venv/lib/python3.11/site-packages/sympy/integrals/manualintegrate.py

"""Integration method that emulates by-hand techniques.

This module also provides functionality to get the steps used to evaluate a
particular integral, in the ``integral_steps`` function. This will return
nested ``Rule`` s representing the integration rules used.

Each ``Rule`` class represents a (maybe parametrized) integration rule, e.g.
``SinRule`` for integrating ``sin(x)`` and ``ReciprocalSqrtQuadraticRule``
for integrating ``1/sqrt(a+b*x+c*x**2)``. The ``eval`` method returns the
integration result.

The ``manualintegrate`` function computes the integral by calling ``eval``
on the rule returned by ``integral_steps``.

The integrator can be extended with new heuristics and evaluation
techniques. To do so, extend the ``Rule`` class, implement ``eval`` method,
then write a function that accepts an ``IntegralInfo`` object and returns
either a ``Rule`` instance or ``None``. If the new technique requires a new
match, add the key and call to the antiderivative function to integral_steps.
To enable simple substitutions, add the match to find_substitutions.

"""

from __future__ import annotations
from typing import NamedTuple, Type, Callable, Sequence
from abc import ABC, abstractmethod
from dataclasses import dataclass
from collections import defaultdict
from collections.abc import Mapping

from sympy.core.add import Add
from sympy.core.cache import cacheit
from sympy.core.containers import Dict
from sympy.core.expr import Expr
from sympy.core.function import Derivative
from sympy.core.logic import fuzzy_not
from sympy.core.mul import Mul
from sympy.core.numbers import Integer, Number, E
from sympy.core.power import Pow
from sympy.core.relational import Eq, Ne, Boolean
from sympy.core.singleton import S
from sympy.core.symbol import Dummy, Symbol, Wild
from sympy.functions.elementary.complexes import Abs
from sympy.functions.elementary.exponential import exp, log
from sympy.functions.elementary.hyperbolic import (HyperbolicFunction, csch,
    cosh, coth, sech, sinh, tanh, asinh)
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import (TrigonometricFunction,
    cos, sin, tan, cot, csc, sec, acos, asin, atan, acot, acsc, asec)
from sympy.functions.special.delta_functions import Heaviside, DiracDelta
from sympy.functions.special.error_functions import (erf, erfi, fresnelc,
    fresnels, Ci, Chi, Si, Shi, Ei, li)
from sympy.functions.special.gamma_functions import uppergamma
from sympy.functions.special.elliptic_integrals import elliptic_e, elliptic_f
from sympy.functions.special.polynomials import (chebyshevt, chebyshevu,
    legendre, hermite, laguerre, assoc_laguerre, gegenbauer, jacobi,
    OrthogonalPolynomial)
from sympy.functions.special.zeta_functions import polylog
from .integrals import Integral
from sympy.logic.boolalg import And
from sympy.ntheory.factor_ import primefactors
from sympy.polys.polytools import degree, lcm_list, gcd_list, Poly
from sympy.simplify.radsimp import fraction
from sympy.simplify.simplify import simplify
from sympy.solvers.solvers import solve
from sympy.strategies.core import switch, do_one, null_safe, condition
from sympy.utilities.iterables import iterable
from sympy.utilities.misc import debug


@dataclass
class Rule(ABC):
    integrand: Expr
    variable: Symbol

    @abstractmethod
    def eval(self) -> Expr:
        pass

    @abstractmethod
    def contains_dont_know(self) -> bool:
        pass


@dataclass
class AtomicRule(Rule, ABC):
    """A simple rule that does not depend on other rules"""
    def contains_dont_know(self) -> bool:
        return False


@dataclass
class ConstantRule(AtomicRule):
    """integrate(a, x)  ->  a*x"""
    def eval(self) -> Expr:
        return self.integrand * self.variable


@dataclass
class ConstantTimesRule(Rule):
    """integrate(a*f(x), x)  ->  a*integrate(f(x), x)"""
    constant: Expr
    other: Expr
    substep: Rule

    def eval(self) -> Expr:
        return self.constant * self.substep.eval()

    def contains_dont_know(self) -> bool:
        return self.substep.contains_dont_know()


@dataclass
class PowerRule(AtomicRule):
    """integrate(x**a, x)"""
    base: Expr
    exp: Expr

    def eval(self) -> Expr:
        return Piecewise(
            ((self.base**(self.exp + 1))/(self.exp + 1), Ne(self.exp, -1)),
            (log(self.base), True),
        )


@dataclass
class NestedPowRule(AtomicRule):
    """integrate((x**a)**b, x)"""
    base: Expr
    exp: Expr

    def eval(self) -> Expr:
        m = self.base * self.integrand
        return Piecewise((m / (self.exp + 1), Ne(self.exp, -1)),
                         (m * log(self.base), True))


@dataclass
class AddRule(Rule):
    """integrate(f(x) + g(x), x) -> integrate(f(x), x) + integrate(g(x), x)"""
    substeps: list[Rule]

    def eval(self) -> Expr:
        return Add(*(substep.eval() for substep in self.substeps))

    def contains_dont_know(self) -> bool:
        return any(substep.contains_dont_know() for substep in self.substeps)


@dataclass
class URule(Rule):
    """integrate(f(g(x))*g'(x), x) -> integrate(f(u), u), u = g(x)"""
    u_var: Symbol
    u_func: Expr
    substep: Rule

    def eval(self) -> Expr:
        result = self.substep.eval()
        if self.u_func.is_Pow:
            base, exp_ = self.u_func.as_base_exp()
            if exp_ == -1:
                # avoid needless -log(1/x) from substitution
                result = result.subs(log(self.u_var), -log(base))
        return result.subs(self.u_var, self.u_func)

    def contains_dont_know(self) -> bool:
        return self.substep.contains_dont_know()


@dataclass
class PartsRule(Rule):
    """integrate(u(x)*v'(x), x) -> u(x)*v(x) - integrate(u'(x)*v(x), x)"""
    u: Symbol
    dv: Expr
    v_step: Rule
    second_step: Rule | None  # None when is a substep of CyclicPartsRule

    def eval(self) -> Expr:
        assert self.second_step is not None
        v = self.v_step.eval()
        return self.u * v - self.second_step.eval()

    def contains_dont_know(self) -> bool:
        return self.v_step.contains_dont_know() or (
            self.second_step is not None and self.second_step.contains_dont_know())


@dataclass
class CyclicPartsRule(Rule):
    """Apply PartsRule multiple times to integrate exp(x)*sin(x)"""
    parts_rules: list[PartsRule]
    coefficient: Expr

    def eval(self) -> Expr:
        result = []
        sign = 1
        for rule in self.parts_rules:
            result.append(sign * rule.u * rule.v_step.eval())
            sign *= -1
        return Add(*result) / (1 - self.coefficient)

    def contains_dont_know(self) -> bool:
        return any(substep.contains_dont_know() for substep in self.parts_rules)


@dataclass
class TrigRule(AtomicRule, ABC):
    pass


@dataclass
class SinRule(TrigRule):
    """integrate(sin(x), x) -> -cos(x)"""
    def eval(self) -> Expr:
        return -cos(self.variable)


@dataclass
class CosRule(TrigRule):
    """integrate(cos(x), x) -> sin(x)"""
    def eval(self) -> Expr:
        return sin(self.variable)


@dataclass
class SecTanRule(TrigRule):
    """integrate(sec(x)*tan(x), x) -> sec(x)"""
    def eval(self) -> Expr:
        return sec(self.variable)


@dataclass
class CscCotRule(TrigRule):
    """integrate(csc(x)*cot(x), x) -> -csc(x)"""
    def eval(self) -> Expr:
        return -csc(self.variable)


@dataclass
class Sec2Rule(TrigRule):
    """integrate(sec(x)**2, x) -> tan(x)"""
    def eval(self) -> Expr:
        return tan(self.variable)


@dataclass
class Csc2Rule(TrigRule):
    """integrate(csc(x)**2, x) -> -cot(x)"""
    def eval(self) -> Expr:
        return -cot(self.variable)


@dataclass
class HyperbolicRule(AtomicRule, ABC):
    pass


@dataclass
class SinhRule(HyperbolicRule):
    """integrate(sinh(x), x) -> cosh(x)"""
    def eval(self) -> Expr:
        return cosh(self.variable)


@dataclass
class CoshRule(HyperbolicRule):
    """integrate(cosh(x), x) -> sinh(x)"""
    def eval(self):
        return sinh(self.variable)


@dataclass
class ExpRule(AtomicRule):
    """integrate(a**x, x) -> a**x/ln(a)"""
    base: Expr
    exp: Expr

    def eval(self) -> Expr:
        return self.integrand / log(self.base)


@dataclass
class ReciprocalRule(AtomicRule):
    """integrate(1/x, x) -> ln(x)"""
    base: Expr

    def eval(self) -> Expr:
        return log(self.base)


@dataclass
class ArcsinRule(AtomicRule):
    """integrate(1/sqrt(1-x**2), x) -> asin(x)"""
    def eval(self) -> Expr:
        return asin(self.variable)


@dataclass
class ArcsinhRule(AtomicRule):
    """integrate(1/sqrt(1+x**2), x) -> asin(x)"""
    def eval(self) -> Expr:
        return asinh(self.variable)


@dataclass
class ReciprocalSqrtQuadraticRule(AtomicRule):
    """integrate(1/sqrt(a+b*x+c*x**2), x) -> log(2*sqrt(c)*sqrt(a+b*x+c*x**2)+b+2*c*x)/sqrt(c)"""
    a: Expr
    b: Expr
    c: Expr

    def eval(self) -> Expr:
        a, b, c, x = self.a, self.b, self.c, self.variable
        return log(2*sqrt(c)*sqrt(a+b*x+c*x**2)+b+2*c*x)/sqrt(c)


@dataclass
class SqrtQuadraticDenomRule(AtomicRule):
    """integrate(poly(x)/sqrt(a+b*x+c*x**2), x)"""
    a: Expr
    b: Expr
    c: Expr
    coeffs: list[Expr]

    def eval(self) -> Expr:
        a, b, c, coeffs, x = self.a, self.b, self.c, self.coeffs.copy(), self.variable
        # Integrate poly/sqrt(a+b*x+c*x**2) using recursion.
        # coeffs are coefficients of the polynomial.
        # Let I_n = x**n/sqrt(a+b*x+c*x**2), then
        # I_n = A * x**(n-1)*sqrt(a+b*x+c*x**2) - B * I_{n-1} - C * I_{n-2}
        # where A = 1/(n*c), B = (2*n-1)*b/(2*n*c), C = (n-1)*a/(n*c)
        # See https://github.com/sympy/sympy/pull/23608 for proof.
        result_coeffs = []
        coeffs = coeffs.copy()
        for i in range(len(coeffs)-2):
            n = len(coeffs)-1-i
            coeff = coeffs[i]/(c*n)
            result_coeffs.append(coeff)
            coeffs[i+1] -= (2*n-1)*b/2*coeff
            coeffs[i+2] -= (n-1)*a*coeff
        d, e = coeffs[-1], coeffs[-2]
        s = sqrt(a+b*x+c*x**2)
        constant = d-b*e/(2*c)
        if constant == 0:
            I0 = 0
        else:
            step = inverse_trig_rule(IntegralInfo(1/s, x), degenerate=False)
            I0 = constant*step.eval()
        return Add(*(result_coeffs[i]*x**(len(coeffs)-2-i)
                     for i in range(len(result_coeffs))), e/c)*s + I0


@dataclass
class SqrtQuadraticRule(AtomicRule):
    """integrate(sqrt(a+b*x+c*x**2), x)"""
    a: Expr
    b: Expr
    c: Expr

    def eval(self) -> Expr:
        step = sqrt_quadratic_rule(IntegralInfo(self.integrand, self.variable), degenerate=False)
        return step.eval()


@dataclass
class AlternativeRule(Rule):
    """Multiple ways to do integration."""
    alternatives: list[Rule]

    def eval(self) -> Expr:
        return self.alternatives[0].eval()

    def contains_dont_know(self) -> bool:
        return any(substep.contains_dont_know() for substep in self.alternatives)


@dataclass
class DontKnowRule(Rule):
    """Leave the integral as is."""
    def eval(self) -> Expr:
        return Integral(self.integrand, self.variable)

    def contains_dont_know(self) -> bool:
        return True


@dataclass
class DerivativeRule(AtomicRule):
    """integrate(f'(x), x) -> f(x)"""
    def eval(self) -> Expr:
        assert isinstance(self.integrand, Derivative)
        variable_count = list(self.integrand.variable_count)
        for i, (var, count) in enumerate(variable_count):
            if var == self.variable:
                variable_count[i] = (var, count - 1)
                break
        return Derivative(self.integrand.expr, *variable_count)


@dataclass
class RewriteRule(Rule):
    """Rewrite integrand to another form that is easier to handle."""
    rewritten: Expr
    substep: Rule

    def eval(self) -> Expr:
        return self.substep.eval()

    def contains_dont_know(self) -> bool:
        return self.substep.contains_dont_know()


@dataclass
class CompleteSquareRule(RewriteRule):
    """Rewrite a+b*x+c*x**2 to a-b**2/(4*c) + c*(x+b/(2*c))**2"""
    pass


@dataclass
class PiecewiseRule(Rule):
    subfunctions: Sequence[tuple[Rule, bool | Boolean]]

    def eval(self) -> Expr:
        return Piecewise(*[(substep.eval(), cond)
                           for substep, cond in self.subfunctions])

    def contains_dont_know(self) -> bool:
        return any(substep.contains_dont_know() for substep, _ in self.subfunctions)


@dataclass
class HeavisideRule(Rule):
    harg: Expr
    ibnd: Expr
    substep: Rule

    def eval(self) -> Expr:
        # If we are integrating over x and the integrand has the form
        #       Heaviside(m*x+b)*g(x) == Heaviside(harg)*g(symbol)
        # then there needs to be continuity at -b/m == ibnd,
        # so we subtract the appropriate term.
        result = self.substep.eval()
        return Heaviside(self.harg) * (result - result.subs(self.variable, self.ibnd))

    def contains_dont_know(self) -> bool:
        return self.substep.contains_dont_know()


@dataclass
class DiracDeltaRule(AtomicRule):
    n: Expr
    a: Expr
    b: Expr

    def eval(self) -> Expr:
        n, a, b, x = self.n, self.a, self.b, self.variable
        if n == 0:
            return Heaviside(a+b*x)/b
        return DiracDelta(a+b*x, n-1)/b


@dataclass
class TrigSubstitutionRule(Rule):
    theta: Expr
    func: Expr
    rewritten: Expr
    substep: Rule
    restriction: bool | Boolean

    def eval(self) -> Expr:
        theta, func, x = self.theta, self.func, self.variable
        func = func.subs(sec(theta), 1/cos(theta))
        func = func.subs(csc(theta), 1/sin(theta))
        func = func.subs(cot(theta), 1/tan(theta))

        trig_function = list(func.find(TrigonometricFunction))
        assert len(trig_function) == 1
        trig_function = trig_function[0]
        relation = solve(x - func, trig_function)
        assert len(relation) == 1
        numer, denom = fraction(relation[0])

        if isinstance(trig_function, sin):
            opposite = numer
            hypotenuse = denom
            adjacent = sqrt(denom**2 - numer**2)
            inverse = asin(relation[0])
        elif isinstance(trig_function, cos):
            adjacent = numer
            hypotenuse = denom
            opposite = sqrt(denom**2 - numer**2)
            inverse = acos(relation[0])
        else:  # tan
            opposite = numer
            adjacent = denom
            hypotenuse = sqrt(denom**2 + numer**2)
            inverse = atan(relation[0])

        substitution = [
            (sin(theta), opposite/hypotenuse),
            (cos(theta), adjacent/hypotenuse),
            (tan(theta), opposite/adjacent),
            (theta, inverse)
        ]
        return Piecewise(
                (self.substep.eval().subs(substitution).trigsimp(), self.restriction) # type: ignore
        )

    def contains_dont_know(self) -> bool:
        return self.substep.contains_dont_know()


@dataclass
class ArctanRule(AtomicRule):
    """integrate(a/(b*x**2+c), x) -> a/b / sqrt(c/b) * atan(x/sqrt(c/b))"""
    a: Expr
    b: Expr
    c: Expr

    def eval(self) -> Expr:
        a, b, c, x = self.a, self.b, self.c, self.variable
        return a/b / sqrt(c/b) * atan(x/sqrt(c/b))


@dataclass
class OrthogonalPolyRule(AtomicRule, ABC):
    n: Expr


@dataclass
class JacobiRule(OrthogonalPolyRule):
    a: Expr
    b: Expr

    def eval(self) -> Expr:
        n, a, b, x = self.n, self.a, self.b, self.variable
        return Piecewise(
            (2*jacobi(n + 1, a - 1, b - 1, x)/(n + a + b), Ne(n + a + b, 0)),
            (x, Eq(n, 0)),
            ((a + b + 2)*x**2/4 + (a - b)*x/2, Eq(n, 1)))


@dataclass
class GegenbauerRule(OrthogonalPolyRule):
    a: Expr

    def eval(self) -> Expr:
        n, a, x = self.n, self.a, self.variable
        return Piecewise(
            (gegenbauer(n + 1, a - 1, x)/(2*(a - 1)), Ne(a, 1)),
            (chebyshevt(n + 1, x)/(n + 1), Ne(n, -1)),
            (S.Zero, True))


@dataclass
class ChebyshevTRule(OrthogonalPolyRule):
    def eval(self) -> Expr:
        n, x = self.n, self.variable
        return Piecewise(
            ((chebyshevt(n + 1, x)/(n + 1) -
              chebyshevt(n - 1, x)/(n - 1))/2, Ne(Abs(n), 1)),
            (x**2/2, True))


@dataclass
class ChebyshevURule(OrthogonalPolyRule):
    def eval(self) -> Expr:
        n, x = self.n, self.variable
        return Piecewise(
            (chebyshevt(n + 1, x)/(n + 1), Ne(n, -1)),
            (S.Zero, True))


@dataclass
class LegendreRule(OrthogonalPolyRule):
    def eval(self) -> Expr:
        n, x = self.n, self.variable
        return(legendre(n + 1, x) - legendre(n - 1, x))/(2*n + 1)


@dataclass
class HermiteRule(OrthogonalPolyRule):
    def eval(self) -> Expr:
        n, x = self.n, self.variable
        return hermite(n + 1, x)/(2*(n + 1))


@dataclass
class LaguerreRule(OrthogonalPolyRule):
    def eval(self) -> Expr:
        n, x = self.n, self.variable
        return laguerre(n, x) - laguerre(n + 1, x)


@dataclass
class AssocLaguerreRule(OrthogonalPolyRule):
    a: Expr

    def eval(self) -> Expr:
        return -assoc_laguerre(self.n + 1, self.a - 1, self.variable)


@dataclass
class IRule(AtomicRule, ABC):
    a: Expr
    b: Expr


@dataclass
class CiRule(IRule):
    def eval(self) -> Expr:
        a, b, x = self.a, self.b, self.variable
        return cos(b)*Ci(a*x) - sin(b)*Si(a*x)


@dataclass
class ChiRule(IRule):
    def eval(self) -> Expr:
        a, b, x = self.a, self.b, self.variable
        return cosh(b)*Chi(a*x) + sinh(b)*Shi(a*x)


@dataclass
class EiRule(IRule):
    def eval(self) -> Expr:
        a, b, x = self.a, self.b, self.variable
        return exp(b)*Ei(a*x)


@dataclass
class SiRule(IRule):
    def eval(self) -> Expr:
        a, b, x = self.a, self.b, self.variable
        return sin(b)*Ci(a*x) + cos(b)*Si(a*x)


@dataclass
class ShiRule(IRule):
    def eval(self) -> Expr:
        a, b, x = self.a, self.b, self.variable
        return sinh(b)*Chi(a*x) + cosh(b)*Shi(a*x)


@dataclass
class LiRule(IRule):
    def eval(self) -> Expr:
        a, b, x = self.a, self.b, self.variable
        return li(a*x + b)/a


@dataclass
class ErfRule(AtomicRule):
    a: Expr
    b: Expr
    c: Expr

    def eval(self) -> Expr:
        a, b, c, x = self.a, self.b, self.c, self.variable
        if a.is_extended_real:
            return Piecewise(
                (sqrt(S.Pi)/sqrt(-a)/2 * exp(c - b**2/(4*a)) *
                    erf((-2*a*x - b)/(2*sqrt(-a))), a < 0),
                (sqrt(S.Pi)/sqrt(a)/2 * exp(c - b**2/(4*a)) *
                    erfi((2*a*x + b)/(2*sqrt(a))), True))
        return sqrt(S.Pi)/sqrt(a)/2 * exp(c - b**2/(4*a)) * \
                erfi((2*a*x + b)/(2*sqrt(a)))


@dataclass
class FresnelCRule(AtomicRule):
    a: Expr
    b: Expr
    c: Expr

    def eval(self) -> Expr:
        a, b, c, x = self.a, self.b, self.c, self.variable
        return sqrt(S.Pi)/sqrt(2*a) * (
            cos(b**2/(4*a) - c)*fresnelc((2*a*x + b)/sqrt(2*a*S.Pi)) +
            sin(b**2/(4*a) - c)*fresnels((2*a*x + b)/sqrt(2*a*S.Pi)))


@dataclass
class FresnelSRule(AtomicRule):
    a: Expr
    b: Expr
    c: Expr

    def eval(self) -> Expr:
        a, b, c, x = self.a, self.b, self.c, self.variable
        return sqrt(S.Pi)/sqrt(2*a) * (
            cos(b**2/(4*a) - c)*fresnels((2*a*x + b)/sqrt(2*a*S.Pi)) -
            sin(b**2/(4*a) - c)*fresnelc((2*a*x + b)/sqrt(2*a*S.Pi)))


@dataclass
class PolylogRule(AtomicRule):
    a: Expr
    b: Expr

    def eval(self) -> Expr:
        return polylog(self.b + 1, self.a * self.variable)


@dataclass
class UpperGammaRule(AtomicRule):
    a: Expr
    e: Expr

    def eval(self) -> Expr:
        a, e, x = self.a, self.e, self.variable
        return x**e * (-a*x)**(-e) * uppergamma(e + 1, -a*x)/a


@dataclass
class EllipticFRule(AtomicRule):
    a: Expr
    d: Expr

    def eval(self) -> Expr:
        return elliptic_f(self.variable, self.d/self.a)/sqrt(self.a)


@dataclass
class EllipticERule(AtomicRule):
    a: Expr
    d: Expr

    def eval(self) -> Expr:
        return elliptic_e(self.variable, self.d/self.a)*sqrt(self.a)


class IntegralInfo(NamedTuple):
    integrand: Expr
    symbol: Symbol


def manual_diff(f, symbol):
    """Derivative of f in form expected by find_substitutions

    SymPy's derivatives for some trig functions (like cot) are not in a form
    that works well with finding substitutions; this replaces the
    derivatives for those particular forms with something that works better.

    """
    if f.args:
        arg = f.args[0]
        if isinstance(f, tan):
            return arg.diff(symbol) * sec(arg)**2
        elif isinstance(f, cot):
            return -arg.diff(symbol) * csc(arg)**2
        elif isinstance(f, sec):
            return arg.diff(symbol) * sec(arg) * tan(arg)
        elif isinstance(f, csc):
            return -arg.diff(symbol) * csc(arg) * cot(arg)
        elif isinstance(f, Add):
            return sum(manual_diff(arg, symbol) for arg in f.args)
        elif isinstance(f, Mul):
            if len(f.args) == 2 and isinstance(f.args[0], Number):
                return f.args[0] * manual_diff(f.args[1], symbol)
    return f.diff(symbol)

def manual_subs(expr, *args):
    """
    A wrapper for `expr.subs(*args)` with additional logic for substitution
    of invertible functions.
    """
    if len(args) == 1:
        sequence = args[0]
        if isinstance(sequence, (Dict, Mapping)):
            sequence = sequence.items()
        elif not iterable(sequence):
            raise ValueError("Expected an iterable of (old, new) pairs")
    elif len(args) == 2:
        sequence = [args]
    else:
        raise ValueError("subs accepts either 1 or 2 arguments")

    new_subs = []
    for old, new in sequence:
        if isinstance(old, log):
            # If log(x) = y, then exp(a*log(x)) = exp(a*y)
            # that is, x**a = exp(a*y). Replace nontrivial powers of x
            # before subs turns them into `exp(y)**a`, but
            # do not replace x itself yet, to avoid `log(exp(y))`.
            x0 = old.args[0]
            expr = expr.replace(lambda x: x.is_Pow and x.base == x0,
                lambda x: exp(x.exp*new))
            new_subs.append((x0, exp(new)))

    return expr.subs(list(sequence) + new_subs)

# Method based on that on SIN, described in "Symbolic Integration: The
# Stormy Decade"

inverse_trig_functions = (atan, asin, acos, acot, acsc, asec)


def find_substitutions(integrand, symbol, u_var):
    results = []

    def test_subterm(u, u_diff):
        if u_diff == 0:
            return False
        substituted = integrand / u_diff
        debug("substituted: {}, u: {}, u_var: {}".format(substituted, u, u_var))
        substituted = manual_subs(substituted, u, u_var).cancel()

        if substituted.has_free(symbol):
            return False
        # avoid increasing the degree of a rational function
        if integrand.is_rational_function(symbol) and substituted.is_rational_function(u_var):
            deg_before = max(degree(t, symbol) for t in integrand.as_numer_denom())
            deg_after = max(degree(t, u_var) for t in substituted.as_numer_denom())
            if deg_after > deg_before:
                return False
        return substituted.as_independent(u_var, as_Add=False)

    def exp_subterms(term: Expr):
        linear_coeffs = []
        terms = []
        n = Wild('n', properties=[lambda n: n.is_Integer])
        for exp_ in term.find(exp):
            arg = exp_.args[0]
            if symbol not in arg.free_symbols:
                continue
            match = arg.match(n*symbol)
            if match:
                linear_coeffs.append(match[n])
            else:
                terms.append(exp_)
        if linear_coeffs:
            terms.append(exp(gcd_list(linear_coeffs)*symbol))
        return terms

    def possible_subterms(term):
        if isinstance(term, (TrigonometricFunction, HyperbolicFunction,
                             *inverse_trig_functions,
                             exp, log, Heaviside)):
            return [term.args[0]]
        elif isinstance(term, (chebyshevt, chebyshevu,
                        legendre, hermite, laguerre)):
            return [term.args[1]]
        elif isinstance(term, (gegenbauer, assoc_laguerre)):
            return [term.args[2]]
        elif isinstance(term, jacobi):
            return [term.args[3]]
        elif isinstance(term, Mul):
            r = []
            for u in term.args:
                r.append(u)
                r.extend(possible_subterms(u))
            return r
        elif isinstance(term, Pow):
            r = [arg for arg in term.args if arg.has(symbol)]
            if term.exp.is_Integer:
                r.extend([term.base**d for d in primefactors(term.exp)
                    if 1 < d < abs(term.args[1])])
                if term.base.is_Add:
                    r.extend([t for t in possible_subterms(term.base)
                        if t.is_Pow])
            return r
        elif isinstance(term, Add):
            r = []
            for arg in term.args:
                r.append(arg)
                r.extend(possible_subterms(arg))
            return r
        return []

    for u in list(dict.fromkeys(possible_subterms(integrand) + exp_subterms(integrand))):
        if u == symbol:
            continue
        u_diff = manual_diff(u, symbol)
        new_integrand = test_subterm(u, u_diff)
        if new_integrand is not False:
            constant, new_integrand = new_integrand
            if new_integrand == integrand.subs(symbol, u_var):
                continue
            substitution = (u, constant, new_integrand)
            if substitution not in results:
                results.append(substitution)

    return results

def rewriter(condition, rewrite):
    """Strategy that rewrites an integrand."""
    def _rewriter(integral):
        integrand, symbol = integral
        debug("Integral: {} is rewritten with {} on symbol: {}".format(integrand, rewrite, symbol))
        if condition(*integral):
            rewritten = rewrite(*integral)
            if rewritten != integrand:
                substep = integral_steps(rewritten, symbol)
                if not isinstance(substep, DontKnowRule) and substep:
                    return RewriteRule(integrand, symbol, rewritten, substep)
    return _rewriter

def proxy_rewriter(condition, rewrite):
    """Strategy that rewrites an integrand based on some other criteria."""
    def _proxy_rewriter(criteria):
        criteria, integral = criteria
        integrand, symbol = integral
        debug("Integral: {} is rewritten with {} on symbol: {} and criteria: {}".format(integrand, rewrite, symbol, criteria))
        args = criteria + list(integral)
        if condition(*args):
            rewritten = rewrite(*args)
            if rewritten != integrand:
                return RewriteRule(integrand, symbol, rewritten, integral_steps(rewritten, symbol))
    return _proxy_rewriter

def multiplexer(conditions):
    """Apply the rule that matches the condition, else None"""
    def multiplexer_rl(expr):
        for key, rule in conditions.items():
            if key(expr):
                return rule(expr)
    return multiplexer_rl

def alternatives(*rules):
    """Strategy that makes an AlternativeRule out of multiple possible results."""
    def _alternatives(integral):
        alts = []
        count = 0
        debug("List of Alternative Rules")
        for rule in rules:
            count = count + 1
            debug("Rule {}: {}".format(count, rule))

            result = rule(integral)
            if (result and not isinstance(result, DontKnowRule) and
                result != integral and result not in alts):
                alts.append(result)
        if len(alts) == 1:
            return alts[0]
        elif alts:
            doable = [rule for rule in alts if not rule.contains_dont_know()]
            if doable:
                return AlternativeRule(*integral, doable)
            else:
                return AlternativeRule(*integral, alts)
    return _alternatives

def constant_rule(integral):
    return ConstantRule(*integral)

def power_rule(integral):
    integrand, symbol = integral
    base, expt = integrand.as_base_exp()

    if symbol not in expt.free_symbols and isinstance(base, Symbol):
        if simplify(expt + 1) == 0:
            return ReciprocalRule(integrand, symbol, base)
        return PowerRule(integrand, symbol, base, expt)
    elif symbol not in base.free_symbols and isinstance(expt, Symbol):
        rule = ExpRule(integrand, symbol, base, expt)

        if fuzzy_not(log(base).is_zero):
            return rule
        elif log(base).is_zero:
            return ConstantRule(1, symbol)

        return PiecewiseRule(integrand, symbol, [
            (rule, Ne(log(base), 0)),
            (ConstantRule(1, symbol), True)
        ])

def exp_rule(integral):
    integrand, symbol = integral
    if isinstance(integrand.args[0], Symbol):
        return ExpRule(integrand, symbol, E, integrand.args[0])


def orthogonal_poly_rule(integral):
    orthogonal_poly_classes = {
        jacobi: JacobiRule,
        gegenbauer: GegenbauerRule,
        chebyshevt: ChebyshevTRule,
        chebyshevu: ChebyshevURule,
        legendre: LegendreRule,
        hermite: HermiteRule,
        laguerre: LaguerreRule,
        assoc_laguerre: AssocLaguerreRule
        }
    orthogonal_poly_var_index = {
        jacobi: 3,
        gegenbauer: 2,
        assoc_laguerre: 2
        }
    integrand, symbol = integral
    for klass in orthogonal_poly_classes:
        if isinstance(integrand, klass):
            var_index = orthogonal_poly_var_index.get(klass, 1)
            if (integrand.args[var_index] is symbol and not
                any(v.has(symbol) for v in integrand.args[:var_index])):
                    return orthogonal_poly_classes[klass](integrand, symbol, *integrand.args[:var_index])


_special_function_patterns: list[tuple[Type, Expr, Callable | None, tuple]] = []
_wilds = []
_symbol = Dummy('x')


def special_function_rule(integral):
    integrand, symbol = integral
    if not _special_function_patterns:
        a = Wild('a', exclude=[_symbol], properties=[lambda x: not x.is_zero])
        b = Wild('b', exclude=[_symbol])
        c = Wild('c', exclude=[_symbol])
        d = Wild('d', exclude=[_symbol], properties=[lambda x: not x.is_zero])
        e = Wild('e', exclude=[_symbol], properties=[
            lambda x: not (x.is_nonnegative and x.is_integer)])
        _wilds.extend((a, b, c, d, e))
        # patterns consist of a SymPy class, a wildcard expr, an optional
        # condition coded as a lambda (when Wild properties are not enough),
        # followed by an applicable rule
        linear_pattern = a*_symbol + b
        quadratic_pattern = a*_symbol**2 + b*_symbol + c
        _special_function_patterns.extend((
            (Mul, exp(linear_pattern, evaluate=False)/_symbol, None, EiRule),
            (Mul, cos(linear_pattern, evaluate=False)/_symbol, None, CiRule),
            (Mul, cosh(linear_pattern, evaluate=False)/_symbol, None, ChiRule),
            (Mul, sin(linear_pattern, evaluate=False)/_symbol, None, SiRule),
            (Mul, sinh(linear_pattern, evaluate=False)/_symbol, None, ShiRule),
            (Pow, 1/log(linear_pattern, evaluate=False), None, LiRule),
            (exp, exp(quadratic_pattern, evaluate=False), None, ErfRule),
            (sin, sin(quadratic_pattern, evaluate=False), None, FresnelSRule),
            (cos, cos(quadratic_pattern, evaluate=False), None, FresnelCRule),
            (Mul, _symbol**e*exp(a*_symbol, evaluate=False), None, UpperGammaRule),
            (Mul, polylog(b, a*_symbol, evaluate=False)/_symbol, None, PolylogRule),
            (Pow, 1/sqrt(a - d*sin(_symbol, evaluate=False)**2),
                lambda a, d: a != d, EllipticFRule),
            (Pow, sqrt(a - d*sin(_symbol, evaluate=False)**2),
                lambda a, d: a != d, EllipticERule),
        ))
    _integrand = integrand.subs(symbol, _symbol)
    for type_, pattern, constraint, rule in _special_function_patterns:
        if isinstance(_integrand, type_):
            match = _integrand.match(pattern)
            if match:
                wild_vals = tuple(match.get(w) for w in _wilds
                                  if match.get(w) is not None)
                if constraint is None or constraint(*wild_vals):
                    return rule(integrand, symbol, *wild_vals)


def _add_degenerate_step(generic_cond, generic_step: Rule, degenerate_step: Rule | None) -> Rule:
    if degenerate_step is None:
        return generic_step
    if isinstance(generic_step, PiecewiseRule):
        subfunctions = [(substep, (cond & generic_cond).simplify())
                        for substep, cond in generic_step.subfunctions]
    else:
        subfunctions = [(generic_step, generic_cond)]
    if isinstance(degenerate_step, PiecewiseRule):
        subfunctions += degenerate_step.subfunctions
    else:
        subfunctions.append((degenerate_step, S.true))
    return PiecewiseRule(generic_step.integrand, generic_step.variable, subfunctions)


def nested_pow_rule(integral: IntegralInfo):
    # nested (c*(a+b*x)**d)**e
    integrand, x = integral

    a_ = Wild('a', exclude=[x])
    b_ = Wild('b', exclude=[x, 0])
    pattern = a_+b_*x
    generic_cond = S.true

    class NoMatch(Exception):
        pass

    def _get_base_exp(expr: Expr) -> tuple[Expr, Expr]:
        if not expr.has_free(x):
            return S.One, S.Zero
        if expr.is_Mul:
            _, terms = expr.as_coeff_mul()
            if not terms:
                return S.One, S.Zero
            results = [_get_base_exp(term) for term in terms]
            bases = {b for b, _ in results}
            bases.discard(S.One)
            if len(bases) == 1:
                return bases.pop(), Add(*(e for _, e in results))
            raise NoMatch
        if expr.is_Pow:
            b, e = expr.base, expr.exp  # type: ignore
            if e.has_free(x):
                raise NoMatch
            base_, sub_exp = _get_base_exp(b)
            return base_, sub_exp * e
        match = expr.match(pattern)
        if match:
            a, b = match[a_], match[b_]
            base_ = x + a/b
            nonlocal generic_cond
            generic_cond = Ne(b, 0)
            return base_, S.One
        raise NoMatch

    try:
        base, exp_ = _get_base_exp(integrand)
    except NoMatch:
        return
    if generic_cond is S.true:
        degenerate_step = None
    else:
        # equivalent with subs(b, 0) but no need to find b
        degenerate_step = ConstantRule(integrand.subs(x, 0), x)
    generic_step = NestedPowRule(integrand, x, base, exp_)
    return _add_degenerate_step(generic_cond, generic_step, degenerate_step)


def inverse_trig_rule(integral: IntegralInfo, degenerate=True):
    """
    Set degenerate=False on recursive call where coefficient of quadratic term
    is assumed non-zero.
    """
    integrand, symbol = integral
    base, exp = integrand.as_base_exp()
    a = Wild('a', exclude=[symbol])
    b = Wild('b', exclude=[symbol])
    c = Wild('c', exclude=[symbol, 0])
    match = base.match(a + b*symbol + c*symbol**2)

    if not match:
        return

    def make_inverse_trig(RuleClass, a, sign_a, c, sign_c, h) -> Rule:
        u_var = Dummy("u")
        rewritten = 1/sqrt(sign_a*a + sign_c*c*(symbol-h)**2)  # a>0, c>0
        quadratic_base = sqrt(c/a)*(symbol-h)
        constant = 1/sqrt(c)
        u_func = None
        if quadratic_base is not symbol:
            u_func = quadratic_base
            quadratic_base = u_var
        standard_form = 1/sqrt(sign_a + sign_c*quadratic_base**2)
        substep = RuleClass(standard_form, quadratic_base)
        if constant != 1:
            substep = ConstantTimesRule(constant*standard_form, symbol, constant, standard_form, substep)
        if u_func is not None:
            substep = URule(rewritten, symbol, u_var, u_func, substep)
        if h != 0:
            substep = CompleteSquareRule(integrand, symbol, rewritten, substep)
        return substep

    a, b, c = [match.get(i, S.Zero) for i in (a, b, c)]
    generic_cond = Ne(c, 0)
    if not degenerate or generic_cond is S.true:
        degenerate_step = None
    elif b.is_zero:
        degenerate_step = ConstantRule(a ** exp, symbol)
    else:
        degenerate_step = sqrt_linear_rule(IntegralInfo((a + b * symbol) ** exp, symbol))

    if simplify(2*exp + 1) == 0:
        h, k = -b/(2*c), a - b**2/(4*c)  # rewrite base to k + c*(symbol-h)**2
        non_square_cond = Ne(k, 0)
        square_step = None
        if non_square_cond is not S.true:
            square_step = NestedPowRule(1/sqrt(c*(symbol-h)**2), symbol, symbol-h, S.NegativeOne)
        if non_square_cond is S.false:
            return square_step
        generic_step = ReciprocalSqrtQuadraticRule(integrand, symbol, a, b, c)
        step = _add_degenerate_step(non_square_cond, generic_step, square_step)
        if k.is_real and c.is_real:
            # list of ((rule, base_exp, a, sign_a, b, sign_b), condition)
            rules = []
            for args, cond in (  # don't apply ArccoshRule to x**2-1
                ((ArcsinRule, k, 1, -c, -1, h), And(k > 0, c < 0)),  # 1-x**2
                ((ArcsinhRule, k, 1, c, 1, h), And(k > 0, c > 0)),  # 1+x**2
            ):
                if cond is S.true:
                    return make_inverse_trig(*args)
                if cond is not S.false:
                    rules.append((make_inverse_trig(*args), cond))
            if rules:
                if not k.is_positive:  # conditions are not thorough, need fall back rule
                    rules.append((generic_step, S.true))
                step = PiecewiseRule(integrand, symbol, rules)
            else:
                step = generic_step
        return _add_degenerate_step(generic_cond, step, degenerate_step)
    if exp == S.Half:
        step = SqrtQuadraticRule(integrand, symbol, a, b, c)
        return _add_degenerate_step(generic_cond, step, degenerate_step)


def add_rule(integral):
    integrand, symbol = integral
    results = [integral_steps(g, symbol)
              for g in integrand.as_ordered_terms()]
    return None if None in results else AddRule(integrand, symbol, results)


def mul_rule(integral: IntegralInfo):
    integrand, symbol = integral

    # Constant times function case
    coeff, f = integrand.as_independent(symbol)
    if coeff != 1:
        next_step = integral_steps(f, symbol)
        if next_step is not None:
            return ConstantTimesRule(integrand, symbol, coeff, f, next_step)


def _parts_rule(integrand, symbol) -> tuple[Expr, Expr, Expr, Expr, Rule] | None:
    # LIATE rule:
    # log, inverse trig, algebraic, trigonometric, exponential
    def pull_out_algebraic(integrand):
        integrand = integrand.cancel().together()
        # iterating over Piecewise args would not work here
        algebraic = ([] if isinstance(integrand, Piecewise) or not integrand.is_Mul
            else [arg for arg in integrand.args if arg.is_algebraic_expr(symbol)])
        if algebraic:
            u = Mul(*algebraic)
            dv = (integrand / u).cancel()
            return u, dv

    def pull_out_u(*functions) -> Callable[[Expr], tuple[Expr, Expr] | None]:
        def pull_out_u_rl(integrand: Expr) -> tuple[Expr, Expr] | None:
            if any(integrand.has(f) for f in functions):
                args = [arg for arg in integrand.args
                        if any(isinstance(arg, cls) for cls in functions)]
                if args:
                    u = Mul(*args) # type: ignore
                    dv = integrand / u
                    return u, dv
            return None

        return pull_out_u_rl

    liate_rules = [pull_out_u(log), pull_out_u(*inverse_trig_functions),
                   pull_out_algebraic, pull_out_u(sin, cos),
                   pull_out_u(exp)]


    dummy = Dummy("temporary")
    # we can integrate log(x) and atan(x) by setting dv = 1
    if isinstance(integrand, (log, *inverse_trig_functions)):
        integrand = dummy * integrand

    for index, rule in enumerate(liate_rules):
        result = rule(integrand)

        if result:
            u, dv = result

            # Don't pick u to be a constant if possible
            if symbol not in u.free_symbols and not u.has(dummy):
                return None

            u = u.subs(dummy, 1)
            dv = dv.subs(dummy, 1)

            # Don't pick a non-polynomial algebraic to be differentiated
            if rule == pull_out_algebraic and not u.is_polynomial(symbol):
                return None
            # Don't trade one logarithm for another
            if isinstance(u, log):
                rec_dv = 1/dv
                if (rec_dv.is_polynomial(symbol) and
                    degree(rec_dv, symbol) == 1):
                    return None

            # Can integrate a polynomial times OrthogonalPolynomial
            if rule == pull_out_algebraic:
                if dv.is_Derivative or dv.has(TrigonometricFunction) or \
                        isinstance(dv, OrthogonalPolynomial):
                    v_step = integral_steps(dv, symbol)
                    if v_step.contains_dont_know():
                        return None
                    else:
                        du = u.diff(symbol)
                        v = v_step.eval()
                        return u, dv, v, du, v_step

            # make sure dv is amenable to integration
            accept = False
            if index < 2:  # log and inverse trig are usually worth trying
                accept = True
            elif (rule == pull_out_algebraic and dv.args and
                all(isinstance(a, (sin, cos, exp))
                for a in dv.args)):
                    accept = True
            else:
                for lrule in liate_rules[index + 1:]:
                    r = lrule(integrand)
                    if r and r[0].subs(dummy, 1).equals(dv):
                        accept = True
                        break

            if accept:
                du = u.diff(symbol)
                v_step = integral_steps(simplify(dv), symbol)
                if not v_step.contains_dont_know():
                    v = v_step.eval()
                    return u, dv, v, du, v_step
    return None


def parts_rule(integral):
    integrand, symbol = integral
    constant, integrand = integrand.as_coeff_Mul()

    result = _parts_rule(integrand, symbol)

    steps = []
    if result:
        u, dv, v, du, v_step = result
        debug("u : {}, dv : {}, v : {}, du : {}, v_step: {}".format(u, dv, v, du, v_step))
        steps.append(result)

        if isinstance(v, Integral):
            return

        # Set a limit on the number of times u can be used
        if isinstance(u, (sin, cos, exp, sinh, cosh)):
            cachekey = u.xreplace({symbol: _cache_dummy})
            if _parts_u_cache[cachekey] > 2:
                return
            _parts_u_cache[cachekey] += 1

        # Try cyclic integration by parts a few times
        for _ in range(4):
            debug("Cyclic integration {} with v: {}, du: {}, integrand: {}".format(_, v, du, integrand))
            coefficient = ((v * du) / integrand).cancel()
            if coefficient == 1:
                break
            if symbol not in coefficient.free_symbols:
                rule = CyclicPartsRule(integrand, symbol,
                    [PartsRule(None, None, u, dv, v_step, None)
                     for (u, dv, v, du, v_step) in steps],
                    (-1) ** len(steps) * coefficient)
                if (constant != 1) and rule:
                    rule = ConstantTimesRule(constant * integrand, symbol, constant, integrand, rule)
                return rule

            # _parts_rule is sensitive to constants, factor it out
            next_constant, next_integrand = (v * du).as_coeff_Mul()
            result = _parts_rule(next_integrand, symbol)

            if result:
                u, dv, v, du, v_step = result
                u *= next_constant
                du *= next_constant
                steps.append((u, dv, v, du, v_step))
            else:
                break

    def make_second_step(steps, integrand):
        if steps:
            u, dv, v, du, v_step = steps[0]
            return PartsRule(integrand, symbol, u, dv, v_step, make_second_step(steps[1:], v * du))
        return integral_steps(integrand, symbol)

    if steps:
        u, dv, v, du, v_step = steps[0]
        rule = PartsRule(integrand, symbol, u, dv, v_step, make_second_step(steps[1:], v * du))
        if (constant != 1) and rule:
            rule = ConstantTimesRule(constant * integrand, symbol, constant, integrand, rule)
        return rule


def trig_rule(integral):
    integrand, symbol = integral
    if integrand == sin(symbol):
        return SinRule(integrand, symbol)
    if integrand == cos(symbol):
        return CosRule(integrand, symbol)
    if integrand == sec(symbol)**2:
        return Sec2Rule(integrand, symbol)
    if integrand == csc(symbol)**2:
        return Csc2Rule(integrand, symbol)

    if isinstance(integrand, tan):
        rewritten = sin(*integrand.args) / cos(*integrand.args)
    elif isinstance(integrand, cot):
        rewritten = cos(*integrand.args) / sin(*integrand.args)
    elif isinstance(integrand, sec):
        arg = integrand.args[0]
        rewritten = ((sec(arg)**2 + tan(arg) * sec(arg)) /
                     (sec(arg) + tan(arg)))
    elif isinstance(integrand, csc):
        arg = integrand.args[0]
        rewritten = ((csc(arg)**2 + cot(arg) * csc(arg)) /
                     (csc(arg) + cot(arg)))
    else:
        return

    return RewriteRule(integrand, symbol, rewritten, integral_steps(rewritten, symbol))

def trig_product_rule(integral: IntegralInfo):
    integrand, symbol = integral
    if integrand == sec(symbol) * tan(symbol):
        return SecTanRule(integrand, symbol)
    if integrand == csc(symbol) * cot(symbol):
        return CscCotRule(integrand, symbol)


def quadratic_denom_rule(integral):
    integrand, symbol = integral
    a = Wild('a', exclude=[symbol])
    b = Wild('b', exclude=[symbol])
    c = Wild('c', exclude=[symbol])

    match = integrand.match(a / (b * symbol ** 2 + c))

    if match:
        a, b, c = match[a], match[b], match[c]
        general_rule = ArctanRule(integrand, symbol, a, b, c)
        if b.is_extended_real and c.is_extended_real:
            positive_cond = c/b > 0
            if positive_cond is S.true:
                return general_rule
            coeff = a/(2*sqrt(-c)*sqrt(b))
            constant = sqrt(-c/b)
            r1 = 1/(symbol-constant)
            r2 = 1/(symbol+constant)
            log_steps = [ReciprocalRule(r1, symbol, symbol-constant),
                         ConstantTimesRule(-r2, symbol, -1, r2, ReciprocalRule(r2, symbol, symbol+constant))]
            rewritten = sub = r1 - r2
            negative_step = AddRule(sub, symbol, log_steps)
            if coeff != 1:
                rewritten = Mul(coeff, sub, evaluate=False)
                negative_step = ConstantTimesRule(rewritten, symbol, coeff, sub, negative_step)
            negative_step = RewriteRule(integrand, symbol, rewritten, negative_step)
            if positive_cond is S.false:
                return negative_step
            return PiecewiseRule(integrand, symbol, [(general_rule, positive_cond), (negative_step, S.true)])

        power = PowerRule(integrand, symbol, symbol, -2)
        if b != 1:
            power = ConstantTimesRule(integrand, symbol, 1/b, symbol**-2, power)

        return PiecewiseRule(integrand, symbol, [(general_rule, Ne(c, 0)), (power, True)])

    d = Wild('d', exclude=[symbol])
    match2 = integrand.match(a / (b * symbol ** 2 + c * symbol + d))
    if match2:
        b, c =  match2[b], match2[c]
        if b.is_zero:
            return
        u = Dummy('u')
        u_func = symbol + c/(2*b)
        integrand2 = integrand.subs(symbol, u - c / (2*b))
        next_step = integral_steps(integrand2, u)
        if next_step:
            return URule(integrand2, symbol, u, u_func, next_step)
        else:
            return
    e = Wild('e', exclude=[symbol])
    match3 = integrand.match((a* symbol + b) / (c * symbol ** 2 + d * symbol + e))
    if match3:
        a, b, c, d, e = match3[a], match3[b], match3[c], match3[d], match3[e]
        if c.is_zero:
            return
        denominator = c * symbol**2 + d * symbol + e
        const =  a/(2*c)
        numer1 =  (2*c*symbol+d)
        numer2 = - const*d + b
        u = Dummy('u')
        step1 = URule(integrand, symbol,
                      u, denominator, integral_steps(u**(-1), u))
        if const != 1:
            step1 = ConstantTimesRule(const*numer1/denominator, symbol,
                                      const, numer1/denominator, step1)
        if numer2.is_zero:
            return step1
        step2 = integral_steps(numer2/denominator, symbol)
        substeps = AddRule(integrand, symbol, [step1, step2])
        rewriten = const*numer1/denominator+numer2/denominator
        return RewriteRule(integrand, symbol, rewriten, substeps)

    return


def sqrt_linear_rule(integral: IntegralInfo):
    """
    Substitute common (a+b*x)**(1/n)
    """
    integrand, x = integral
    a = Wild('a', exclude=[x])
    b = Wild('b', exclude=[x, 0])
    a0 = b0 = 0
    bases, qs, bs = [], [], []
    for pow_ in integrand.find(Pow):  # collect all (a+b*x)**(p/q)
        base, exp_ = pow_.base, pow_.exp
        if exp_.is_Integer or x not in base.free_symbols:  # skip 1/x and sqrt(2)
            continue
        if not exp_.is_Rational:  # exclude x**pi
            return
        match = base.match(a+b*x)
        if not match:  # skip non-linear
            continue  # for sqrt(x+sqrt(x)), although base is non-linear, we can still substitute sqrt(x)
        a1, b1 = match[a], match[b]
        if a0*b1 != a1*b0 or not (b0/b1).is_nonnegative:  # cannot transform sqrt(x) to sqrt(x+1) or sqrt(-x)
            return
        if b0 == 0 or (b0/b1 > 1) is S.true:  # choose the latter of sqrt(2*x) and sqrt(x) as representative
            a0, b0 = a1, b1
        bases.append(base)
        bs.append(b1)
        qs.append(exp_.q)
    if b0 == 0:  # no such pattern found
        return
    q0: Integer = lcm_list(qs)
    u_x = (a0 + b0*x)**(1/q0)
    u = Dummy("u")
    substituted = integrand.subs({base**(S.One/q): (b/b0)**(S.One/q)*u**(q0/q)
                                  for base, b, q in zip(bases, bs, qs)}).subs(x, (u**q0-a0)/b0)
    substep = integral_steps(substituted*u**(q0-1)*q0/b0, u)
    if not substep.contains_dont_know():
        step: Rule = URule(integrand, x, u, u_x, substep)
        generic_cond = Ne(b0, 0)
        if generic_cond is not S.true:  # possible degenerate case
            simplified = integrand.subs(dict.fromkeys(bs, 0))
            degenerate_step = integral_steps(simplified, x)
            step = PiecewiseRule(integrand, x, [(step, generic_cond), (degenerate_step, S.true)])
        return step


def sqrt_quadratic_rule(integral: IntegralInfo, degenerate=True):
    integrand, x = integral
    a = Wild('a', exclude=[x])
    b = Wild('b', exclude=[x])
    c = Wild('c', exclude=[x, 0])
    f = Wild('f')
    n = Wild('n', properties=[lambda n: n.is_Integer and n.is_odd])
    match = integrand.match(f*sqrt(a+b*x+c*x**2)**n)
    if not match:
        return
    a, b, c, f, n = match[a], match[b], match[c], match[f], match[n]
    f_poly = f.as_poly(x)
    if f_poly is None:
        return

    generic_cond = Ne(c, 0)
    if not degenerate or generic_cond is S.true:
        degenerate_step = None
    elif b.is_zero:
        degenerate_step = integral_steps(f*sqrt(a)**n, x)
    else:
        degenerate_step = sqrt_linear_rule(IntegralInfo(f*sqrt(a+b*x)**n, x))

    def sqrt_quadratic_denom_rule(numer_poly: Poly, integrand: Expr):
        denom = sqrt(a+b*x+c*x**2)
        deg = numer_poly.degree()
        if deg <= 1:
            # integrand == (d+e*x)/sqrt(a+b*x+c*x**2)
            e, d = numer_poly.all_coeffs() if deg == 1 else (S.Zero, numer_poly.as_expr())
            # rewrite numerator to A*(2*c*x+b) + B
            A = e/(2*c)
            B = d-A*b
            pre_substitute = (2*c*x+b)/denom
            constant_step: Rule | None = None
            linear_step: Rule | None = None
            if A != 0:
                u = Dummy("u")
                pow_rule = PowerRule(1/sqrt(u), u, u, -S.Half)
                linear_step = URule(pre_substitute, x, u, a+b*x+c*x**2, pow_rule)
                if A != 1:
                    linear_step = ConstantTimesRule(A*pre_substitute, x, A, pre_substitute, linear_step)
            if B != 0:
                constant_step = inverse_trig_rule(IntegralInfo(1/denom, x), degenerate=False)
                if B != 1:
                    constant_step = ConstantTimesRule(B/denom, x, B, 1/denom, constant_step)  # type: ignore
            if linear_step and constant_step:
                add = Add(A*pre_substitute, B/denom, evaluate=False)
                step: Rule | None = RewriteRule(integrand, x, add, AddRule(add, x, [linear_step, constant_step]))
            else:
                step = linear_step or constant_step
        else:
            coeffs = numer_poly.all_coeffs()
            step = SqrtQuadraticDenomRule(integrand, x, a, b, c, coeffs)
        return step

    if n > 0:  # rewrite poly * sqrt(s)**(2*k-1) to poly*s**k / sqrt(s)
        numer_poly = f_poly * (a+b*x+c*x**2)**((n+1)/2)
        rewritten = numer_poly.as_expr()/sqrt(a+b*x+c*x**2)
        substep = sqrt_quadratic_denom_rule(numer_poly, rewritten)
        generic_step = RewriteRule(integrand, x, rewritten, substep)
    elif n == -1:
        generic_step = sqrt_quadratic_denom_rule(f_poly, integrand)
    else:
        return  # todo: handle n < -1 case
    return _add_degenerate_step(generic_cond, generic_step, degenerate_step)


def hyperbolic_rule(integral: tuple[Expr, Symbol]):
    integrand, symbol = integral
    if isinstance(integrand, HyperbolicFunction) and integrand.args[0] == symbol:
        if integrand.func == sinh:
            return SinhRule(integrand, symbol)
        if integrand.func == cosh:
            return CoshRule(integrand, symbol)
        u = Dummy('u')
        if integrand.func == tanh:
            rewritten = sinh(symbol)/cosh(symbol)
            return RewriteRule(integrand, symbol, rewritten,
                   URule(rewritten, symbol, u, cosh(symbol), ReciprocalRule(1/u, u, u)))
        if integrand.func == coth:
            rewritten = cosh(symbol)/sinh(symbol)
            return RewriteRule(integrand, symbol, rewritten,
                   URule(rewritten, symbol, u, sinh(symbol), ReciprocalRule(1/u, u, u)))
        else:
            rewritten = integrand.rewrite(tanh)
            if integrand.func == sech:
                return RewriteRule(integrand, symbol, rewritten,
                       URule(rewritten, symbol, u, tanh(symbol/2),
                       ArctanRule(2/(u**2 + 1), u, S(2), S.One, S.One)))
            if integrand.func == csch:
                return RewriteRule(integrand, symbol, rewritten,
                       URule(rewritten, symbol, u, tanh(symbol/2),
                       ReciprocalRule(1/u, u, u)))

@cacheit
def make_wilds(symbol):
    a = Wild('a', exclude=[symbol])
    b = Wild('b', exclude=[symbol])
    m = Wild('m', exclude=[symbol], properties=[lambda n: isinstance(n, Integer)])
    n = Wild('n', exclude=[symbol], properties=[lambda n: isinstance(n, Integer)])

    return a, b, m, n

@cacheit
def sincos_pattern(symbol):
    a, b, m, n = make_wilds(symbol)
    pattern = sin(a*symbol)**m * cos(b*symbol)**n

    return pattern, a, b, m, n

@cacheit
def tansec_pattern(symbol):
    a, b, m, n = make_wilds(symbol)
    pattern = tan(a*symbol)**m * sec(b*symbol)**n

    return pattern, a, b, m, n

@cacheit
def cotcsc_pattern(symbol):
    a, b, m, n = make_wilds(symbol)
    pattern = cot(a*symbol)**m * csc(b*symbol)**n

    return pattern, a, b, m, n

@cacheit
def heaviside_pattern(symbol):
    m = Wild('m', exclude=[symbol])
    b = Wild('b', exclude=[symbol])
    g = Wild('g')
    pattern = Heaviside(m*symbol + b) * g

    return pattern, m, b, g

def uncurry(func):
    def uncurry_rl(args):
        return func(*args)
    return uncurry_rl

def trig_rewriter(rewrite):
    def trig_rewriter_rl(args):
        a, b, m, n, integrand, symbol = args
        rewritten = rewrite(a, b, m, n, integrand, symbol)
        if rewritten != integrand:
            return RewriteRule(integrand, symbol, rewritten, integral_steps(rewritten, symbol))
    return trig_rewriter_rl

sincos_botheven_condition = uncurry(
    lambda a, b, m, n, i, s: m.is_even and n.is_even and
    m.is_nonnegative and n.is_nonnegative)

sincos_botheven = trig_rewriter(
    lambda a, b, m, n, i, symbol: ( (((1 - cos(2*a*symbol)) / 2) ** (m / 2)) *
                                    (((1 + cos(2*b*symbol)) / 2) ** (n / 2)) ))

sincos_sinodd_condition = uncurry(lambda a, b, m, n, i, s: m.is_odd and m >= 3)

sincos_sinodd = trig_rewriter(
    lambda a, b, m, n, i, symbol: ( (1 - cos(a*symbol)**2)**((m - 1) / 2) *
                                    sin(a*symbol) *
                                    cos(b*symbol) ** n))

sincos_cosodd_condition = uncurry(lambda a, b, m, n, i, s: n.is_odd and n >= 3)

sincos_cosodd = trig_rewriter(
    lambda a, b, m, n, i, symbol: ( (1 - sin(b*symbol)**2)**((n - 1) / 2) *
                                    cos(b*symbol) *
                                    sin(a*symbol) ** m))

tansec_seceven_condition = uncurry(lambda a, b, m, n, i, s: n.is_even and n >= 4)
tansec_seceven = trig_rewriter(
    lambda a, b, m, n, i, symbol: ( (1 + tan(b*symbol)**2) ** (n/2 - 1) *
                                    sec(b*symbol)**2 *
                                    tan(a*symbol) ** m ))

tansec_tanodd_condition = uncurry(lambda a, b, m, n, i, s: m.is_odd)
tansec_tanodd = trig_rewriter(
    lambda a, b, m, n, i, symbol: ( (sec(a*symbol)**2 - 1) ** ((m - 1) / 2) *
                                     tan(a*symbol) *
                                     sec(b*symbol) ** n ))

tan_tansquared_condition = uncurry(lambda a, b, m, n, i, s: m == 2 and n == 0)
tan_tansquared = trig_rewriter(
    lambda a, b, m, n, i, symbol: ( sec(a*symbol)**2 - 1))

cotcsc_csceven_condition = uncurry(lambda a, b, m, n, i, s: n.is_even and n >= 4)
cotcsc_csceven = trig_rewriter(
    lambda a, b, m, n, i, symbol: ( (1 + cot(b*symbol)**2) ** (n/2 - 1) *
                                    csc(b*symbol)**2 *
                                    cot(a*symbol) ** m ))

cotcsc_cotodd_condition = uncurry(lambda a, b, m, n, i, s: m.is_odd)
cotcsc_cotodd = trig_rewriter(
    lambda a, b, m, n, i, symbol: ( (csc(a*symbol)**2 - 1) ** ((m - 1) / 2) *
                                    cot(a*symbol) *
                                    csc(b*symbol) ** n ))

def trig_sincos_rule(integral):
    integrand, symbol = integral

    if any(integrand.has(f) for f in (sin, cos)):
        pattern, a, b, m, n = sincos_pattern(symbol)
        match = integrand.match(pattern)
        if not match:
            return

        return multiplexer({
            sincos_botheven_condition: sincos_botheven,
            sincos_sinodd_condition: sincos_sinodd,
            sincos_cosodd_condition: sincos_cosodd
        })(tuple(
            [match.get(i, S.Zero) for i in (a, b, m, n)] +
            [integrand, symbol]))

def trig_tansec_rule(integral):
    integrand, symbol = integral

    integrand = integrand.subs({
        1 / cos(symbol): sec(symbol)
    })

    if any(integrand.has(f) for f in (tan, sec)):
        pattern, a, b, m, n = tansec_pattern(symbol)
        match = integrand.match(pattern)
        if not match:
            return

        return multiplexer({
            tansec_tanodd_condition: tansec_tanodd,
            tansec_seceven_condition: tansec_seceven,
            tan_tansquared_condition: tan_tansquared
        })(tuple(
            [match.get(i, S.Zero) for i in (a, b, m, n)] +
            [integrand, symbol]))

def trig_cotcsc_rule(integral):
    integrand, symbol = integral
    integrand = integrand.subs({
        1 / sin(symbol): csc(symbol),
        1 / tan(symbol): cot(symbol),
        cos(symbol) / tan(symbol): cot(symbol)
    })

    if any(integrand.has(f) for f in (cot, csc)):
        pattern, a, b, m, n = cotcsc_pattern(symbol)
        match = integrand.match(pattern)
        if not match:
            return

        return multiplexer({
            cotcsc_cotodd_condition: cotcsc_cotodd,
            cotcsc_csceven_condition: cotcsc_csceven
        })(tuple(
            [match.get(i, S.Zero) for i in (a, b, m, n)] +
            [integrand, symbol]))

def trig_sindouble_rule(integral):
    integrand, symbol = integral
    a = Wild('a', exclude=[sin(2*symbol)])
    match = integrand.match(sin(2*symbol)*a)
    if match:
        sin_double = 2*sin(symbol)*cos(symbol)/sin(2*symbol)
        return integral_steps(integrand * sin_double, symbol)

def trig_powers_products_rule(integral):
    return do_one(null_safe(trig_sincos_rule),
                  null_safe(trig_tansec_rule),
                  null_safe(trig_cotcsc_rule),
                  null_safe(trig_sindouble_rule))(integral)

def trig_substitution_rule(integral):
    integrand, symbol = integral
    A = Wild('a', exclude=[0, symbol])
    B = Wild('b', exclude=[0, symbol])
    theta = Dummy("theta")
    target_pattern = A + B*symbol**2

    matches = integrand.find(target_pattern)
    for expr in matches:
        match = expr.match(target_pattern)
        a = match.get(A, S.Zero)
        b = match.get(B, S.Zero)

        a_positive = ((a.is_number and a > 0) or a.is_positive)
        b_positive = ((b.is_number and b > 0) or b.is_positive)
        a_negative = ((a.is_number and a < 0) or a.is_negative)
        b_negative = ((b.is_number and b < 0) or b.is_negative)
        x_func = None
        if a_positive and b_positive:
            # a**2 + b*x**2. Assume sec(theta) > 0, -pi/2 < theta < pi/2
            x_func = (sqrt(a)/sqrt(b)) * tan(theta)
            # Do not restrict the domain: tan(theta) takes on any real
            # value on the interval -pi/2 < theta < pi/2 so x takes on
            # any value
            restriction = True
        elif a_positive and b_negative:
            # a**2 - b*x**2. Assume cos(theta) > 0, -pi/2 < theta < pi/2
            constant = sqrt(a)/sqrt(-b)
            x_func = constant * sin(theta)
            restriction = And(symbol > -constant, symbol < constant)
        elif a_negative and b_positive:
            # b*x**2 - a**2. Assume sin(theta) > 0, 0 < theta < pi
            constant = sqrt(-a)/sqrt(b)
            x_func = constant * sec(theta)
            restriction = And(symbol > -constant, symbol < constant)
        if x_func:
            # Manually simplify sqrt(trig(theta)**2) to trig(theta)
            # Valid due to assumed domain restriction
            substitutions = {}
            for f in [sin, cos, tan,
                      sec, csc, cot]:
                substitutions[sqrt(f(theta)**2)] = f(theta)
                substitutions[sqrt(f(theta)**(-2))] = 1/f(theta)

            replaced = integrand.subs(symbol, x_func).trigsimp()
            replaced = manual_subs(replaced, substitutions)
            if not replaced.has(symbol):
                replaced *= manual_diff(x_func, theta)
                replaced = replaced.trigsimp()
                secants = replaced.find(1/cos(theta))
                if secants:
                    replaced = replaced.xreplace({
                        1/cos(theta): sec(theta)
                    })

                substep = integral_steps(replaced, theta)
                if not substep.contains_dont_know():
                    return TrigSubstitutionRule(integrand, symbol,
                        theta, x_func, replaced, substep, restriction)

def heaviside_rule(integral):
    integrand, symbol = integral
    pattern, m, b, g = heaviside_pattern(symbol)
    match = integrand.match(pattern)
    if match and 0 != match[g]:
        # f = Heaviside(m*x + b)*g
        substep = integral_steps(match[g], symbol)
        m, b = match[m], match[b]
        return HeavisideRule(integrand, symbol, m*symbol + b, -b/m, substep)


def dirac_delta_rule(integral: IntegralInfo):
    integrand, x = integral
    if len(integrand.args) == 1:
        n = S.Zero
    else:
        n = integrand.args[1] # type: ignore
    if not n.is_Integer or n < 0:
        return
    a, b = Wild('a', exclude=[x]), Wild('b', exclude=[x, 0])
    match = integrand.args[0].match(a+b*x)
    if not match:
        return
    a, b = match[a], match[b]
    generic_cond = Ne(b, 0)
    if generic_cond is S.true:
        degenerate_step = None
    else:
        degenerate_step = ConstantRule(DiracDelta(a, n), x)
    generic_step = DiracDeltaRule(integrand, x, n, a, b)
    return _add_degenerate_step(generic_cond, generic_step, degenerate_step)


def substitution_rule(integral):
    integrand, symbol = integral

    u_var = Dummy("u")
    substitutions = find_substitutions(integrand, symbol, u_var)
    count = 0
    if substitutions:
        debug("List of Substitution Rules")
        ways = []
        for u_func, c, substituted in substitutions:
            subrule = integral_steps(substituted, u_var)
            count = count + 1
            debug("Rule {}: {}".format(count, subrule))

            if subrule.contains_dont_know():
                continue

            if simplify(c - 1) != 0:
                _, denom = c.as_numer_denom()
                if subrule:
                    subrule = ConstantTimesRule(c * substituted, u_var, c, substituted, subrule)

                if denom.free_symbols:
                    piecewise = []
                    could_be_zero = []

                    if isinstance(denom, Mul):
                        could_be_zero = denom.args
                    else:
                        could_be_zero.append(denom)

                    for expr in could_be_zero:
                        if not fuzzy_not(expr.is_zero):
                            substep = integral_steps(manual_subs(integrand, expr, 0), symbol)

                            if substep:
                                piecewise.append((
                                    substep,
                                    Eq(expr, 0)
                                ))
                    piecewise.append((subrule, True))
                    subrule = PiecewiseRule(substituted, symbol, piecewise)

            ways.append(URule(integrand, symbol, u_var, u_func, subrule))

        if len(ways) > 1:
            return AlternativeRule(integrand, symbol, ways)
        elif ways:
            return ways[0]


partial_fractions_rule = rewriter(
    lambda integrand, symbol: integrand.is_rational_function(),
    lambda integrand, symbol: integrand.apart(symbol))

cancel_rule = rewriter(
    # lambda integrand, symbol: integrand.is_algebraic_expr(),
    # lambda integrand, symbol: isinstance(integrand, Mul),
    lambda integrand, symbol: True,
    lambda integrand, symbol: integrand.cancel())

distribute_expand_rule = rewriter(
    lambda integrand, symbol: (
        isinstance(integrand, (Pow, Mul)) or all(arg.is_Pow or arg.is_polynomial(symbol) for arg in integrand.args)),
    lambda integrand, symbol: integrand.expand())

trig_expand_rule = rewriter(
    # If there are trig functions with different arguments, expand them
    lambda integrand, symbol: (
        len({a.args[0] for a in integrand.atoms(TrigonometricFunction)}) > 1),
    lambda integrand, symbol: integrand.expand(trig=True))

def derivative_rule(integral):
    integrand = integral[0]
    diff_variables = integrand.variables
    undifferentiated_function = integrand.expr
    integrand_variables = undifferentiated_function.free_symbols

    if integral.symbol in integrand_variables:
        if integral.symbol in diff_variables:
            return DerivativeRule(*integral)
        else:
            return DontKnowRule(integrand, integral.symbol)
    else:
        return ConstantRule(*integral)

def rewrites_rule(integral):
    integrand, symbol = integral

    if integrand.match(1/cos(symbol)):
        rewritten = integrand.subs(1/cos(symbol), sec(symbol))
        return RewriteRule(integrand, symbol, rewritten, integral_steps(rewritten, symbol))

def fallback_rule(integral):
    return DontKnowRule(*integral)

# Cache is used to break cyclic integrals.
# Need to use the same dummy variable in cached expressions for them to match.
# Also record "u" of integration by parts, to avoid infinite repetition.
_integral_cache: dict[Expr, Expr | None] = {}
_parts_u_cache: dict[Expr, int] = defaultdict(int)
_cache_dummy = Dummy("z")

def integral_steps(integrand, symbol, **options):
    """Returns the steps needed to compute an integral.

    Explanation
    ===========

    This function attempts to mirror what a student would do by hand as
    closely as possible.

    SymPy Gamma uses this to provide a step-by-step explanation of an
    integral. The code it uses to format the results of this function can be
    found at
    https://github.com/sympy/sympy_gamma/blob/master/app/logic/intsteps.py.

    Examples
    ========

    >>> from sympy import exp, sin
    >>> from sympy.integrals.manualintegrate import integral_steps
    >>> from sympy.abc import x
    >>> print(repr(integral_steps(exp(x) / (1 + exp(2 * x)), x))) \
    # doctest: +NORMALIZE_WHITESPACE
    URule(integrand=exp(x)/(exp(2*x) + 1), variable=x, u_var=_u, u_func=exp(x),
    substep=ArctanRule(integrand=1/(_u**2 + 1), variable=_u, a=1, b=1, c=1))
    >>> print(repr(integral_steps(sin(x), x))) \
    # doctest: +NORMALIZE_WHITESPACE
    SinRule(integrand=sin(x), variable=x)
    >>> print(repr(integral_steps((x**2 + 3)**2, x))) \
    # doctest: +NORMALIZE_WHITESPACE
    RewriteRule(integrand=(x**2 + 3)**2, variable=x, rewritten=x**4 + 6*x**2 + 9,
    substep=AddRule(integrand=x**4 + 6*x**2 + 9, variable=x,
    substeps=[PowerRule(integrand=x**4, variable=x, base=x, exp=4),
    ConstantTimesRule(integrand=6*x**2, variable=x, constant=6, other=x**2,
    substep=PowerRule(integrand=x**2, variable=x, base=x, exp=2)),
    ConstantRule(integrand=9, variable=x)]))

    Returns
    =======

    rule : Rule
        The first step; most rules have substeps that must also be
        considered. These substeps can be evaluated using ``manualintegrate``
        to obtain a result.

    """
    cachekey = integrand.xreplace({symbol: _cache_dummy})
    if cachekey in _integral_cache:
        if _integral_cache[cachekey] is None:
            # Stop this attempt, because it leads around in a loop
            return DontKnowRule(integrand, symbol)
        else:
            # TODO: This is for future development, as currently
            # _integral_cache gets no values other than None
            return (_integral_cache[cachekey].xreplace(_cache_dummy, symbol),
                symbol)
    else:
        _integral_cache[cachekey] = None

    integral = IntegralInfo(integrand, symbol)

    def key(integral):
        integrand = integral.integrand

        if symbol not in integrand.free_symbols:
            return Number
        for cls in (Symbol, TrigonometricFunction, OrthogonalPolynomial):
            if isinstance(integrand, cls):
                return cls
        return type(integrand)

    def integral_is_subclass(*klasses):
        def _integral_is_subclass(integral):
            k = key(integral)
            return k and issubclass(k, klasses)
        return _integral_is_subclass

    result = do_one(
        null_safe(special_function_rule),
        null_safe(switch(key, {
            Pow: do_one(null_safe(power_rule), null_safe(inverse_trig_rule),
                        null_safe(sqrt_linear_rule),
                        null_safe(quadratic_denom_rule)),
            Symbol: power_rule,
            exp: exp_rule,
            Add: add_rule,
            Mul: do_one(null_safe(mul_rule), null_safe(trig_product_rule),
                        null_safe(heaviside_rule), null_safe(quadratic_denom_rule),
                        null_safe(sqrt_linear_rule),
                        null_safe(sqrt_quadratic_rule)),
            Derivative: derivative_rule,
            TrigonometricFunction: trig_rule,
            Heaviside: heaviside_rule,
            DiracDelta: dirac_delta_rule,
            OrthogonalPolynomial: orthogonal_poly_rule,
            Number: constant_rule
        })),
        do_one(
            null_safe(trig_rule),
            null_safe(hyperbolic_rule),
            null_safe(alternatives(
                rewrites_rule,
                substitution_rule,
                condition(
                    integral_is_subclass(Mul, Pow),
                    partial_fractions_rule),
                condition(
                    integral_is_subclass(Mul, Pow),
                    cancel_rule),
                condition(
                    integral_is_subclass(Mul, log,
                    *inverse_trig_functions),
                    parts_rule),
                condition(
                    integral_is_subclass(Mul, Pow),
                    distribute_expand_rule),
                trig_powers_products_rule,
                trig_expand_rule
            )),
            null_safe(condition(integral_is_subclass(Mul, Pow), nested_pow_rule)),
            null_safe(trig_substitution_rule)
        ),
        fallback_rule)(integral)
    del _integral_cache[cachekey]
    return result


def manualintegrate(f, var):
    """manualintegrate(f, var)

    Explanation
    ===========

    Compute indefinite integral of a single variable using an algorithm that
    resembles what a student would do by hand.

    Unlike :func:`~.integrate`, var can only be a single symbol.

    Examples
    ========

    >>> from sympy import sin, cos, tan, exp, log, integrate
    >>> from sympy.integrals.manualintegrate import manualintegrate
    >>> from sympy.abc import x
    >>> manualintegrate(1 / x, x)
    log(x)
    >>> integrate(1/x)
    log(x)
    >>> manualintegrate(log(x), x)
    x*log(x) - x
    >>> integrate(log(x))
    x*log(x) - x
    >>> manualintegrate(exp(x) / (1 + exp(2 * x)), x)
    atan(exp(x))
    >>> integrate(exp(x) / (1 + exp(2 * x)))
    RootSum(4*_z**2 + 1, Lambda(_i, _i*log(2*_i + exp(x))))
    >>> manualintegrate(cos(x)**4 * sin(x), x)
    -cos(x)**5/5
    >>> integrate(cos(x)**4 * sin(x), x)
    -cos(x)**5/5
    >>> manualintegrate(cos(x)**4 * sin(x)**3, x)
    cos(x)**7/7 - cos(x)**5/5
    >>> integrate(cos(x)**4 * sin(x)**3, x)
    cos(x)**7/7 - cos(x)**5/5
    >>> manualintegrate(tan(x), x)
    -log(cos(x))
    >>> integrate(tan(x), x)
    -log(cos(x))

    See Also
    ========

    sympy.integrals.integrals.integrate
    sympy.integrals.integrals.Integral.doit
    sympy.integrals.integrals.Integral
    """
    result = integral_steps(f, var).eval()
    # Clear the cache of u-parts
    _parts_u_cache.clear()
    # If we got Piecewise with two parts, put generic first
    if isinstance(result, Piecewise) and len(result.args) == 2:
        cond = result.args[0][1]
        if isinstance(cond, Eq) and result.args[1][1] == True:
            result = result.func(
                (result.args[1][0], Ne(*cond.args)),
                (result.args[0][0], True))
    return result