760 lines
25 KiB
Python
760 lines
25 KiB
Python
from typing import Dict, List
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from itertools import permutations
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from functools import reduce
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from sympy.core.add import Add
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from sympy.core.basic import Basic
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from sympy.core.mul import Mul
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from sympy.core.symbol import Wild, Dummy
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from sympy.core.basic import sympify
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from sympy.core.numbers import Rational, pi, I
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from sympy.core.relational import Eq, Ne
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from sympy.core.singleton import S
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from sympy.functions import exp, sin, cos, tan, cot, asin, atan
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from sympy.functions import log, sinh, cosh, tanh, coth, asinh, acosh
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from sympy.functions import sqrt, erf, erfi, li, Ei
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from sympy.functions import besselj, bessely, besseli, besselk
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from sympy.functions import hankel1, hankel2, jn, yn
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from sympy.functions.elementary.complexes import Abs, re, im, sign, arg
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from sympy.functions.elementary.exponential import LambertW
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from sympy.functions.elementary.integers import floor, ceiling
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from sympy.functions.elementary.piecewise import Piecewise
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from sympy.functions.special.delta_functions import Heaviside, DiracDelta
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from sympy.simplify.radsimp import collect
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from sympy.logic.boolalg import And, Or
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from sympy.utilities.iterables import uniq
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from sympy.polys import quo, gcd, lcm, factor, cancel, PolynomialError
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from sympy.polys.monomials import itermonomials
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from sympy.polys.polyroots import root_factors
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from sympy.polys.rings import PolyRing
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from sympy.polys.solvers import solve_lin_sys
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from sympy.polys.constructor import construct_domain
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from sympy.core.compatibility import ordered
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from sympy.integrals.integrals import integrate
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def components(f, x):
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"""
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Returns a set of all functional components of the given expression
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which includes symbols, function applications and compositions and
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non-integer powers. Fractional powers are collected with
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minimal, positive exponents.
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Examples
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========
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>>> from sympy import cos, sin
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>>> from sympy.abc import x
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>>> from sympy.integrals.heurisch import components
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>>> components(sin(x)*cos(x)**2, x)
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{x, sin(x), cos(x)}
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See Also
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========
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heurisch
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"""
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result = set()
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if x in f.free_symbols:
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if f.is_symbol and f.is_commutative:
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result.add(f)
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elif f.is_Function or f.is_Derivative:
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for g in f.args:
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result |= components(g, x)
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result.add(f)
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elif f.is_Pow:
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result |= components(f.base, x)
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if not f.exp.is_Integer:
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if f.exp.is_Rational:
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result.add(f.base**Rational(1, f.exp.q))
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else:
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result |= components(f.exp, x) | {f}
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else:
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for g in f.args:
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result |= components(g, x)
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return result
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# name -> [] of symbols
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_symbols_cache = {} # type: Dict[str, List[Dummy]]
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# NB @cacheit is not convenient here
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def _symbols(name, n):
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"""get vector of symbols local to this module"""
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try:
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lsyms = _symbols_cache[name]
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except KeyError:
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lsyms = []
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_symbols_cache[name] = lsyms
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while len(lsyms) < n:
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lsyms.append( Dummy('%s%i' % (name, len(lsyms))) )
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return lsyms[:n]
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def heurisch_wrapper(f, x, rewrite=False, hints=None, mappings=None, retries=3,
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degree_offset=0, unnecessary_permutations=None,
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_try_heurisch=None):
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"""
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A wrapper around the heurisch integration algorithm.
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Explanation
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===========
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This method takes the result from heurisch and checks for poles in the
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denominator. For each of these poles, the integral is reevaluated, and
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the final integration result is given in terms of a Piecewise.
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Examples
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========
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>>> from sympy.core import symbols
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>>> from sympy.functions import cos
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>>> from sympy.integrals.heurisch import heurisch, heurisch_wrapper
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>>> n, x = symbols('n x')
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>>> heurisch(cos(n*x), x)
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sin(n*x)/n
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>>> heurisch_wrapper(cos(n*x), x)
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Piecewise((sin(n*x)/n, Ne(n, 0)), (x, True))
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See Also
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========
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heurisch
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"""
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from sympy.solvers.solvers import solve, denoms
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f = sympify(f)
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if x not in f.free_symbols:
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return f*x
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res = heurisch(f, x, rewrite, hints, mappings, retries, degree_offset,
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unnecessary_permutations, _try_heurisch)
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if not isinstance(res, Basic):
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return res
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# We consider each denominator in the expression, and try to find
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# cases where one or more symbolic denominator might be zero. The
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# conditions for these cases are stored in the list slns.
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slns = []
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for d in denoms(res):
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try:
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slns += solve(d, dict=True, exclude=(x,))
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except NotImplementedError:
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pass
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if not slns:
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return res
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slns = list(uniq(slns))
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# Remove the solutions corresponding to poles in the original expression.
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slns0 = []
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for d in denoms(f):
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try:
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slns0 += solve(d, dict=True, exclude=(x,))
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except NotImplementedError:
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pass
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slns = [s for s in slns if s not in slns0]
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if not slns:
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return res
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if len(slns) > 1:
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eqs = []
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for sub_dict in slns:
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eqs.extend([Eq(key, value) for key, value in sub_dict.items()])
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slns = solve(eqs, dict=True, exclude=(x,)) + slns
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# For each case listed in the list slns, we reevaluate the integral.
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pairs = []
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for sub_dict in slns:
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expr = heurisch(f.subs(sub_dict), x, rewrite, hints, mappings, retries,
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degree_offset, unnecessary_permutations,
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_try_heurisch)
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cond = And(*[Eq(key, value) for key, value in sub_dict.items()])
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generic = Or(*[Ne(key, value) for key, value in sub_dict.items()])
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if expr is None:
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expr = integrate(f.subs(sub_dict),x)
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pairs.append((expr, cond))
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# If there is one condition, put the generic case first. Otherwise,
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# doing so may lead to longer Piecewise formulas
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if len(pairs) == 1:
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pairs = [(heurisch(f, x, rewrite, hints, mappings, retries,
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degree_offset, unnecessary_permutations,
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_try_heurisch),
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generic),
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(pairs[0][0], True)]
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else:
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pairs.append((heurisch(f, x, rewrite, hints, mappings, retries,
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degree_offset, unnecessary_permutations,
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_try_heurisch),
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True))
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return Piecewise(*pairs)
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class BesselTable:
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"""
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Derivatives of Bessel functions of orders n and n-1
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in terms of each other.
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See the docstring of DiffCache.
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"""
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def __init__(self):
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self.table = {}
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self.n = Dummy('n')
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self.z = Dummy('z')
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self._create_table()
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def _create_table(t):
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table, n, z = t.table, t.n, t.z
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for f in (besselj, bessely, hankel1, hankel2):
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table[f] = (f(n-1, z) - n*f(n, z)/z,
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(n-1)*f(n-1, z)/z - f(n, z))
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f = besseli
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table[f] = (f(n-1, z) - n*f(n, z)/z,
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(n-1)*f(n-1, z)/z + f(n, z))
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f = besselk
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table[f] = (-f(n-1, z) - n*f(n, z)/z,
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(n-1)*f(n-1, z)/z - f(n, z))
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for f in (jn, yn):
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table[f] = (f(n-1, z) - (n+1)*f(n, z)/z,
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(n-1)*f(n-1, z)/z - f(n, z))
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def diffs(t, f, n, z):
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if f in t.table:
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diff0, diff1 = t.table[f]
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repl = [(t.n, n), (t.z, z)]
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return (diff0.subs(repl), diff1.subs(repl))
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def has(t, f):
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return f in t.table
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_bessel_table = None
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class DiffCache:
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"""
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Store for derivatives of expressions.
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Explanation
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===========
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The standard form of the derivative of a Bessel function of order n
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contains two Bessel functions of orders n-1 and n+1, respectively.
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Such forms cannot be used in parallel Risch algorithm, because
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there is a linear recurrence relation between the three functions
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while the algorithm expects that functions and derivatives are
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represented in terms of algebraically independent transcendentals.
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The solution is to take two of the functions, e.g., those of orders
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n and n-1, and to express the derivatives in terms of the pair.
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To guarantee that the proper form is used the two derivatives are
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cached as soon as one is encountered.
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Derivatives of other functions are also cached at no extra cost.
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All derivatives are with respect to the same variable `x`.
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"""
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def __init__(self, x):
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self.cache = {}
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self.x = x
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global _bessel_table
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if not _bessel_table:
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_bessel_table = BesselTable()
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def get_diff(self, f):
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cache = self.cache
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if f in cache:
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pass
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elif (not hasattr(f, 'func') or
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not _bessel_table.has(f.func)):
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cache[f] = cancel(f.diff(self.x))
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else:
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n, z = f.args
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d0, d1 = _bessel_table.diffs(f.func, n, z)
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dz = self.get_diff(z)
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cache[f] = d0*dz
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cache[f.func(n-1, z)] = d1*dz
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return cache[f]
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def heurisch(f, x, rewrite=False, hints=None, mappings=None, retries=3,
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degree_offset=0, unnecessary_permutations=None,
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_try_heurisch=None):
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"""
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Compute indefinite integral using heuristic Risch algorithm.
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Explanation
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===========
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This is a heuristic approach to indefinite integration in finite
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terms using the extended heuristic (parallel) Risch algorithm, based
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on Manuel Bronstein's "Poor Man's Integrator".
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The algorithm supports various classes of functions including
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transcendental elementary or special functions like Airy,
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Bessel, Whittaker and Lambert.
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Note that this algorithm is not a decision procedure. If it isn't
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able to compute the antiderivative for a given function, then this is
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not a proof that such a functions does not exist. One should use
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recursive Risch algorithm in such case. It's an open question if
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this algorithm can be made a full decision procedure.
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This is an internal integrator procedure. You should use toplevel
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'integrate' function in most cases, as this procedure needs some
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preprocessing steps and otherwise may fail.
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Specification
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=============
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heurisch(f, x, rewrite=False, hints=None)
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where
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f : expression
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x : symbol
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rewrite -> force rewrite 'f' in terms of 'tan' and 'tanh'
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hints -> a list of functions that may appear in anti-derivate
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- hints = None --> no suggestions at all
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- hints = [ ] --> try to figure out
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- hints = [f1, ..., fn] --> we know better
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Examples
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========
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>>> from sympy import tan
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>>> from sympy.integrals.heurisch import heurisch
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>>> from sympy.abc import x, y
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>>> heurisch(y*tan(x), x)
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y*log(tan(x)**2 + 1)/2
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See Manuel Bronstein's "Poor Man's Integrator":
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References
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==========
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.. [1] http://www-sop.inria.fr/cafe/Manuel.Bronstein/pmint/index.html
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For more information on the implemented algorithm refer to:
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.. [2] K. Geddes, L. Stefanus, On the Risch-Norman Integration
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Method and its Implementation in Maple, Proceedings of
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ISSAC'89, ACM Press, 212-217.
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.. [3] J. H. Davenport, On the Parallel Risch Algorithm (I),
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Proceedings of EUROCAM'82, LNCS 144, Springer, 144-157.
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.. [4] J. H. Davenport, On the Parallel Risch Algorithm (III):
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Use of Tangents, SIGSAM Bulletin 16 (1982), 3-6.
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.. [5] J. H. Davenport, B. M. Trager, On the Parallel Risch
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Algorithm (II), ACM Transactions on Mathematical
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Software 11 (1985), 356-362.
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See Also
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========
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sympy.integrals.integrals.Integral.doit
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sympy.integrals.integrals.Integral
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sympy.integrals.heurisch.components
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"""
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f = sympify(f)
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# There are some functions that Heurisch cannot currently handle,
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# so do not even try.
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# Set _try_heurisch=True to skip this check
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if _try_heurisch is not True:
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if f.has(Abs, re, im, sign, Heaviside, DiracDelta, floor, ceiling, arg):
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return
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if x not in f.free_symbols:
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return f*x
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if not f.is_Add:
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indep, f = f.as_independent(x)
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else:
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indep = S.One
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rewritables = {
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(sin, cos, cot): tan,
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(sinh, cosh, coth): tanh,
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}
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if rewrite:
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for candidates, rule in rewritables.items():
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f = f.rewrite(candidates, rule)
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else:
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for candidates in rewritables.keys():
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if f.has(*candidates):
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break
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else:
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rewrite = True
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terms = components(f, x)
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if hints is not None:
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if not hints:
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a = Wild('a', exclude=[x])
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b = Wild('b', exclude=[x])
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c = Wild('c', exclude=[x])
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for g in set(terms): # using copy of terms
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if g.is_Function:
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if isinstance(g, li):
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M = g.args[0].match(a*x**b)
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if M is not None:
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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])) )
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#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])) )
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#terms.add( x*(li(M[a]*x**M[b]) - x*Ei((M[b]+1)*log(M[a]*x**M[b])/M[b])) )
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#terms.add( li(M[a]*x**M[b]) - Ei((M[b]+1)*log(M[a]*x**M[b])/M[b]) )
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elif isinstance(g, exp):
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M = g.args[0].match(a*x**2)
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if M is not None:
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if M[a].is_positive:
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terms.add(erfi(sqrt(M[a])*x))
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else: # M[a].is_negative or unknown
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terms.add(erf(sqrt(-M[a])*x))
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M = g.args[0].match(a*x**2 + b*x + c)
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if M is not None:
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if M[a].is_positive:
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terms.add(sqrt(pi/4*(-M[a]))*exp(M[c] - M[b]**2/(4*M[a]))*
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erfi(sqrt(M[a])*x + M[b]/(2*sqrt(M[a]))))
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elif M[a].is_negative:
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terms.add(sqrt(pi/4*(-M[a]))*exp(M[c] - M[b]**2/(4*M[a]))*
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erf(sqrt(-M[a])*x - M[b]/(2*sqrt(-M[a]))))
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M = g.args[0].match(a*log(x)**2)
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if M is not None:
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if M[a].is_positive:
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terms.add(erfi(sqrt(M[a])*log(x) + 1/(2*sqrt(M[a]))))
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if M[a].is_negative:
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terms.add(erf(sqrt(-M[a])*log(x) - 1/(2*sqrt(-M[a]))))
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elif g.is_Pow:
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if g.exp.is_Rational and g.exp.q == 2:
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M = g.base.match(a*x**2 + b)
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if M is not None and M[b].is_positive:
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if M[a].is_positive:
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terms.add(asinh(sqrt(M[a]/M[b])*x))
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elif M[a].is_negative:
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terms.add(asin(sqrt(-M[a]/M[b])*x))
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M = g.base.match(a*x**2 - b)
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if M is not None and M[b].is_positive:
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if M[a].is_positive:
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terms.add(acosh(sqrt(M[a]/M[b])*x))
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elif M[a].is_negative:
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terms.add(-M[b]/2*sqrt(-M[a])*
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atan(sqrt(-M[a])*x/sqrt(M[a]*x**2 - M[b])))
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else:
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terms |= set(hints)
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dcache = DiffCache(x)
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for g in set(terms): # using copy of terms
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terms |= components(dcache.get_diff(g), x)
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# TODO: caching is significant factor for why permutations work at all. Change this.
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V = _symbols('x', len(terms))
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# sort mapping expressions from largest to smallest (last is always x).
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mapping = list(reversed(list(zip(*ordered( #
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[(a[0].as_independent(x)[1], a) for a in zip(terms, V)])))[1])) #
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rev_mapping = {v: k for k, v in mapping} #
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if mappings is None: #
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# optimizing the number of permutations of mapping #
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assert mapping[-1][0] == x # if not, find it and correct this comment
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unnecessary_permutations = [mapping.pop(-1)]
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mappings = permutations(mapping)
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else:
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unnecessary_permutations = unnecessary_permutations or []
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def _substitute(expr):
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return expr.subs(mapping)
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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 polys:
|
|
if poly.has(*V):
|
|
try:
|
|
factorization = factor(poly, greedy=True)
|
|
except PolynomialError:
|
|
factorization = poly
|
|
|
|
if factorization.is_Mul:
|
|
factors = factorization.args
|
|
else:
|
|
factors = (factorization, )
|
|
|
|
for fact in factors:
|
|
if fact.is_Pow:
|
|
reducibles.add(fact.base)
|
|
else:
|
|
reducibles.add(fact)
|
|
|
|
def _integrate(field=None):
|
|
irreducibles = set()
|
|
atans = set()
|
|
pairs = set()
|
|
|
|
for poly in reducibles:
|
|
for z in poly.free_symbols:
|
|
if z in V:
|
|
break # should this be: `irreducibles |= \
|
|
else: # set(root_factors(poly, z, filter=field))`
|
|
continue # and the line below deleted?
|
|
# |
|
|
# V
|
|
irreducibles |= set(root_factors(poly, z, filter=field))
|
|
|
|
log_part, atan_part = [], []
|
|
|
|
for poly in list(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.free_symbols & 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 not (F.free_symbols - set(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
|