.simplify.hyperexpand import hyperexpand
+from sympy.simplify.trigsimp import trigsimp
+from sympy.testing.pytest import XFAIL, slow, skip, raises
+from sympy.abc import x, s, a, b, c, d
+
+
+nu, beta, rho = symbols('nu beta rho')
+
+
+def test_undefined_function():
+    from sympy.integrals.transforms import MellinTransform
+    f = Function('f')
+    assert mellin_transform(f(x), x, s) == MellinTransform(f(x), x, s)
+    assert mellin_transform(f(x) + exp(-x), x, s) == \
+        (MellinTransform(f(x), x, s) + gamma(s + 1)/s, (0, oo), True)
+
+
+def test_free_symbols():
+    f = Function('f')
+    assert mellin_transform(f(x), x, s).free_symbols == {s}
+    assert mellin_transform(f(x)*a, x, s).free_symbols == {s, a}
+
+
+def test_as_integral():
+    from sympy.integrals.integrals import Integral
+    f = Function('f')
+    assert mellin_transform(f(x), x, s).rewrite('Integral') == \
+        Integral(x**(s - 1)*f(x), (x, 0, oo))
+    assert fourier_transform(f(x), x, s).rewrite('Integral') == \
+        Integral(f(x)*exp(-2*I*pi*s*x), (x, -oo, oo))
+    assert laplace_transform(f(x), x, s, noconds=True).rewrite('Integral') == \
+        Integral(f(x)*exp(-s*x), (x, 0, oo))
+    assert str(2*pi*I*inverse_mellin_transform(f(s), s, x, (a, b)).rewrite('Integral')) \
+        == "Integral(f(s)/x**s, (s, _c - oo*I, _c + oo*I))"
+    assert str(2*pi*I*inverse_laplace_transform(f(s), s, x).rewrite('Integral')) == \
+        "Integral(f(s)*exp(s*x), (s, _c - oo*I, _c + oo*I))"
+    assert inverse_fourier_transform(f(s), s, x).rewrite('Integral') == \
+        Integral(f(s)*exp(2*I*pi*s*x), (s, -oo, oo))
+
+# NOTE this is stuck in risch because meijerint cannot handle it
+
+
+@slow
+@XFAIL
+def test_mellin_transform_fail():
+    skip("Risch takes forever.")
+
+    MT = mellin_transform
+
+    bpos = symbols('b', positive=True)
+    # bneg = symbols('b', negative=True)
+
+    expr = (sqrt(x + b**2) + b)**a/sqrt(x + b**2)
+    # TODO does not work with bneg, argument wrong. Needs changes to matching.
+    assert MT(expr.subs(b, -bpos), x, s) == \
+        ((-1)**(a + 1)*2**(a + 2*s)*bpos**(a + 2*s - 1)*gamma(a + s)
+         *gamma(1 - a - 2*s)/gamma(1 - s),
+            (-re(a), -re(a)/2 + S.Half), True)
+
+    expr = (sqrt(x + b**2) + b)**a
+    assert MT(expr.subs(b, -bpos), x, s) == \
+        (
+            2**(a + 2*s)*a*bpos**(a + 2*s)*gamma(-a - 2*
+                   s)*gamma(a + s)/gamma(-s + 1),
+            (-re(a), -re(a)/2), True)
+
+    # Test exponent 1:
+    assert MT(expr.subs({b: -bpos, a: 1}), x, s) == \
+        (-bpos**(2*s + 1)*gamma(s)*gamma(-s - S.Half)/(2*sqrt(pi)),
+            (-1, Rational(-1, 2)), True)
+
+
+def test_mellin_transform():
+    from sympy.functions.elementary.miscellaneous import (Max, Min)
+    MT = mellin_transform
+
+    bpos = symbols('b', positive=True)
+
+    # 8.4.2
+    assert MT(x**nu*Heaviside(x - 1), x, s) == \
+        (-1/(nu + s), (-oo, -re(nu)), True)
+    assert MT(x**nu*Heaviside(1 - x), x, s) == \
+        (1/(nu + s), (-re(nu), oo), True)
+
+    assert MT((1 - x)**(beta - 1)*Heaviside(1 - x), x, s) == \
+        (gamma(beta)*gamma(s)/gamma(beta + s), (0, oo), re(beta) > 0)
+    assert MT((x - 1)**(beta - 1)*Heaviside(x - 1), x, s) == \
+        (gamma(beta)*gamma(1 - beta - s)/gamma(1 - s),
+            (-oo, 1 - re(beta)), re(beta) > 0)
+
+    assert MT((1 + x)**(-rho), x, s) == \
+        (gamma(s)*gamma(rho - s)/gamma(rho), (0, re(rho)), True)
+
+    assert MT(abs(1 - x)**(-rho), x, s) == (
+        2*sin(pi*rho/2)*gamma(1 - rho)*
+        cos(pi*(s - rho/2))*gamma(s)*gamma(rho-s)/pi,
+        (0, re(rho)), re(rho) < 1)
+    mt = MT((1 - x)**(beta - 1)*Heaviside(1 - x)
+            + a*(x - 1)**(beta - 1)*Heaviside(x - 1), x, s)
+    assert mt[1], mt[2] == ((0, -re(beta) + 1), re(beta) > 0)
+
+    assert MT((x**a - b**a)/(x - b), x, s)[0] == \
+        pi*b**(a + s - 1)*sin(pi*a)/(sin(pi*s)*sin(pi*(a + s)))
+    assert MT((x**a - bpos**a)/(x - bpos), x, s) == \
+        (pi*bpos**(a + s - 1)*sin(pi*a)/(sin(pi*s)*sin(pi*(a + s))),
+            (Max(0, -re(a)), Min(1, 1 - re(a))), True)
+
+    expr = (sqrt(x + b**2) + b)**a
+    assert MT(expr.subs(b, bpos), x, s) == \
+        (-a*(2*bpos)**(a + 2*s)*gamma(s)*gamma(-a - 2*s)/gamma(-a - s + 1),
+            (0, -re(a)/2), True)
+
+    expr = (sqrt(x + b**2) + b)**a/sqrt(x + b**2)
+    assert MT(expr.subs(b, bpos), x, s) == \
+        (2**(a + 2*s)*bpos**(a + 2*s - 1)*gamma(s)
+                                         *gamma(1 - a - 2*s)/gamma(1 - a - s),
+            (0, -re(a)/2 + S.Half), True)
+
+    # 8.4.2
+    assert MT(exp(-x), x, s) == (gamma(s), (0, oo), True)
+    assert MT(exp(-1/x), x, s) == (gamma(-s), (-oo, 0), True)
+
+    # 8.4.5
+    assert MT(log(x)**4*Heaviside(1 - x), x, s) == (24/s**5, (0, oo), True)
+    assert MT(log(x)**3*Heaviside(x - 1), x, s) == (6/s**4, (-oo, 0), True)
+    assert MT(log(x + 1), x, s) == (pi/(s*sin(pi*s)), (-1, 0), True)
+    assert MT(log(1/x + 1), x, s) == (pi/(s*sin(pi*s)), (0, 1), True)
+    assert MT(log(abs(1 - x)), x, s) == (pi/(s*tan(pi*s)), (-1, 0), True)
+    assert MT(log(abs(1 - 1/x)), x, s) == (pi/(s*tan(pi*s)), (0, 1), True)
+
+    # 8.4.14
+    assert MT(erf(sqrt(x)), x, s) == \
+        (-gamma(s + S.Half)/(sqrt(pi)*s), (Rational(-1, 2), 0), True)
+
+
+def test_mellin_transform2():
+    MT = mellin_transform
+    # TODO we cannot currently do these (needs summation of 3F2(-1))
+    #      this also implies that they cannot be written as a single g-function
+    #      (although this is possible)
+    mt = MT(log(x)/(x + 1), x, s)
+    assert mt[1:] == ((0, 1), True)
+    assert not hyperexpand(mt[0], allow_hyper=True).has(meijerg)
+    mt = MT(log(x)**2/(x + 1), x, s)
+    assert mt[1:] == ((0, 1), True)
+    assert not hyperexpand(mt[0], allow_hyper=True).has(meijerg)
+    mt = MT(log(x)/(x + 1)**2, x, s)
+    assert mt[1:] == ((0, 2), True)
+    assert not hyperexpand(mt[0], allow_hyper=True).has(meijerg)
+
+
+@slow
+def test_mellin_transform_bessel():
+    from sympy.functions.elementary.miscellaneous import Max
+    MT = mellin_transform
+
+    # 8.4.19
+    assert MT(besselj(a, 2*sqrt(x)), x, s) == \
+        (gamma(a/2 + s)/gamma(a/2 - s + 1), (-re(a)/2, Rational(3, 4)), True)
+    assert MT(sin(sqrt(x))*besselj(a, sqrt(x)), x, s) == \
+        (2**a*gamma(-2*s + S.Half)*gamma(a/2 + s + S.Half)/(
+        gamma(-a/2 - s + 1)*gamma(a - 2*s + 1)), (
+        -re(a)/2 - S.Half, Rational(1, 4)), True)
+    assert MT(cos(sqrt(x))*besselj(a, sqrt(x)), x, s) == \
+        (2**a*gamma(a/2 + s)*gamma(-2*s + S.Half)/(
+        gamma(-a/2 - s + S.Half)*gamma(a - 2*s + 1)), (
+        -re(a)/2, Rational(1, 4)), True)
+    assert MT(besselj(a, sqrt(x))**2, x, s) == \
+        (gamma(a + s)*gamma(S.Half - s)
+         / (sqrt(pi)*gamma(1 - s)*gamma(1 + a - s)),
+            (-re(a), S.Half), True)
+    assert MT(besselj(a, sqrt(x))*besselj(-a, sqrt(x)), x, s) == \
+        (gamma(s)*gamma(S.Half - s)
+         / (sqrt(pi)*gamma(1 - a - s)*gamma(1 + a - s)),
+            (0, S.Half), True)
+    # NOTE: prudnikov gives the strip below as (1/2 - re(a), 1). As far as
+    #       I can see this is wrong (since besselj(z) ~ 1/sqrt(z) for z large)
+    assert MT(besselj(a - 1, sqrt(x))*besselj(a, sqrt(x)), x, s) == \
+        (gamma(1 - s)*gamma(a + s - S.Half)
+         / (sqrt(pi)*gamma(Rational(3, 2) - s)*gamma(a - s + S.Half)),
+            (S.Half - re(a), S.Half), True)
+    assert MT(besselj(a, sqrt(x))*besselj(b, sqrt(x)), x, s) == \
+        (4**s*gamma(1 - 2*s)*gamma((a + b)/2 + s)
+         / (gamma(1 - s + (b - a)/2)*gamma(1 - s + (a - b)/2)
+            *gamma( 1 - s + (a + b)/2)),
+            (-(re(a) + re(b))/2, S.Half), True)
+    assert MT(besselj(a, sqrt(x))**2 + besselj(-a, sqrt(x))**2, x, s)[1:] == \
+        ((Max(re(a), -re(a)), S.Half), True)
+
+    # Section 8.4.20
+    assert MT(bessely(a, 2*sqrt(x)), x, s) == \
+        (-cos(pi*(a/2 - s))*gamma(s - a/2)*gamma(s + a/2)/pi,
+            (Max(-re(a)/2, re(a)/2), Rational(3, 4)), True)
+    assert MT(sin(sqrt(x))*bessely(a, sqrt(x)), x, s) == \
+        (-4**s*sin(pi*(a/2 - s))*gamma(S.Half - 2*s)
+         * gamma((1 - a)/2 + s)*gamma((1 + a)/2 + s)
+         / (sqrt(pi)*gamma(1 - s - a/2)*gamma(1 - s + a/2)),
+            (Max(-(re(a) + 1)/2, (re(a) - 1)/2), Rational(1, 4)), True)
+    assert MT(cos(sqrt(x))*bessely(a, sqrt(x)), x, s) == \
+        (-4**s*cos(pi*(a/2 - s))*gamma(s - a/2)*gamma(s + a/2)*gamma(S.Half - 2*s)
+         / (sqrt(pi)*gamma(S.Half - s - a/2)*gamma(S.Half - s + a/2)),
+            (Max(-re(a)/2, re(a)/2), Rational(1, 4)), True)
+    assert MT(besselj(a, sqrt(x))*bessely(a, sqrt(x)), x, s) == \
+        (-cos(pi*s)*gamma(s)*gamma(a + s)*gamma(S.Half - s)
+         / (pi**S('3/2')*gamma(1 + a - s)),
+            (Max(-re(a), 0), S.Half), True)
+    assert MT(besselj(a, sqrt(x))*bessely(b, sqrt(x)), x, s) == \
+        (-4**s*cos(pi*(a/2 - b/2 + s))*gamma(1 - 2*s)
+         * gamma(a/2 - b/2 + s)*gamma(a/2 + b/2 + s)
+         / (pi*gamma(a/2 - b/2 - s + 1)*gamma(a/2 + b/2 - s + 1)),
+            (Max((-re(a) + re(b))/2, (-re(a) - re(b))/2), S.Half), True)
+    # NOTE bessely(a, sqrt(x))**2 and bessely(a, sqrt(x))*bessely(b, sqrt(x))
+    # are a mess (no matter what way you look at it ...)
+    assert MT(bessely(a, sqrt(x))**2, x, s)[1:] == \
+             ((Max(-re(a), 0, re(a)), S.Half), True)
+
+    # Section 8.4.22
+    # TODO we can't do any of these (delicate cancellation)
+
+    # Section 8.4.23
+    assert MT(besselk(a, 2*sqrt(x)), x, s) == \
+        (gamma(
+         s - a/2)*gamma(s + a/2)/2, (Max(-re(a)/2, re(a)/2), oo), True)
+    assert MT(besselj(a, 2*sqrt(2*sqrt(x)))*besselk(
+        a, 2*sqrt(2*sqrt(x))), x, s) == (4**(-s)*gamma(2*s)*
+        gamma(a/2 + s)/(2*gamma(a/2 - s + 1)), (Max(0, -re(a)/2), oo), True)
+    # TODO bessely(a, x)*besselk(a, x) is a mess
+    assert MT(besseli(a, sqrt(x))*besselk(a, sqrt(x)), x, s) == \
+        (gamma(s)*gamma(
+        a + s)*gamma(-s + S.Half)/(2*sqrt(pi)*gamma(a - s + 1)),
+        (Max(-re(a), 0), S.Half), True)
+    assert MT(besseli(b, sqrt(x))*besselk(a, sqrt(x)), x, s) == \
+        (2**(2*s - 1)*gamma(-2*s + 1)*gamma(-a/2 + b/2 + s)* \
+        gamma(a/2 + b/2 + s)/(gamma(-a/2 + b/2 - s + 1)* \
+        gamma(a/2 + b/2 - s + 1)), (Max(-re(a)/2 - re(b)/2, \
+        re(a)/2 - re(b)/2), S.Half), True)
+
+    # TODO products of besselk are a mess
+
+    mt = MT(exp(-x/2)*besselk(a, x/2), x, s)
+    mt0 = gammasimp(trigsimp(gammasimp(mt[0].expand(func=True))))
+    assert mt0 == 2*pi**Rational(3, 2)*cos(pi*s)*gamma(S.Half - s)/(
+        (cos(2*pi*a) - cos(2*pi*s))*gamma(-a - s + 1)*gamma(a - s + 1))
+    assert mt[1:] == ((Max(-re(a), re(a)), oo), True)
+    # TODO exp(x/2)*besselk(a, x/2) [etc] cannot currently be done
+    # TODO various strange products of special orders
+
+
+@slow
+def test_expint():
+    from sympy.functions.elementary.miscellaneous import Max
+    from sympy.functions.special.error_functions import Ci, E1, Si
+    from sympy.simplify.simplify import simplify
+
+    aneg = Symbol('a', negative=True)
+    u = Symbol('u', polar=True)
+
+    assert mellin_transform(E1(x), x, s) == (gamma(s)/s, (0, oo), True)
+    assert inverse_mellin_transform(gamma(s)/s, s, x,
+              (0, oo)).rewrite(expint).expand() == E1(x)
+    assert mellin_transform(expint(a, x), x, s) == \
+        (gamma(s)/(a + s - 1), (Max(1 - re(a), 0), oo), True)
+    # XXX IMT has hickups with complicated strips ...
+    assert simplify(unpolarify(
+                    inverse_mellin_transform(gamma(s)/(aneg + s - 1), s, x,
+                  (1 - aneg, oo)).rewrite(expint).expand(func=True))) == \
+        expint(aneg, x)
+
+    assert mellin_transform(Si(x), x, s) == \
+        (-2**s*sqrt(pi)*gamma(s/2 + S.Half)/(
+        2*s*gamma(-s/2 + 1)), (-1, 0), True)
+    assert inverse_mellin_transform(-2**s*sqrt(pi)*gamma((s + 1)/2)
+                                    /(2*s*gamma(-s/2 + 1)), s, x, (-1, 0)) \
+        == Si(x)
+
+    assert mellin_transform(Ci(sqrt(x)), x, s) == \
+        (-2**(2*s - 1)*sqrt(pi)*gamma(s)/(s*gamma(-s + S.Half)), (0, 1), True)
+    assert inverse_mellin_transform(
+        -4**s*sqrt(pi)*gamma(s)/(2*s*gamma(-s + S.Half)),
+        s, u, (0, 1)).expand() == Ci(sqrt(u))
+
+
+@slow
+def test_inverse_mellin_transform():
+    from sympy.core.function import expand
+    from sympy.functions.elementary.miscellaneous import (Max, Min)
+    from sympy.functions.elementary.trigonometric import cot
+    from sympy.simplify.powsimp import powsimp
+    from sympy.simplify.simplify import simplify
+    IMT = inverse_mellin_transform
+
+    assert IMT(gamma(s), s, x, (0, oo)) == exp(-x)
+    assert IMT(gamma(-s), s, x, (-oo, 0)) == exp(-1/x)
+    assert simplify(IMT(s/(2*s**2 - 2), s, x, (2, oo))) == \
+        (x**2 + 1)*Heaviside(1 - x)/(4*x)
+
+    # test passing "None"
+    assert IMT(1/(s**2 - 1), s, x, (-1, None)) == \
+        -x*Heaviside(-x + 1)/2 - Heaviside(x - 1)/(2*x)
+    assert IMT(1/(s**2 - 1), s, x, (None, 1)) == \
+        -x*Heaviside(-x + 1)/2 - Heaviside(x - 1)/(2*x)
+
+    # test expansion of sums
+    assert IMT(gamma(s) + gamma(s - 1), s, x, (1, oo)) == (x + 1)*exp(-x)/x
+
+    # test factorisation of polys
+    r = symbols('r', real=True)
+    assert IMT(1/(s**2 + 1), s, exp(-x), (None, oo)
+              ).subs(x, r).rewrite(sin).simplify() \
+        == sin(r)*Heaviside(1 - exp(-r))
+
+    # test multiplicative substitution
+    _a, _b = symbols('a b', positive=True)
+    assert IMT(_b**(-s/_a)*factorial(s/_a)/s, s, x, (0, oo)) == exp(-_b*x**_a)
+    assert IMT(factorial(_a/_b + s/_b)/(_a + s), s, x, (-_a, oo)) == x**_a*exp(-x**_b)
+
+    def simp_pows(expr):
+        return simplify(powsimp(expand_mul(expr, deep=False), force=True)).replace(exp_polar, exp)
+
+    # Now test the inverses of all direct transforms tested above
+
+    # Section 8.4.2
+    nu = symbols('nu', real=True)
+    assert IMT(-1/(nu + s), s, x, (-oo, None)) == x**nu*Heaviside(x - 1)
+    assert IMT(1/(nu + s), s, x, (None, oo)) == x**nu*Heaviside(1 - x)
+    assert simp_pows(IMT(gamma(beta)*gamma(s)/gamma(s + beta), s, x, (0, oo))) \
+        == (1 - x)**(beta - 1)*Heaviside(1 - x)
+    assert simp_pows(IMT(gamma(beta)*gamma(1 - beta - s)/gamma(1 - s),
+                         s, x, (-oo, None))) \
+        == (x - 1)**(beta - 1)*Heaviside(x - 1)
+    assert simp_pows(IMT(gamma(s)*gamma(rho - s)/gamma(rho), s, x, (0, None))) \
+        == (1/(x + 1))**rho
+    assert simp_pows(IMT(d**c*d**(s - 1)*sin(pi*c)
+                         *gamma(s)*gamma(s + c)*gamma(1 - s)*gamma(1 - s - c)/pi,
+                         s, x, (Max(-re(c), 0), Min(1 - re(c), 1)))) \
+        == (x**c - d**c)/(x - d)
+
+    assert simplify(IMT(1/sqrt(pi)*(-c/2)*gamma(s)*gamma((1 - c)/2 - s)
+                        *gamma(-c/2 - s)/gamma(1 - c - s),
+                        s, x, (0, -re(c)/2))) == \
+        (1 + sqrt(x + 1))**c
+    assert simplify(IMT(2**(a + 2*s)*b**(a + 2*s - 1)*gamma(s)*gamma(1 - a - 2*s)
+                        /gamma(1 - a - s), s, x, (0, (-re(a) + 1)/2))) == \
+        b**(a - 1)*(b**2*(sqrt(1 + x/b**2) + 1)**a + x*(sqrt(1 + x/b**2) + 1
+        )**(a - 1))/(b**2 + x)
+    assert simplify(IMT(-2**(c + 2*s)*c*b**(c + 2*s)*gamma(s)*gamma(-c - 2*s)
+                        / gamma(-c - s + 1), s, x, (0, -re(c)/2))) == \
+        b**c*(sqrt(1 + x/b**2) + 1)**c
+
+    # Section 8.4.5
+    assert IMT(24/s**5, s, x, (0, oo)) == log(x)**4*Heaviside(1 - x)
+    assert expand(IMT(6/s**4, s, x, (-oo, 0)), force=True) == \
+        log(x)**3*Heaviside(x - 1)
+    assert IMT(pi/(s*sin(pi*s)), s, x, (-1, 0)) == log(x + 1)
+    assert IMT(pi/(s*sin(pi*s/2)), s, x, (-2, 0)) == log(x**2 + 1)
+    assert IMT(pi/(s*sin(2*pi*s)), s, x, (Rational(-1, 2), 0)) == log(sqrt(x) + 1)
+    assert IMT(pi/(s*sin(pi*s)), s, x, (0, 1)) == log(1 + 1/x)
+
+    # TODO
+    def mysimp(expr):
+        from sympy.core.function import expand
+        from sympy.simplify.powsimp import powsimp
+        from sympy.simplify.simplify import logcombine
+        return expand(
+            powsimp(logcombine(expr, force=True), force=True, deep=True),
+            force=True).replace(exp_polar, exp)
+
+    assert mysimp(mysimp(IMT(pi/(s*tan(pi*s)), s, x, (-1, 0)))) in [
+        log(1 - x)*Heaviside(1 - x) + log(x - 1)*Heaviside(x - 1),
+        log(x)*Heaviside(x - 1) + log(1 - 1/x)*Heaviside(x - 1) + log(-x +
+        1)*Heaviside(-x + 1)]
+    # test passing cot
+    assert mysimp(IMT(pi*cot(pi*s)/s, s, x, (0, 1))) in [
+        log(1/x - 1)*Heaviside(1 - x) + log(1 - 1/x)*Heaviside(x - 1),
+        -log(x)*Heaviside(-x + 1) + log(1 - 1/x)*Heaviside(x - 1) + log(-x +
+        1)*Heaviside(-x + 1), ]
+
+    # 8.4.14
+    assert IMT(-gamma(s + S.Half)/(sqrt(pi)*s), s, x, (Rational(-1, 2), 0)) == \
+        erf(sqrt(x))
+
+    # 8.4.19
+    assert simplify(IMT(gamma(a/2 + s)/gamma(a/2 - s + 1), s, x, (-re(a)/2, Rational(3, 4)))) \
+        == besselj(a, 2*sqrt(x))
+    assert simplify(IMT(2**a*gamma(S.Half - 2*s)*gamma(s + (a + 1)/2)
+                      / (gamma(1 - s - a/2)*gamma(1 - 2*s + a)),
+                      s, x, (-(re(a) + 1)/2, Rational(1, 4)))) == \
+        sin(sqrt(x))*besselj(a, sqrt(x))
+    assert simplify(IMT(2**a*gamma(a/2 + s)*gamma(S.Half - 2*s)
+                      / (gamma(S.Half - s - a/2)*gamma(1 - 2*s + a)),
+                      s, x, (-re(a)/2, Rational(1, 4)))) == \
+        cos(sqrt(x))*besselj(a, sqrt(x))
+    # TODO this comes out as an amazing mess, but simplifies nicely
+    assert simplify(IMT(gamma(a + s)*gamma(S.Half - s)
+                      / (sqrt(pi)*gamma(1 - s)*gamma(1 + a - s)),
+                      s, x, (-re(a), S.Half))) == \
+        besselj(a, sqrt(x))**2
+    assert simplify(IMT(gamma(s)*gamma(S.Half - s)
+                      / (sqrt(pi)*gamma(1 - s - a)*gamma(1 + a - s)),
+                      s, x, (0, S.Half))) == \
+        besselj(-a, sqrt(x))*besselj(a, sqrt(x))
+    assert simplify(IMT(4**s*gamma(-2*s + 1)*gamma(a/2 + b/2 + s)
+                      / (gamma(-a/2 + b/2 - s + 1)*gamma(a/2 - b/2 - s + 1)
+                         *gamma(a/2 + b/2 - s + 1)),
+                      s, x, (-(re(a) + re(b))/2, S.Half))) == \
+        besselj(a, sqrt(x))*besselj(b, sqrt(x))
+
+    # Section 8.4.20
+    # TODO this can be further simplified!
+    assert simplify(IMT(-2**(2*s)*cos(pi*a/2 - pi*b/2 + pi*s)*gamma(-2*s + 1) *
+                    gamma(a/2 - b/2 + s)*gamma(a/2 + b/2 + s) /
+                    (pi*gamma(a/2 - b/2 - s + 1)*gamma(a/2 + b/2 - s + 1)),
+                    s, x,
+                    (Max(-re(a)/2 - re(b)/2, -re(a)/2 + re(b)/2), S.Half))) == \
+                    besselj(a, sqrt(x))*-(besselj(-b, sqrt(x)) -
+                    besselj(b, sqrt(x))*cos(pi*b))/sin(pi*b)
+    # TODO more
+
+    # for coverage
+
+    assert IMT(pi/cos(pi*s), s, x, (0, S.Half)) == sqrt(x)/(x + 1)
+
+
+def test_fourier_transform():
+    from sympy.core.function import (expand, expand_complex, expand_trig)
+    from sympy.polys.polytools import factor
+    from sympy.simplify.simplify import simplify
+    FT = fourier_transform
+    IFT = inverse_fourier_transform
+
+    def simp(x):
+        return simplify(expand_trig(expand_complex(expand(x))))
+
+    def sinc(x):
+        return sin(pi*x)/(pi*x)
+    k = symbols('k', real=True)
+    f = Function("f")
+
+    # TODO for this to work with real a, need to expand abs(a*x) to abs(a)*abs(x)
+    a = symbols('a', positive=True)
+    b = symbols('b', positive=True)
+
+    posk = symbols('posk', positive=True)
+
+    # Test unevaluated form
+    assert fourier_transform(f(x), x, k) == FourierTransform(f(x), x, k)
+    assert inverse_fourier_transform(
+        f(k), k, x) == InverseFourierTransform(f(k), k, x)
+
+    # basic examples from wikipedia
+    assert simp(FT(Heaviside(1 - abs(2*a*x)), x, k)) == sinc(k/a)/a
+    # TODO IFT is a *mess*
+    assert simp(FT(Heaviside(1 - abs(a*x))*(1 - abs(a*x)), x, k)) == sinc(k/a)**2/a
+    # TODO IFT
+
+    assert factor(FT(exp(-a*x)*Heaviside(x), x, k), extension=I) == \
+        1/(a + 2*pi*I*k)
+    # NOTE: the ift comes out in pieces
+    assert IFT(1/(a + 2*pi*I*x), x, posk,
+            noconds=False) == (exp(-a*posk), True)
+    assert IFT(1/(a + 2*pi*I*x), x, -posk,
+            noconds=False) == (0, True)
+    assert IFT(1/(a + 2*pi*I*x), x, symbols('k', negative=True),
+            noconds=False) == (0, True)
+    # TODO IFT without factoring comes out as meijer g
+
+    assert factor(FT(x*exp(-a*x)*Heaviside(x), x, k), extension=I) == \
+        1/(a + 2*pi*I*k)**2
+    assert FT(exp(-a*x)*sin(b*x)*Heaviside(x), x, k) == \
+        b/(b**2 + (a + 2*I*pi*k)**2)
+
+    assert FT(exp(-a*x**2), x, k) == sqrt(pi)*exp(-pi**2*k**2/a)/sqrt(a)
+    assert IFT(sqrt(pi/a)*exp(-(pi*k)**2/a), k, x) == exp(-a*x**2)
+    assert FT(exp(-a*abs(x)), x, k) == 2*a/(a**2 + 4*pi**2*k**2)
+    # TODO IFT (comes out as meijer G)
+
+    # TODO besselj(n, x), n an integer > 0 actually can be done...
+
+    # TODO are there other common transforms (no distributions!)?
+
+
+def test_sine_transform():
+    t = symbols("t")
+    w = symbols("w")
+    a = symbols("a")
+    f = Function("f")
+
+    # Test unevaluated form
+    assert sine_transform(f(t), t, w) == SineTransform(f(t), t, w)
+    assert inverse_sine_transform(
+        f(w), w, t) == InverseSineTransform(f(w), w, t)
+
+    assert sine_transform(1/sqrt(t), t, w) == 1/sqrt(w)
+    assert inverse_sine_transform(1/sqrt(w), w, t) == 1/sqrt(t)
+
+    assert sine_transform((1/sqrt(t))**3, t, w) == 2*sqrt(w)
+
+    assert sine_transform(t**(-a), t, w) == 2**(
+        -a + S.Half)*w**(a - 1)*gamma(-a/2 + 1)/gamma((a + 1)/2)
+    assert inverse_sine_transform(2**(-a + S(
+        1)/2)*w**(a - 1)*gamma(-a/2 + 1)/gamma(a/2 + S.Half), w, t) == t**(-a)
+
+    assert sine_transform(
+        exp(-a*t), t, w) == sqrt(2)*w/(sqrt(pi)*(a**2 + w**2))
+    assert inverse_sine_transform(
+        sqrt(2)*w/(sqrt(pi)*(a**2 + w**2)), w, t) == exp(-a*t)
+
+    assert sine_transform(
+        log(t)/t, t, w) == sqrt(2)*sqrt(pi)*-(log(w**2) + 2*EulerGamma)/4
+
+    assert sine_transform(
+        t*exp(-a*t**2), t, w) == sqrt(2)*w*exp(-w**2/(4*a))/(4*a**Rational(3, 2))
+    assert inverse_sine_transform(
+        sqrt(2)*w*exp(-w**2/(4*a))/(4*a**Rational(3, 2)), w, t) == t*exp(-a*t**2)
+
+
+def test_cosine_transform():
+    from sympy.functions.special.error_functions import (Ci, Si)
+
+    t = symbols("t")
+    w = symbols("w")
+    a = symbols("a")
+    f = Function("f")
+
+    # Test unevaluated form
+    assert cosine_transform(f(t), t, w) == CosineTransform(f(t), t, w)
+    assert inverse_cosine_transform(
+        f(w), w, t) == InverseCosineTransform(f(w), w, t)
+
+    assert cosine_transform(1/sqrt(t), t, w) == 1/sqrt(w)
+    assert inverse_cosine_transform(1/sqrt(w), w, t) == 1/sqrt(t)
+
+    assert cosine_transform(1/(
+        a**2 + t**2), t, w) == sqrt(2)*sqrt(pi)*exp(-a*w)/(2*a)
+
+    assert cosine_transform(t**(
+        -a), t, w) == 2**(-a + S.Half)*w**(a - 1)*gamma((-a + 1)/2)/gamma(a/2)
+    assert inverse_cosine_transform(2**(-a + S(
+        1)/2)*w**(a - 1)*gamma(-a/2 + S.Half)/gamma(a/2), w, t) == t**(-a)
+
+    assert cosine_transform(
+        exp(-a*t), t, w) == sqrt(2)*a/(sqrt(pi)*(a**2 + w**2))
+    assert inverse_cosine_transform(
+        sqrt(2)*a/(sqrt(pi)*(a**2 + w**2)), w, t) == exp(-a*t)
+
+    assert cosine_transform(exp(-a*sqrt(t))*cos(a*sqrt(
+        t)), t, w) == a*exp(-a**2/(2*w))/(2*w**Rational(3, 2))
+
+    assert cosine_transform(1/(a + t), t, w) == sqrt(2)*(
+        (-2*Si(a*w) + pi)*sin(a*w)/2 - cos(a*w)*Ci(a*w))/sqrt(pi)
+    assert inverse_cosine_transform(sqrt(2)*meijerg(((S.Half, 0), ()), (
+        (S.Half, 0, 0), (S.Half,)), a**2*w**2/4)/(2*pi), w, t) == 1/(a + t)
+
+    assert cosine_transform(1/sqrt(a**2 + t**2), t, w) == sqrt(2)*meijerg(
+        ((S.Half,), ()), ((0, 0), (S.Half,)), a**2*w**2/4)/(2*sqrt(pi))
+    assert inverse_cosine_transform(sqrt(2)*meijerg(((S.Half,), ()), ((0, 0), (S.Half,)), a**2*w**2/4)/(2*sqrt(pi)), w, t) == 1/(t*sqrt(a**2/t**2 + 1))
+
+
+def test_hankel_transform():
+    r = Symbol("r")
+    k = Symbol("k")
+    nu = Symbol("nu")
+    m = Symbol("m")
+    a = symbols("a")
+
+    assert hankel_transform(1/r, r, k, 0) == 1/k
+    assert inverse_hankel_transform(1/k, k, r, 0) == 1/r
+
+    assert hankel_transform(
+        1/r**m, r, k, 0) == 2**(-m + 1)*k**(m - 2)*gamma(-m/2 + 1)/gamma(m/2)
+    assert inverse_hankel_transform(
+        2**(-m + 1)*k**(m - 2)*gamma(-m/2 + 1)/gamma(m/2), k, r, 0) == r**(-m)
+
+    assert hankel_transform(1/r**m, r, k, nu) == (
+        2*2**(-m)*k**(m - 2)*gamma(-m/2 + nu/2 + 1)/gamma(m/2 + nu/2))
+    assert inverse_hankel_transform(2**(-m + 1)*k**(
+        m - 2)*gamma(-m/2 + nu/2 + 1)/gamma(m/2 + nu/2), k, r, nu) == r**(-m)
+
+    assert hankel_transform(r**nu*exp(-a*r), r, k, nu) == \
+        2**(nu + 1)*a*k**(-nu - 3)*(a**2/k**2 + 1)**(-nu - S(
+                                                     3)/2)*gamma(nu + Rational(3, 2))/sqrt(pi)
+    assert inverse_hankel_transform(
+        2**(nu + 1)*a*k**(-nu - 3)*(a**2/k**2 + 1)**(-nu - Rational(3, 2))*gamma(
+        nu + Rational(3, 2))/sqrt(pi), k, r, nu) == r**nu*exp(-a*r)
+
+
+def test_issue_7181():
+    assert mellin_transform(1/(1 - x), x, s) != None
+
+
+def test_issue_8882():
+    # This is the original test.
+    # from sympy import diff, Integral, integrate
+    # r = Symbol('r')
+    # psi = 1/r*sin(r)*exp(-(a0*r))
+    # h = -1/2*diff(psi, r, r) - 1/r*psi
+    # f = 4*pi*psi*h*r**2
+    # assert integrate(f, (r, -oo, 3), meijerg=True).has(Integral) == True
+
+    # To save time, only the critical part is included.
+    F = -a**(-s + 1)*(4 + 1/a**2)**(-s/2)*sqrt(1/a**2)*exp(-s*I*pi)* \
+        sin(s*atan(sqrt(1/a**2)/2))*gamma(s)
+    raises(IntegralTransformError, lambda:
+        inverse_mellin_transform(F, s, x, (-1, oo),
+        **{'as_meijerg': True, 'needeval': True}))
+
+
+def test_issue_12591():
+    x, y = symbols("x y", real=True)
+    assert fourier_transform(exp(x), x, y) == FourierTransform(exp(x), x, y)

rl/Lib/site-packages/sympy/integrals/tests/test_trigonometry.py

@@ -0,0 +1,98 @@
+from sympy.core import Ne, Rational, Symbol
+from sympy.functions import sin, cos, tan, csc, sec, cot, log, Piecewise
+from sympy.integrals.trigonometry import trigintegrate
+
+x = Symbol('x')
+
+
+def test_trigintegrate_odd():
+    assert trigintegrate(Rational(1), x) == x
+    assert trigintegrate(x, x) is None
+    assert trigintegrate(x**2, x) is None
+
+    assert trigintegrate(sin(x), x) == -cos(x)
+    assert trigintegrate(cos(x), x) == sin(x)
+
+    assert trigintegrate(sin(3*x), x) == -cos(3*x)/3
+    assert trigintegrate(cos(3*x), x) == sin(3*x)/3
+
+    y = Symbol('y')
+    assert trigintegrate(sin(y*x), x) == Piecewise(
+        (-cos(y*x)/y, Ne(y, 0)), (0, True))
+    assert trigintegrate(cos(y*x), x) == Piecewise(
+        (sin(y*x)/y, Ne(y, 0)), (x, True))
+    assert trigintegrate(sin(y*x)**2, x) == Piecewise(
+        ((x*y/2 - sin(x*y)*cos(x*y)/2)/y, Ne(y, 0)), (0, True))
+    assert trigintegrate(sin(y*x)*cos(y*x), x) == Piecewise(
+        (sin(x*y)**2/(2*y), Ne(y, 0)), (0, True))
+    assert trigintegrate(cos(y*x)**2, x) == Piecewise(
+        ((x*y/2 + sin(x*y)*cos(x*y)/2)/y, Ne(y, 0)), (x, True))
+
+    y = Symbol('y', positive=True)
+    # TODO: remove conds='none' below. For this to work we would have to rule
+    #       out (e.g. by trying solve) the condition y = 0, incompatible with
+    #       y.is_positive being True.
+    assert trigintegrate(sin(y*x), x, conds='none') == -cos(y*x)/y
+    assert trigintegrate(cos(y*x), x, conds='none') == sin(y*x)/y
+
+    assert trigintegrate(sin(x)*cos(x), x) == sin(x)**2/2
+    assert trigintegrate(sin(x)*cos(x)**2, x) == -cos(x)**3/3
+    assert trigintegrate(sin(x)**2*cos(x), x) == sin(x)**3/3
+
+    # check if it selects right function to substitute,
+    # so the result is kept simple
+    assert trigintegrate(sin(x)**7 * cos(x), x) == sin(x)**8/8
+    assert trigintegrate(sin(x) * cos(x)**7, x) == -cos(x)**8/8
+
+    assert trigintegrate(sin(x)**7 * cos(x)**3, x) == \
+        -sin(x)**10/10 + sin(x)**8/8
+    assert trigintegrate(sin(x)**3 * cos(x)**7, x) == \
+        cos(x)**10/10 - cos(x)**8/8
+
+    # both n, m are odd and -ve, and not necessarily equal
+    assert trigintegrate(sin(x)**-1*cos(x)**-1, x) == \
+        -log(sin(x)**2 - 1)/2 + log(sin(x))
+
+
+def test_trigintegrate_even():
+    assert trigintegrate(sin(x)**2, x) == x/2 - cos(x)*sin(x)/2
+    assert trigintegrate(cos(x)**2, x) == x/2 + cos(x)*sin(x)/2
+
+    assert trigintegrate(sin(3*x)**2, x) == x/2 - cos(3*x)*sin(3*x)/6
+    assert trigintegrate(cos(3*x)**2, x) == x/2 + cos(3*x)*sin(3*x)/6
+    assert trigintegrate(sin(x)**2 * cos(x)**2, x) == \
+        x/8 - sin(2*x)*cos(2*x)/16
+
+    assert trigintegrate(sin(x)**4 * cos(x)**2, x) == \
+        x/16 - sin(x) *cos(x)/16 - sin(x)**3*cos(x)/24 + \
+        sin(x)**5*cos(x)/6
+
+    assert trigintegrate(sin(x)**2 * cos(x)**4, x) == \
+        x/16 + cos(x) *sin(x)/16 + cos(x)**3*sin(x)/24 - \
+        cos(x)**5*sin(x)/6
+
+    assert trigintegrate(sin(x)**(-4), x) == -2*cos(x)/(3*sin(x)) \
+        - cos(x)/(3*sin(x)**3)
+
+    assert trigintegrate(cos(x)**(-6), x) == sin(x)/(5*cos(x)**5) \
+        + 4*sin(x)/(15*cos(x)**3) + 8*sin(x)/(15*cos(x))
+
+
+def test_trigintegrate_mixed():
+    assert trigintegrate(sin(x)*sec(x), x) == -log(cos(x))
+    assert trigintegrate(sin(x)*csc(x), x) == x
+    assert trigintegrate(sin(x)*cot(x), x) == sin(x)
+
+    assert trigintegrate(cos(x)*sec(x), x) == x
+    assert trigintegrate(cos(x)*csc(x), x) == log(sin(x))
+    assert trigintegrate(cos(x)*tan(x), x) == -cos(x)
+    assert trigintegrate(cos(x)*cot(x), x) == log(cos(x) - 1)/2 \
+        - log(cos(x) + 1)/2 + cos(x)
+    assert trigintegrate(cot(x)*cos(x)**2, x) == log(sin(x)) - sin(x)**2/2
+
+
+def test_trigintegrate_symbolic():
+    n = Symbol('n', integer=True)
+    assert trigintegrate(cos(x)**n, x) is None
+    assert trigintegrate(sin(x)**n, x) is None
+    assert trigintegrate(cot(x)**n, x) is None

rl/Lib/site-packages/sympy/integrals/trigonometry.py

@@ -0,0 +1,335 @@
+from sympy.core import cacheit, Dummy, Ne, Integer, Rational, S, Wild
+from sympy.functions import binomial, sin, cos, Piecewise, Abs
+from .integrals import integrate
+
+# TODO sin(a*x)*cos(b*x) -> sin((a+b)x) + sin((a-b)x) ?
+
+# creating, each time, Wild's and sin/cos/Mul is expensive. Also, our match &
+# subs are very slow when not cached, and if we create Wild each time, we
+# effectively block caching.
+#
+# so we cache the pattern
+
+# need to use a function instead of lamda since hash of lambda changes on
+# each call to _pat_sincos
+def _integer_instance(n):
+    return isinstance(n, Integer)
+
+@cacheit
+def _pat_sincos(x):
+    a = Wild('a', exclude=[x])
+    n, m = [Wild(s, exclude=[x], properties=[_integer_instance])
+                for s in 'nm']
+    pat = sin(a*x)**n * cos(a*x)**m
+    return pat, a, n, m
+
+_u = Dummy('u')
+
+
+def trigintegrate(f, x, conds='piecewise'):
+    """
+    Integrate f = Mul(trig) over x.
+
+    Examples
+    ========
+
+    >>> from sympy import sin, cos, tan, sec
+    >>> from sympy.integrals.trigonometry import trigintegrate
+    >>> from sympy.abc import x
+
+    >>> trigintegrate(sin(x)*cos(x), x)
+    sin(x)**2/2
+
+    >>> trigintegrate(sin(x)**2, x)
+    x/2 - sin(x)*cos(x)/2
+
+    >>> trigintegrate(tan(x)*sec(x), x)
+    1/cos(x)
+
+    >>> trigintegrate(sin(x)*tan(x), x)
+    -log(sin(x) - 1)/2 + log(sin(x) + 1)/2 - sin(x)
+
+    References
+    ==========
+
+    .. [1] https://en.wikibooks.org/wiki/Calculus/Integration_techniques
+
+    See Also
+    ========
+
+    sympy.integrals.integrals.Integral.doit
+    sympy.integrals.integrals.Integral
+    """
+    pat, a, n, m = _pat_sincos(x)
+
+    f = f.rewrite('sincos')
+    M = f.match(pat)
+
+    if M is None:
+        return
+
+    n, m = M[n], M[m]
+    if n.is_zero and m.is_zero:
+        return x
+    zz = x if n.is_zero else S.Zero
+
+    a = M[a]
+
+    if n.is_odd or m.is_odd:
+        u = _u
+        n_, m_ = n.is_odd, m.is_odd
+
+        # take smallest n or m -- to choose simplest substitution
+        if n_ and m_:
+
+            # Make sure to choose the positive one
+            # otherwise an incorrect integral can occur.
+            if n < 0 and m > 0:
+                m_ = True
+                n_ = False
+            elif m < 0 and n > 0:
+                n_ = True
+                m_ = False
+            # Both are negative so choose the smallest n or m
+            # in absolute value for simplest substitution.
+            elif (n < 0 and m < 0):
+                n_ = n > m
+                m_ = not (n > m)
+
+            # Both n and m are odd and positive
+            else:
+                n_ = (n < m)      # NB: careful here, one of the
+                m_ = not (n < m)  # conditions *must* be true
+
+        #  n      m       u=C        (n-1)/2    m
+        # S(x) * C(x) dx  --> -(1-u^2)       * u  du
+        if n_:
+            ff = -(1 - u**2)**((n - 1)/2) * u**m
+            uu = cos(a*x)
+
+        #  n      m       u=S   n         (m-1)/2
+        # S(x) * C(x) dx  -->  u  * (1-u^2)       du
+        elif m_:
+            ff = u**n * (1 - u**2)**((m - 1)/2)
+            uu = sin(a*x)
+
+        fi = integrate(ff, u)  # XXX cyclic deps
+        fx = fi.subs(u, uu)
+        if conds == 'piecewise':
+            return Piecewise((fx / a, Ne(a, 0)), (zz, True))
+        return fx / a
+
+    # n & m are both even
+    #
+    #               2k      2m                         2l       2l
+    # we transform S (x) * C (x) into terms with only S (x) or C (x)
+    #
+    # example:
+    #  100     4       100        2    2    100          4         2
+    # S (x) * C (x) = S (x) * (1-S (x))  = S (x) * (1 + S (x) - 2*S (x))
+    #
+    #                  104       102     100
+    #               = S (x) - 2*S (x) + S (x)
+    #       2k
+    # then S   is integrated with recursive formula
+
+    # take largest n or m -- to choose simplest substitution
+    n_ = (Abs(n) > Abs(m))
+    m_ = (Abs(m) > Abs(n))
+    res = S.Zero
+
+    if n_:
+        #  2k         2 k             i             2i
+        # C   = (1 - S )  = sum(i, (-) * B(k, i) * S  )
+        if m > 0:
+            for i in range(0, m//2 + 1):
+                res += (S.NegativeOne**i * binomial(m//2, i) *
+                        _sin_pow_integrate(n + 2*i, x))
+
+        elif m == 0:
+            res = _sin_pow_integrate(n, x)
+        else:
+
+            # m < 0 , |n| > |m|
+            #  /
+            # |
+            # |    m       n
+            # | cos (x) sin (x) dx =
+            # |
+            # |
+            #/
+            #                                      /
+            #                                     |
+            #   -1        m+1     n-1     n - 1   |     m+2     n-2
+            # ________ cos (x) sin (x) + _______  |  cos (x) sin (x) dx
+            #                                     |
+            #   m + 1                     m + 1   |
+            #                                    /
+
+            res = (Rational(-1, m + 1) * cos(x)**(m + 1) * sin(x)**(n - 1) +
+                   Rational(n - 1, m + 1) *
+                   trigintegrate(cos(x)**(m + 2)*sin(x)**(n - 2), x))
+
+    elif m_:
+        #  2k         2 k            i             2i
+        # S   = (1 - C ) = sum(i, (-) * B(k, i) * C  )
+        if n > 0:
+
+            #      /                            /
+            #     |                            |
+            #     |    m       n               |    -m         n
+            #     | cos (x)*sin (x) dx  or     | cos (x) * sin (x) dx
+            #     |                            |
+            #    /                            /
+            #
+            #    |m| > |n| ; m, n >0 ; m, n belong to Z - {0}
+            #       n                                         2
+            #    sin (x) term is expanded here in terms of cos (x),
+            #    and then integrated.
+            #
+
+            for i in range(0, n//2 + 1):
+                res += (S.NegativeOne**i * binomial(n//2, i) *
+                        _cos_pow_integrate(m + 2*i, x))
+
+        elif n == 0:
+
+            #   /
+            #  |
+            #  |  1
+            #  | _ _ _
+            #  |    m
+            #  | cos (x)
+            # /
+            #
+
+            res = _cos_pow_integrate(m, x)
+        else:
+
+            # n < 0 , |m| > |n|
+            #  /
+            # |
+            # |    m       n
+            # | cos (x) sin (x) dx =
+            # |
+            # |
+            #/
+            #                                      /
+            #                                     |
+            #    1        m-1     n+1     m - 1   |     m-2     n+2
+            #  _______ cos (x) sin (x) + _______  |  cos (x) sin (x) dx
+            #                                     |
+            #   n + 1                     n + 1   |
+            #                                    /
+
+            res = (Rational(1, n + 1) * cos(x)**(m - 1)*sin(x)**(n + 1) +
+                   Rational(m - 1, n + 1) *
+                   trigintegrate(cos(x)**(m - 2)*sin(x)**(n + 2), x))
+
+    else:
+        if m == n:
+            ##Substitute sin(2x)/2 for sin(x)cos(x) and then Integrate.
+            res = integrate((sin(2*x)*S.Half)**m, x)
+        elif (m == -n):
+            if n < 0:
+                # Same as the scheme described above.
+                # the function argument to integrate in the end will
+                # be 1, this cannot be integrated by trigintegrate.
+                # Hence use sympy.integrals.integrate.
+                res = (Rational(1, n + 1) * cos(x)**(m - 1) * sin(x)**(n + 1) +
+                       Rational(m - 1, n + 1) *
+                       integrate(cos(x)**(m - 2) * sin(x)**(n + 2), x))
+            else:
+                res = (Rational(-1, m + 1) * cos(x)**(m + 1) * sin(x)**(n - 1) +
+                       Rational(n - 1, m + 1) *
+                       integrate(cos(x)**(m + 2)*sin(x)**(n - 2), x))
+    if conds == 'piecewise':
+        return Piecewise((res.subs(x, a*x) / a, Ne(a, 0)), (zz, True))
+    return res.subs(x, a*x) / a
+
+
+def _sin_pow_integrate(n, x):
+    if n > 0:
+        if n == 1:
+            #Recursion break
+            return -cos(x)
+
+        # n > 0
+        #  /                                                 /
+        # |                                                 |
+        # |    n           -1               n-1     n - 1   |     n-2
+        # | sin (x) dx =  ______ cos (x) sin (x) + _______  |  sin (x) dx
+        # |                                                 |
+        # |                 n                         n     |
+        #/                                                 /
+        #
+        #
+
+        return (Rational(-1, n) * cos(x) * sin(x)**(n - 1) +
+                Rational(n - 1, n) * _sin_pow_integrate(n - 2, x))
+
+    if n < 0:
+        if n == -1:
+            ##Make sure this does not come back here again.
+            ##Recursion breaks here or at n==0.
+            return trigintegrate(1/sin(x), x)
+
+        # n < 0
+        #  /                                                 /
+        # |                                                 |
+        # |    n            1               n+1     n + 2   |     n+2
+        # | sin (x) dx = _______ cos (x) sin (x) + _______  |  sin (x) dx
+        # |                                                 |
+        # |               n + 1                     n + 1   |
+        #/                                                 /
+        #
+
+        return (Rational(1, n + 1) * cos(x) * sin(x)**(n + 1) +
+                Rational(n + 2, n + 1) * _sin_pow_integrate(n + 2, x))
+
+    else:
+        #n == 0
+        #Recursion break.
+        return x
+
+
+def _cos_pow_integrate(n, x):
+    if n > 0:
+        if n == 1:
+            #Recursion break.
+            return sin(x)
+
+        # n > 0
+        #  /                                                 /
+        # |                                                 |
+        # |    n            1               n-1     n - 1   |     n-2
+        # | sin (x) dx =  ______ sin (x) cos (x) + _______  |  cos (x) dx
+        # |                                                 |
+        # |                 n                         n     |
+        #/                                                 /
+        #
+
+        return (Rational(1, n) * sin(x) * cos(x)**(n - 1) +
+                Rational(n - 1, n) * _cos_pow_integrate(n - 2, x))
+
+    if n < 0:
+        if n == -1:
+            ##Recursion break
+            return trigintegrate(1/cos(x), x)
+
+        # n < 0
+        #  /                                                 /
+        # |                                                 |
+        # |    n            -1              n+1     n + 2   |     n+2
+        # | cos (x) dx = _______ sin (x) cos (x) + _______  |  cos (x) dx
+        # |                                                 |
+        # |               n + 1                     n + 1   |
+        #/                                                 /
+        #
+
+        return (Rational(-1, n + 1) * sin(x) * cos(x)**(n + 1) +
+                Rational(n + 2, n + 1) * _cos_pow_integrate(n + 2, x))
+    else:
+        # n == 0
+        #Recursion Break.
+        return x