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ai-content-maker/.venv/Lib/site-packages/sympy/integrals/integrals.py

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from typing import Tuple as tTuple
from sympy.concrete.expr_with_limits import AddWithLimits
from sympy.core.add import Add
from sympy.core.basic import Basic
from sympy.core.containers import Tuple
from sympy.core.expr import Expr
from sympy.core.exprtools import factor_terms
from sympy.core.function import diff
from sympy.core.logic import fuzzy_bool
from sympy.core.mul import Mul
from sympy.core.numbers import oo, pi
from sympy.core.relational import Ne
from sympy.core.singleton import S
from sympy.core.symbol import (Dummy, Symbol, Wild)
from sympy.core.sympify import sympify
from sympy.functions import Piecewise, sqrt, piecewise_fold, tan, cot, atan
from sympy.functions.elementary.exponential import log
from sympy.functions.elementary.integers import floor
from sympy.functions.elementary.complexes import Abs, sign
from sympy.functions.elementary.miscellaneous import Min, Max
from .rationaltools import ratint
from sympy.matrices import MatrixBase
from sympy.polys import Poly, PolynomialError
from sympy.series.formal import FormalPowerSeries
from sympy.series.limits import limit
from sympy.series.order import Order
from sympy.tensor.functions import shape
from sympy.utilities.exceptions import sympy_deprecation_warning
from sympy.utilities.iterables import is_sequence
from sympy.utilities.misc import filldedent
class Integral(AddWithLimits):
"""Represents unevaluated integral."""
__slots__ = ()
args: tTuple[Expr, Tuple]
def __new__(cls, function, *symbols, **assumptions):
"""Create an unevaluated integral.
Explanation
===========
Arguments are an integrand followed by one or more limits.
If no limits are given and there is only one free symbol in the
expression, that symbol will be used, otherwise an error will be
raised.
>>> from sympy import Integral
>>> from sympy.abc import x, y
>>> Integral(x)
Integral(x, x)
>>> Integral(y)
Integral(y, y)
When limits are provided, they are interpreted as follows (using
``x`` as though it were the variable of integration):
(x,) or x - indefinite integral
(x, a) - "evaluate at" integral is an abstract antiderivative
(x, a, b) - definite integral
The ``as_dummy`` method can be used to see which symbols cannot be
targeted by subs: those with a prepended underscore cannot be
changed with ``subs``. (Also, the integration variables themselves --
the first element of a limit -- can never be changed by subs.)
>>> i = Integral(x, x)
>>> at = Integral(x, (x, x))
>>> i.as_dummy()
Integral(x, x)
>>> at.as_dummy()
Integral(_0, (_0, x))
"""
#This will help other classes define their own definitions
#of behaviour with Integral.
if hasattr(function, '_eval_Integral'):
return function._eval_Integral(*symbols, **assumptions)
if isinstance(function, Poly):
sympy_deprecation_warning(
"""
integrate(Poly) and Integral(Poly) are deprecated. Instead,
use the Poly.integrate() method, or convert the Poly to an
Expr first with the Poly.as_expr() method.
""",
deprecated_since_version="1.6",
active_deprecations_target="deprecated-integrate-poly")
obj = AddWithLimits.__new__(cls, function, *symbols, **assumptions)
return obj
def __getnewargs__(self):
return (self.function,) + tuple([tuple(xab) for xab in self.limits])
@property
def free_symbols(self):
"""
This method returns the symbols that will exist when the
integral is evaluated. This is useful if one is trying to
determine whether an integral depends on a certain
symbol or not.
Examples
========
>>> from sympy import Integral
>>> from sympy.abc import x, y
>>> Integral(x, (x, y, 1)).free_symbols
{y}
See Also
========
sympy.concrete.expr_with_limits.ExprWithLimits.function
sympy.concrete.expr_with_limits.ExprWithLimits.limits
sympy.concrete.expr_with_limits.ExprWithLimits.variables
"""
return super().free_symbols
def _eval_is_zero(self):
# This is a very naive and quick test, not intended to do the integral to
# answer whether it is zero or not, e.g. Integral(sin(x), (x, 0, 2*pi))
# is zero but this routine should return None for that case. But, like
# Mul, there are trivial situations for which the integral will be
# zero so we check for those.
if self.function.is_zero:
return True
got_none = False
for l in self.limits:
if len(l) == 3:
z = (l[1] == l[2]) or (l[1] - l[2]).is_zero
if z:
return True
elif z is None:
got_none = True
free = self.function.free_symbols
for xab in self.limits:
if len(xab) == 1:
free.add(xab[0])
continue
if len(xab) == 2 and xab[0] not in free:
if xab[1].is_zero:
return True
elif xab[1].is_zero is None:
got_none = True
# take integration symbol out of free since it will be replaced
# with the free symbols in the limits
free.discard(xab[0])
# add in the new symbols
for i in xab[1:]:
free.update(i.free_symbols)
if self.function.is_zero is False and got_none is False:
return False
def transform(self, x, u):
r"""
Performs a change of variables from `x` to `u` using the relationship
given by `x` and `u` which will define the transformations `f` and `F`
(which are inverses of each other) as follows:
1) If `x` is a Symbol (which is a variable of integration) then `u`
will be interpreted as some function, f(u), with inverse F(u).
This, in effect, just makes the substitution of x with f(x).
2) If `u` is a Symbol then `x` will be interpreted as some function,
F(x), with inverse f(u). This is commonly referred to as
u-substitution.
Once f and F have been identified, the transformation is made as
follows:
.. math:: \int_a^b x \mathrm{d}x \rightarrow \int_{F(a)}^{F(b)} f(x)
\frac{\mathrm{d}}{\mathrm{d}x}
where `F(x)` is the inverse of `f(x)` and the limits and integrand have
been corrected so as to retain the same value after integration.
Notes
=====
The mappings, F(x) or f(u), must lead to a unique integral. Linear
or rational linear expression, ``2*x``, ``1/x`` and ``sqrt(x)``, will
always work; quadratic expressions like ``x**2 - 1`` are acceptable
as long as the resulting integrand does not depend on the sign of
the solutions (see examples).
The integral will be returned unchanged if ``x`` is not a variable of
integration.
``x`` must be (or contain) only one of of the integration variables. If
``u`` has more than one free symbol then it should be sent as a tuple
(``u``, ``uvar``) where ``uvar`` identifies which variable is replacing
the integration variable.
XXX can it contain another integration variable?
Examples
========
>>> from sympy.abc import a, x, u
>>> from sympy import Integral, cos, sqrt
>>> i = Integral(x*cos(x**2 - 1), (x, 0, 1))
transform can change the variable of integration
>>> i.transform(x, u)
Integral(u*cos(u**2 - 1), (u, 0, 1))
transform can perform u-substitution as long as a unique
integrand is obtained:
>>> i.transform(x**2 - 1, u)
Integral(cos(u)/2, (u, -1, 0))
This attempt fails because x = +/-sqrt(u + 1) and the
sign does not cancel out of the integrand:
>>> Integral(cos(x**2 - 1), (x, 0, 1)).transform(x**2 - 1, u)
Traceback (most recent call last):
...
ValueError:
The mapping between F(x) and f(u) did not give a unique integrand.
transform can do a substitution. Here, the previous
result is transformed back into the original expression
using "u-substitution":
>>> ui = _
>>> _.transform(sqrt(u + 1), x) == i
True
We can accomplish the same with a regular substitution:
>>> ui.transform(u, x**2 - 1) == i
True
If the `x` does not contain a symbol of integration then
the integral will be returned unchanged. Integral `i` does
not have an integration variable `a` so no change is made:
>>> i.transform(a, x) == i
True
When `u` has more than one free symbol the symbol that is
replacing `x` must be identified by passing `u` as a tuple:
>>> Integral(x, (x, 0, 1)).transform(x, (u + a, u))
Integral(a + u, (u, -a, 1 - a))
>>> Integral(x, (x, 0, 1)).transform(x, (u + a, a))
Integral(a + u, (a, -u, 1 - u))
See Also
========
sympy.concrete.expr_with_limits.ExprWithLimits.variables : Lists the integration variables
as_dummy : Replace integration variables with dummy ones
"""
d = Dummy('d')
xfree = x.free_symbols.intersection(self.variables)
if len(xfree) > 1:
raise ValueError(
'F(x) can only contain one of: %s' % self.variables)
xvar = xfree.pop() if xfree else d
if xvar not in self.variables:
return self
u = sympify(u)
if isinstance(u, Expr):
ufree = u.free_symbols
if len(ufree) == 0:
raise ValueError(filldedent('''
f(u) cannot be a constant'''))
if len(ufree) > 1:
raise ValueError(filldedent('''
When f(u) has more than one free symbol, the one replacing x
must be identified: pass f(u) as (f(u), u)'''))
uvar = ufree.pop()
else:
u, uvar = u
if uvar not in u.free_symbols:
raise ValueError(filldedent('''
Expecting a tuple (expr, symbol) where symbol identified
a free symbol in expr, but symbol is not in expr's free
symbols.'''))
if not isinstance(uvar, Symbol):
# This probably never evaluates to True
raise ValueError(filldedent('''
Expecting a tuple (expr, symbol) but didn't get
a symbol; got %s''' % uvar))
if x.is_Symbol and u.is_Symbol:
return self.xreplace({x: u})
if not x.is_Symbol and not u.is_Symbol:
raise ValueError('either x or u must be a symbol')
if uvar == xvar:
return self.transform(x, (u.subs(uvar, d), d)).xreplace({d: uvar})
if uvar in self.limits:
raise ValueError(filldedent('''
u must contain the same variable as in x
or a variable that is not already an integration variable'''))
from sympy.solvers.solvers import solve
if not x.is_Symbol:
F = [x.subs(xvar, d)]
soln = solve(u - x, xvar, check=False)
if not soln:
raise ValueError('no solution for solve(F(x) - f(u), x)')
f = [fi.subs(uvar, d) for fi in soln]
else:
f = [u.subs(uvar, d)]
from sympy.simplify.simplify import posify
pdiff, reps = posify(u - x)
puvar = uvar.subs([(v, k) for k, v in reps.items()])
soln = [s.subs(reps) for s in solve(pdiff, puvar)]
if not soln:
raise ValueError('no solution for solve(F(x) - f(u), u)')
F = [fi.subs(xvar, d) for fi in soln]
newfuncs = {(self.function.subs(xvar, fi)*fi.diff(d)
).subs(d, uvar) for fi in f}
if len(newfuncs) > 1:
raise ValueError(filldedent('''
The mapping between F(x) and f(u) did not give
a unique integrand.'''))
newfunc = newfuncs.pop()
def _calc_limit_1(F, a, b):
"""
replace d with a, using subs if possible, otherwise limit
where sign of b is considered
"""
wok = F.subs(d, a)
if wok is S.NaN or wok.is_finite is False and a.is_finite:
return limit(sign(b)*F, d, a)
return wok
def _calc_limit(a, b):
"""
replace d with a, using subs if possible, otherwise limit
where sign of b is considered
"""
avals = list({_calc_limit_1(Fi, a, b) for Fi in F})
if len(avals) > 1:
raise ValueError(filldedent('''
The mapping between F(x) and f(u) did not
give a unique limit.'''))
return avals[0]
newlimits = []
for xab in self.limits:
sym = xab[0]
if sym == xvar:
if len(xab) == 3:
a, b = xab[1:]
a, b = _calc_limit(a, b), _calc_limit(b, a)
if fuzzy_bool(a - b > 0):
a, b = b, a
newfunc = -newfunc
newlimits.append((uvar, a, b))
elif len(xab) == 2:
a = _calc_limit(xab[1], 1)
newlimits.append((uvar, a))
else:
newlimits.append(uvar)
else:
newlimits.append(xab)
return self.func(newfunc, *newlimits)
def doit(self, **hints):
"""
Perform the integration using any hints given.
Examples
========
>>> from sympy import Piecewise, S
>>> from sympy.abc import x, t
>>> p = x**2 + Piecewise((0, x/t < 0), (1, True))
>>> p.integrate((t, S(4)/5, 1), (x, -1, 1))
1/3
See Also
========
sympy.integrals.trigonometry.trigintegrate
sympy.integrals.heurisch.heurisch
sympy.integrals.rationaltools.ratint
as_sum : Approximate the integral using a sum
"""
if not hints.get('integrals', True):
return self
deep = hints.get('deep', True)
meijerg = hints.get('meijerg', None)
conds = hints.get('conds', 'piecewise')
risch = hints.get('risch', None)
heurisch = hints.get('heurisch', None)
manual = hints.get('manual', None)
if len(list(filter(None, (manual, meijerg, risch, heurisch)))) > 1:
raise ValueError("At most one of manual, meijerg, risch, heurisch can be True")
elif manual:
meijerg = risch = heurisch = False
elif meijerg:
manual = risch = heurisch = False
elif risch:
manual = meijerg = heurisch = False
elif heurisch:
manual = meijerg = risch = False
eval_kwargs = {"meijerg": meijerg, "risch": risch, "manual": manual, "heurisch": heurisch,
"conds": conds}
if conds not in ('separate', 'piecewise', 'none'):
raise ValueError('conds must be one of "separate", "piecewise", '
'"none", got: %s' % conds)
if risch and any(len(xab) > 1 for xab in self.limits):
raise ValueError('risch=True is only allowed for indefinite integrals.')
# check for the trivial zero
if self.is_zero:
return S.Zero
# hacks to handle integrals of
# nested summations
from sympy.concrete.summations import Sum
if isinstance(self.function, Sum):
if any(v in self.function.limits[0] for v in self.variables):
raise ValueError('Limit of the sum cannot be an integration variable.')
if any(l.is_infinite for l in self.function.limits[0][1:]):
return self
_i = self
_sum = self.function
return _sum.func(_i.func(_sum.function, *_i.limits).doit(), *_sum.limits).doit()
# now compute and check the function
function = self.function
if deep:
function = function.doit(**hints)
if function.is_zero:
return S.Zero
# hacks to handle special cases
if isinstance(function, MatrixBase):
return function.applyfunc(
lambda f: self.func(f, *self.limits).doit(**hints))
if isinstance(function, FormalPowerSeries):
if len(self.limits) > 1:
raise NotImplementedError
xab = self.limits[0]
if len(xab) > 1:
return function.integrate(xab, **eval_kwargs)
else:
return function.integrate(xab[0], **eval_kwargs)
# There is no trivial answer and special handling
# is done so continue
# first make sure any definite limits have integration
# variables with matching assumptions
reps = {}
for xab in self.limits:
if len(xab) != 3:
# it makes sense to just make
# all x real but in practice with the
# current state of integration...this
# doesn't work out well
# x = xab[0]
# if x not in reps and not x.is_real:
# reps[x] = Dummy(real=True)
continue
x, a, b = xab
l = (a, b)
if all(i.is_nonnegative for i in l) and not x.is_nonnegative:
d = Dummy(positive=True)
elif all(i.is_nonpositive for i in l) and not x.is_nonpositive:
d = Dummy(negative=True)
elif all(i.is_real for i in l) and not x.is_real:
d = Dummy(real=True)
else:
d = None
if d:
reps[x] = d
if reps:
undo = {v: k for k, v in reps.items()}
did = self.xreplace(reps).doit(**hints)
if isinstance(did, tuple): # when separate=True
did = tuple([i.xreplace(undo) for i in did])
else:
did = did.xreplace(undo)
return did
# continue with existing assumptions
undone_limits = []
# ulj = free symbols of any undone limits' upper and lower limits
ulj = set()
for xab in self.limits:
# compute uli, the free symbols in the
# Upper and Lower limits of limit I
if len(xab) == 1:
uli = set(xab[:1])
elif len(xab) == 2:
uli = xab[1].free_symbols
elif len(xab) == 3:
uli = xab[1].free_symbols.union(xab[2].free_symbols)
# this integral can be done as long as there is no blocking
# limit that has been undone. An undone limit is blocking if
# it contains an integration variable that is in this limit's
# upper or lower free symbols or vice versa
if xab[0] in ulj or any(v[0] in uli for v in undone_limits):
undone_limits.append(xab)
ulj.update(uli)
function = self.func(*([function] + [xab]))
factored_function = function.factor()
if not isinstance(factored_function, Integral):
function = factored_function
continue
if function.has(Abs, sign) and (
(len(xab) < 3 and all(x.is_extended_real for x in xab)) or
(len(xab) == 3 and all(x.is_extended_real and not x.is_infinite for
x in xab[1:]))):
# some improper integrals are better off with Abs
xr = Dummy("xr", real=True)
function = (function.xreplace({xab[0]: xr})
.rewrite(Piecewise).xreplace({xr: xab[0]}))
elif function.has(Min, Max):
function = function.rewrite(Piecewise)
if (function.has(Piecewise) and
not isinstance(function, Piecewise)):
function = piecewise_fold(function)
if isinstance(function, Piecewise):
if len(xab) == 1:
antideriv = function._eval_integral(xab[0],
**eval_kwargs)
else:
antideriv = self._eval_integral(
function, xab[0], **eval_kwargs)
else:
# There are a number of tradeoffs in using the
# Meijer G method. It can sometimes be a lot faster
# than other methods, and sometimes slower. And
# there are certain types of integrals for which it
# is more likely to work than others. These
# heuristics are incorporated in deciding what
# integration methods to try, in what order. See the
# integrate() docstring for details.
def try_meijerg(function, xab):
ret = None
if len(xab) == 3 and meijerg is not False:
x, a, b = xab
try:
res = meijerint_definite(function, x, a, b)
except NotImplementedError:
_debug('NotImplementedError '
'from meijerint_definite')
res = None
if res is not None:
f, cond = res
if conds == 'piecewise':
u = self.func(function, (x, a, b))
# if Piecewise modifies cond too
# much it may not be recognized by
# _condsimp pattern matching so just
# turn off all evaluation
return Piecewise((f, cond), (u, True),
evaluate=False)
elif conds == 'separate':
if len(self.limits) != 1:
raise ValueError(filldedent('''
conds=separate not supported in
multiple integrals'''))
ret = f, cond
else:
ret = f
return ret
meijerg1 = meijerg
if (meijerg is not False and
len(xab) == 3 and xab[1].is_extended_real and xab[2].is_extended_real
and not function.is_Poly and
(xab[1].has(oo, -oo) or xab[2].has(oo, -oo))):
ret = try_meijerg(function, xab)
if ret is not None:
function = ret
continue
meijerg1 = False
# If the special meijerg code did not succeed in
# finding a definite integral, then the code using
# meijerint_indefinite will not either (it might
# find an antiderivative, but the answer is likely
# to be nonsensical). Thus if we are requested to
# only use Meijer G-function methods, we give up at
# this stage. Otherwise we just disable G-function
# methods.
if meijerg1 is False and meijerg is True:
antideriv = None
else:
antideriv = self._eval_integral(
function, xab[0], **eval_kwargs)
if antideriv is None and meijerg is True:
ret = try_meijerg(function, xab)
if ret is not None:
function = ret
continue
final = hints.get('final', True)
# dotit may be iterated but floor terms making atan and acot
# continuous should only be added in the final round
if (final and not isinstance(antideriv, Integral) and
antideriv is not None):
for atan_term in antideriv.atoms(atan):
atan_arg = atan_term.args[0]
# Checking `atan_arg` to be linear combination of `tan` or `cot`
for tan_part in atan_arg.atoms(tan):
x1 = Dummy('x1')
tan_exp1 = atan_arg.subs(tan_part, x1)
# The coefficient of `tan` should be constant
coeff = tan_exp1.diff(x1)
if x1 not in coeff.free_symbols:
a = tan_part.args[0]
antideriv = antideriv.subs(atan_term, Add(atan_term,
sign(coeff)*pi*floor((a-pi/2)/pi)))
for cot_part in atan_arg.atoms(cot):
x1 = Dummy('x1')
cot_exp1 = atan_arg.subs(cot_part, x1)
# The coefficient of `cot` should be constant
coeff = cot_exp1.diff(x1)
if x1 not in coeff.free_symbols:
a = cot_part.args[0]
antideriv = antideriv.subs(atan_term, Add(atan_term,
sign(coeff)*pi*floor((a)/pi)))
if antideriv is None:
undone_limits.append(xab)
function = self.func(*([function] + [xab])).factor()
factored_function = function.factor()
if not isinstance(factored_function, Integral):
function = factored_function
continue
else:
if len(xab) == 1:
function = antideriv
else:
if len(xab) == 3:
x, a, b = xab
elif len(xab) == 2:
x, b = xab
a = None
else:
raise NotImplementedError
if deep:
if isinstance(a, Basic):
a = a.doit(**hints)
if isinstance(b, Basic):
b = b.doit(**hints)
if antideriv.is_Poly:
gens = list(antideriv.gens)
gens.remove(x)
antideriv = antideriv.as_expr()
function = antideriv._eval_interval(x, a, b)
function = Poly(function, *gens)
else:
def is_indef_int(g, x):
return (isinstance(g, Integral) and
any(i == (x,) for i in g.limits))
def eval_factored(f, x, a, b):
# _eval_interval for integrals with
# (constant) factors
# a single indefinite integral is assumed
args = []
for g in Mul.make_args(f):
if is_indef_int(g, x):
args.append(g._eval_interval(x, a, b))
else:
args.append(g)
return Mul(*args)
integrals, others, piecewises = [], [], []
for f in Add.make_args(antideriv):
if any(is_indef_int(g, x)
for g in Mul.make_args(f)):
integrals.append(f)
elif any(isinstance(g, Piecewise)
for g in Mul.make_args(f)):
piecewises.append(piecewise_fold(f))
else:
others.append(f)
uneval = Add(*[eval_factored(f, x, a, b)
for f in integrals])
try:
evalued = Add(*others)._eval_interval(x, a, b)
evalued_pw = piecewise_fold(Add(*piecewises))._eval_interval(x, a, b)
function = uneval + evalued + evalued_pw
except NotImplementedError:
# This can happen if _eval_interval depends in a
# complicated way on limits that cannot be computed
undone_limits.append(xab)
function = self.func(*([function] + [xab]))
factored_function = function.factor()
if not isinstance(factored_function, Integral):
function = factored_function
return function
def _eval_derivative(self, sym):
"""Evaluate the derivative of the current Integral object by
differentiating under the integral sign [1], using the Fundamental
Theorem of Calculus [2] when possible.
Explanation
===========
Whenever an Integral is encountered that is equivalent to zero or
has an integrand that is independent of the variable of integration
those integrals are performed. All others are returned as Integral
instances which can be resolved with doit() (provided they are integrable).
References
==========
.. [1] https://en.wikipedia.org/wiki/Differentiation_under_the_integral_sign
.. [2] https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus
Examples
========
>>> from sympy import Integral
>>> from sympy.abc import x, y
>>> i = Integral(x + y, y, (y, 1, x))
>>> i.diff(x)
Integral(x + y, (y, x)) + Integral(1, y, (y, 1, x))
>>> i.doit().diff(x) == i.diff(x).doit()
True
>>> i.diff(y)
0
The previous must be true since there is no y in the evaluated integral:
>>> i.free_symbols
{x}
>>> i.doit()
2*x**3/3 - x/2 - 1/6
"""
# differentiate under the integral sign; we do not
# check for regularity conditions (TODO), see issue 4215
# get limits and the function
f, limits = self.function, list(self.limits)
# the order matters if variables of integration appear in the limits
# so work our way in from the outside to the inside.
limit = limits.pop(-1)
if len(limit) == 3:
x, a, b = limit
elif len(limit) == 2:
x, b = limit
a = None
else:
a = b = None
x = limit[0]
if limits: # f is the argument to an integral
f = self.func(f, *tuple(limits))
# assemble the pieces
def _do(f, ab):
dab_dsym = diff(ab, sym)
if not dab_dsym:
return S.Zero
if isinstance(f, Integral):
limits = [(x, x) if (len(l) == 1 and l[0] == x) else l
for l in f.limits]
f = self.func(f.function, *limits)
return f.subs(x, ab)*dab_dsym
rv = S.Zero
if b is not None:
rv += _do(f, b)
if a is not None:
rv -= _do(f, a)
if len(limit) == 1 and sym == x:
# the dummy variable *is* also the real-world variable
arg = f
rv += arg
else:
# the dummy variable might match sym but it's
# only a dummy and the actual variable is determined
# by the limits, so mask off the variable of integration
# while differentiating
u = Dummy('u')
arg = f.subs(x, u).diff(sym).subs(u, x)
if arg:
rv += self.func(arg, (x, a, b))
return rv
def _eval_integral(self, f, x, meijerg=None, risch=None, manual=None,
heurisch=None, conds='piecewise',final=None):
"""
Calculate the anti-derivative to the function f(x).
Explanation
===========
The following algorithms are applied (roughly in this order):
1. Simple heuristics (based on pattern matching and integral table):
- most frequently used functions (e.g. polynomials, products of
trig functions)
2. Integration of rational functions:
- A complete algorithm for integrating rational functions is
implemented (the Lazard-Rioboo-Trager algorithm). The algorithm
also uses the partial fraction decomposition algorithm
implemented in apart() as a preprocessor to make this process
faster. Note that the integral of a rational function is always
elementary, but in general, it may include a RootSum.
3. Full Risch algorithm:
- The Risch algorithm is a complete decision
procedure for integrating elementary functions, which means that
given any elementary function, it will either compute an
elementary antiderivative, or else prove that none exists.
Currently, part of transcendental case is implemented, meaning
elementary integrals containing exponentials, logarithms, and
(soon!) trigonometric functions can be computed. The algebraic
case, e.g., functions containing roots, is much more difficult
and is not implemented yet.
- If the routine fails (because the integrand is not elementary, or
because a case is not implemented yet), it continues on to the
next algorithms below. If the routine proves that the integrals
is nonelementary, it still moves on to the algorithms below,
because we might be able to find a closed-form solution in terms
of special functions. If risch=True, however, it will stop here.
4. The Meijer G-Function algorithm:
- This algorithm works by first rewriting the integrand in terms of
very general Meijer G-Function (meijerg in SymPy), integrating
it, and then rewriting the result back, if possible. This
algorithm is particularly powerful for definite integrals (which
is actually part of a different method of Integral), since it can
compute closed-form solutions of definite integrals even when no
closed-form indefinite integral exists. But it also is capable
of computing many indefinite integrals as well.
- Another advantage of this method is that it can use some results
about the Meijer G-Function to give a result in terms of a
Piecewise expression, which allows to express conditionally
convergent integrals.
- Setting meijerg=True will cause integrate() to use only this
method.
5. The "manual integration" algorithm:
- This algorithm tries to mimic how a person would find an
antiderivative by hand, for example by looking for a
substitution or applying integration by parts. This algorithm
does not handle as many integrands but can return results in a
more familiar form.
- Sometimes this algorithm can evaluate parts of an integral; in
this case integrate() will try to evaluate the rest of the
integrand using the other methods here.
- Setting manual=True will cause integrate() to use only this
method.
6. The Heuristic Risch algorithm:
- This is a heuristic version of the Risch algorithm, meaning that
it is not deterministic. This is tried as a last resort because
it can be very slow. It is still used because not enough of the
full Risch algorithm is implemented, so that there are still some
integrals that can only be computed using this method. The goal
is to implement enough of the Risch and Meijer G-function methods
so that this can be deleted.
Setting heurisch=True will cause integrate() to use only this
method. Set heurisch=False to not use it.
"""
from sympy.integrals.risch import risch_integrate, NonElementaryIntegral
from sympy.integrals.manualintegrate import manualintegrate
if risch:
try:
return risch_integrate(f, x, conds=conds)
except NotImplementedError:
return None
if manual:
try:
result = manualintegrate(f, x)
if result is not None and result.func != Integral:
return result
except (ValueError, PolynomialError):
pass
eval_kwargs = {"meijerg": meijerg, "risch": risch, "manual": manual,
"heurisch": heurisch, "conds": conds}
# if it is a poly(x) then let the polynomial integrate itself (fast)
#
# It is important to make this check first, otherwise the other code
# will return a SymPy expression instead of a Polynomial.
#
# see Polynomial for details.
if isinstance(f, Poly) and not (manual or meijerg or risch):
# Note: this is deprecated, but the deprecation warning is already
# issued in the Integral constructor.
return f.integrate(x)
# Piecewise antiderivatives need to call special integrate.
if isinstance(f, Piecewise):
return f.piecewise_integrate(x, **eval_kwargs)
# let's cut it short if `f` does not depend on `x`; if
# x is only a dummy, that will be handled below
if not f.has(x):
return f*x
# try to convert to poly(x) and then integrate if successful (fast)
poly = f.as_poly(x)
if poly is not None and not (manual or meijerg or risch):
return poly.integrate().as_expr()
if risch is not False:
try:
result, i = risch_integrate(f, x, separate_integral=True,
conds=conds)
except NotImplementedError:
pass
else:
if i:
# There was a nonelementary integral. Try integrating it.
# if no part of the NonElementaryIntegral is integrated by
# the Risch algorithm, then use the original function to
# integrate, instead of re-written one
if result == 0:
return NonElementaryIntegral(f, x).doit(risch=False)
else:
return result + i.doit(risch=False)
else:
return result
# since Integral(f=g1+g2+...) == Integral(g1) + Integral(g2) + ...
# we are going to handle Add terms separately,
# if `f` is not Add -- we only have one term
# Note that in general, this is a bad idea, because Integral(g1) +
# Integral(g2) might not be computable, even if Integral(g1 + g2) is.
# For example, Integral(x**x + x**x*log(x)). But many heuristics only
# work term-wise. So we compute this step last, after trying
# risch_integrate. We also try risch_integrate again in this loop,
# because maybe the integral is a sum of an elementary part and a
# nonelementary part (like erf(x) + exp(x)). risch_integrate() is
# quite fast, so this is acceptable.
from sympy.simplify.fu import sincos_to_sum
parts = []
args = Add.make_args(f)
for g in args:
coeff, g = g.as_independent(x)
# g(x) = const
if g is S.One and not meijerg:
parts.append(coeff*x)
continue
# g(x) = expr + O(x**n)
order_term = g.getO()
if order_term is not None:
h = self._eval_integral(g.removeO(), x, **eval_kwargs)
if h is not None:
h_order_expr = self._eval_integral(order_term.expr, x, **eval_kwargs)
if h_order_expr is not None:
h_order_term = order_term.func(
h_order_expr, *order_term.variables)
parts.append(coeff*(h + h_order_term))
continue
# NOTE: if there is O(x**n) and we fail to integrate then
# there is no point in trying other methods because they
# will fail, too.
return None
# c
# g(x) = (a*x+b)
if g.is_Pow and not g.exp.has(x) and not meijerg:
a = Wild('a', exclude=[x])
b = Wild('b', exclude=[x])
M = g.base.match(a*x + b)
if M is not None:
if g.exp == -1:
h = log(g.base)
elif conds != 'piecewise':
h = g.base**(g.exp + 1) / (g.exp + 1)
else:
h1 = log(g.base)
h2 = g.base**(g.exp + 1) / (g.exp + 1)
h = Piecewise((h2, Ne(g.exp, -1)), (h1, True))
parts.append(coeff * h / M[a])
continue
# poly(x)
# g(x) = -------
# poly(x)
if g.is_rational_function(x) and not (manual or meijerg or risch):
parts.append(coeff * ratint(g, x))
continue
if not (manual or meijerg or risch):
# g(x) = Mul(trig)
h = trigintegrate(g, x, conds=conds)
if h is not None:
parts.append(coeff * h)
continue
# g(x) has at least a DiracDelta term
h = deltaintegrate(g, x)
if h is not None:
parts.append(coeff * h)
continue
from .singularityfunctions import singularityintegrate
# g(x) has at least a Singularity Function term
h = singularityintegrate(g, x)
if h is not None:
parts.append(coeff * h)
continue
# Try risch again.
if risch is not False:
try:
h, i = risch_integrate(g, x,
separate_integral=True, conds=conds)
except NotImplementedError:
h = None
else:
if i:
h = h + i.doit(risch=False)
parts.append(coeff*h)
continue
# fall back to heurisch
if heurisch is not False:
from sympy.integrals.heurisch import (heurisch as heurisch_,
heurisch_wrapper)
try:
if conds == 'piecewise':
h = heurisch_wrapper(g, x, hints=[])
else:
h = heurisch_(g, x, hints=[])
except PolynomialError:
# XXX: this exception means there is a bug in the
# implementation of heuristic Risch integration
# algorithm.
h = None
else:
h = None
if meijerg is not False and h is None:
# rewrite using G functions
try:
h = meijerint_indefinite(g, x)
except NotImplementedError:
_debug('NotImplementedError from meijerint_definite')
if h is not None:
parts.append(coeff * h)
continue
if h is None and manual is not False:
try:
result = manualintegrate(g, x)
if result is not None and not isinstance(result, Integral):
if result.has(Integral) and not manual:
# Try to have other algorithms do the integrals
# manualintegrate can't handle,
# unless we were asked to use manual only.
# Keep the rest of eval_kwargs in case another
# method was set to False already
new_eval_kwargs = eval_kwargs
new_eval_kwargs["manual"] = False
new_eval_kwargs["final"] = False
result = result.func(*[
arg.doit(**new_eval_kwargs) if
arg.has(Integral) else arg
for arg in result.args
]).expand(multinomial=False,
log=False,
power_exp=False,
power_base=False)
if not result.has(Integral):
parts.append(coeff * result)
continue
except (ValueError, PolynomialError):
# can't handle some SymPy expressions
pass
# if we failed maybe it was because we had
# a product that could have been expanded,
# so let's try an expansion of the whole
# thing before giving up; we don't try this
# at the outset because there are things
# that cannot be solved unless they are
# NOT expanded e.g., x**x*(1+log(x)). There
# should probably be a checker somewhere in this
# routine to look for such cases and try to do
# collection on the expressions if they are already
# in an expanded form
if not h and len(args) == 1:
f = sincos_to_sum(f).expand(mul=True, deep=False)
if f.is_Add:
# Note: risch will be identical on the expanded
# expression, but maybe it will be able to pick out parts,
# like x*(exp(x) + erf(x)).
return self._eval_integral(f, x, **eval_kwargs)
if h is not None:
parts.append(coeff * h)
else:
return None
return Add(*parts)
def _eval_lseries(self, x, logx=None, cdir=0):
expr = self.as_dummy()
symb = x
for l in expr.limits:
if x in l[1:]:
symb = l[0]
break
for term in expr.function.lseries(symb, logx):
yield integrate(term, *expr.limits)
def _eval_nseries(self, x, n, logx=None, cdir=0):
expr = self.as_dummy()
symb = x
for l in expr.limits:
if x in l[1:]:
symb = l[0]
break
terms, order = expr.function.nseries(
x=symb, n=n, logx=logx).as_coeff_add(Order)
order = [o.subs(symb, x) for o in order]
return integrate(terms, *expr.limits) + Add(*order)*x
def _eval_as_leading_term(self, x, logx=None, cdir=0):
series_gen = self.args[0].lseries(x)
for leading_term in series_gen:
if leading_term != 0:
break
return integrate(leading_term, *self.args[1:])
def _eval_simplify(self, **kwargs):
expr = factor_terms(self)
if isinstance(expr, Integral):
from sympy.simplify.simplify import simplify
return expr.func(*[simplify(i, **kwargs) for i in expr.args])
return expr.simplify(**kwargs)
def as_sum(self, n=None, method="midpoint", evaluate=True):
"""
Approximates a definite integral by a sum.
Parameters
==========
n :
The number of subintervals to use, optional.
method :
One of: 'left', 'right', 'midpoint', 'trapezoid'.
evaluate : bool
If False, returns an unevaluated Sum expression. The default
is True, evaluate the sum.
Notes
=====
These methods of approximate integration are described in [1].
Examples
========
>>> from sympy import Integral, sin, sqrt
>>> from sympy.abc import x, n
>>> e = Integral(sin(x), (x, 3, 7))
>>> e
Integral(sin(x), (x, 3, 7))
For demonstration purposes, this interval will only be split into 2
regions, bounded by [3, 5] and [5, 7].
The left-hand rule uses function evaluations at the left of each
interval:
>>> e.as_sum(2, 'left')
2*sin(5) + 2*sin(3)
The midpoint rule uses evaluations at the center of each interval:
>>> e.as_sum(2, 'midpoint')
2*sin(4) + 2*sin(6)
The right-hand rule uses function evaluations at the right of each
interval:
>>> e.as_sum(2, 'right')
2*sin(5) + 2*sin(7)
The trapezoid rule uses function evaluations on both sides of the
intervals. This is equivalent to taking the average of the left and
right hand rule results:
>>> e.as_sum(2, 'trapezoid')
2*sin(5) + sin(3) + sin(7)
>>> (e.as_sum(2, 'left') + e.as_sum(2, 'right'))/2 == _
True
Here, the discontinuity at x = 0 can be avoided by using the
midpoint or right-hand method:
>>> e = Integral(1/sqrt(x), (x, 0, 1))
>>> e.as_sum(5).n(4)
1.730
>>> e.as_sum(10).n(4)
1.809
>>> e.doit().n(4) # the actual value is 2
2.000
The left- or trapezoid method will encounter the discontinuity and
return infinity:
>>> e.as_sum(5, 'left')
zoo
The number of intervals can be symbolic. If omitted, a dummy symbol
will be used for it.
>>> e = Integral(x**2, (x, 0, 2))
>>> e.as_sum(n, 'right').expand()
8/3 + 4/n + 4/(3*n**2)
This shows that the midpoint rule is more accurate, as its error
term decays as the square of n:
>>> e.as_sum(method='midpoint').expand()
8/3 - 2/(3*_n**2)
A symbolic sum is returned with evaluate=False:
>>> e.as_sum(n, 'midpoint', evaluate=False)
2*Sum((2*_k/n - 1/n)**2, (_k, 1, n))/n
See Also
========
Integral.doit : Perform the integration using any hints
References
==========
.. [1] https://en.wikipedia.org/wiki/Riemann_sum#Riemann_summation_methods
"""
from sympy.concrete.summations import Sum
limits = self.limits
if len(limits) > 1:
raise NotImplementedError(
"Multidimensional midpoint rule not implemented yet")
else:
limit = limits[0]
if (len(limit) != 3 or limit[1].is_finite is False or
limit[2].is_finite is False):
raise ValueError("Expecting a definite integral over "
"a finite interval.")
if n is None:
n = Dummy('n', integer=True, positive=True)
else:
n = sympify(n)
if (n.is_positive is False or n.is_integer is False or
n.is_finite is False):
raise ValueError("n must be a positive integer, got %s" % n)
x, a, b = limit
dx = (b - a)/n
k = Dummy('k', integer=True, positive=True)
f = self.function
if method == "left":
result = dx*Sum(f.subs(x, a + (k-1)*dx), (k, 1, n))
elif method == "right":
result = dx*Sum(f.subs(x, a + k*dx), (k, 1, n))
elif method == "midpoint":
result = dx*Sum(f.subs(x, a + k*dx - dx/2), (k, 1, n))
elif method == "trapezoid":
result = dx*((f.subs(x, a) + f.subs(x, b))/2 +
Sum(f.subs(x, a + k*dx), (k, 1, n - 1)))
else:
raise ValueError("Unknown method %s" % method)
return result.doit() if evaluate else result
def principal_value(self, **kwargs):
"""
Compute the Cauchy Principal Value of the definite integral of a real function in the given interval
on the real axis.
Explanation
===========
In mathematics, the Cauchy principal value, is a method for assigning values to certain improper
integrals which would otherwise be undefined.
Examples
========
>>> from sympy import Integral, oo
>>> from sympy.abc import x
>>> Integral(x+1, (x, -oo, oo)).principal_value()
oo
>>> f = 1 / (x**3)
>>> Integral(f, (x, -oo, oo)).principal_value()
0
>>> Integral(f, (x, -10, 10)).principal_value()
0
>>> Integral(f, (x, -10, oo)).principal_value() + Integral(f, (x, -oo, 10)).principal_value()
0
References
==========
.. [1] https://en.wikipedia.org/wiki/Cauchy_principal_value
.. [2] https://mathworld.wolfram.com/CauchyPrincipalValue.html
"""
if len(self.limits) != 1 or len(list(self.limits[0])) != 3:
raise ValueError("You need to insert a variable, lower_limit, and upper_limit correctly to calculate "
"cauchy's principal value")
x, a, b = self.limits[0]
if not (a.is_comparable and b.is_comparable and a <= b):
raise ValueError("The lower_limit must be smaller than or equal to the upper_limit to calculate "
"cauchy's principal value. Also, a and b need to be comparable.")
if a == b:
return S.Zero
from sympy.calculus.singularities import singularities
r = Dummy('r')
f = self.function
singularities_list = [s for s in singularities(f, x) if s.is_comparable and a <= s <= b]
for i in singularities_list:
if i in (a, b):
raise ValueError(
'The principal value is not defined in the given interval due to singularity at %d.' % (i))
F = integrate(f, x, **kwargs)
if F.has(Integral):
return self
if a is -oo and b is oo:
I = limit(F - F.subs(x, -x), x, oo)
else:
I = limit(F, x, b, '-') - limit(F, x, a, '+')
for s in singularities_list:
I += limit(((F.subs(x, s - r)) - F.subs(x, s + r)), r, 0, '+')
return I
def integrate(*args, meijerg=None, conds='piecewise', risch=None, heurisch=None, manual=None, **kwargs):
"""integrate(f, var, ...)
.. deprecated:: 1.6
Using ``integrate()`` with :class:`~.Poly` is deprecated. Use
:meth:`.Poly.integrate` instead. See :ref:`deprecated-integrate-poly`.
Explanation
===========
Compute definite or indefinite integral of one or more variables
using Risch-Norman algorithm and table lookup. This procedure is
able to handle elementary algebraic and transcendental functions
and also a huge class of special functions, including Airy,
Bessel, Whittaker and Lambert.
var can be:
- a symbol -- indefinite integration
- a tuple (symbol, a) -- indefinite integration with result
given with ``a`` replacing ``symbol``
- a tuple (symbol, a, b) -- definite integration
Several variables can be specified, in which case the result is
multiple integration. (If var is omitted and the integrand is
univariate, the indefinite integral in that variable will be performed.)
Indefinite integrals are returned without terms that are independent
of the integration variables. (see examples)
Definite improper integrals often entail delicate convergence
conditions. Pass conds='piecewise', 'separate' or 'none' to have
these returned, respectively, as a Piecewise function, as a separate
result (i.e. result will be a tuple), or not at all (default is
'piecewise').
**Strategy**
SymPy uses various approaches to definite integration. One method is to
find an antiderivative for the integrand, and then use the fundamental
theorem of calculus. Various functions are implemented to integrate
polynomial, rational and trigonometric functions, and integrands
containing DiracDelta terms.
SymPy also implements the part of the Risch algorithm, which is a decision
procedure for integrating elementary functions, i.e., the algorithm can
either find an elementary antiderivative, or prove that one does not
exist. There is also a (very successful, albeit somewhat slow) general
implementation of the heuristic Risch algorithm. This algorithm will
eventually be phased out as more of the full Risch algorithm is
implemented. See the docstring of Integral._eval_integral() for more
details on computing the antiderivative using algebraic methods.
The option risch=True can be used to use only the (full) Risch algorithm.
This is useful if you want to know if an elementary function has an
elementary antiderivative. If the indefinite Integral returned by this
function is an instance of NonElementaryIntegral, that means that the
Risch algorithm has proven that integral to be non-elementary. Note that
by default, additional methods (such as the Meijer G method outlined
below) are tried on these integrals, as they may be expressible in terms
of special functions, so if you only care about elementary answers, use
risch=True. Also note that an unevaluated Integral returned by this
function is not necessarily a NonElementaryIntegral, even with risch=True,
as it may just be an indication that the particular part of the Risch
algorithm needed to integrate that function is not yet implemented.
Another family of strategies comes from re-writing the integrand in
terms of so-called Meijer G-functions. Indefinite integrals of a
single G-function can always be computed, and the definite integral
of a product of two G-functions can be computed from zero to
infinity. Various strategies are implemented to rewrite integrands
as G-functions, and use this information to compute integrals (see
the ``meijerint`` module).
The option manual=True can be used to use only an algorithm that tries
to mimic integration by hand. This algorithm does not handle as many
integrands as the other algorithms implemented but may return results in
a more familiar form. The ``manualintegrate`` module has functions that
return the steps used (see the module docstring for more information).
In general, the algebraic methods work best for computing
antiderivatives of (possibly complicated) combinations of elementary
functions. The G-function methods work best for computing definite
integrals from zero to infinity of moderately complicated
combinations of special functions, or indefinite integrals of very
simple combinations of special functions.
The strategy employed by the integration code is as follows:
- If computing a definite integral, and both limits are real,
and at least one limit is +- oo, try the G-function method of
definite integration first.
- Try to find an antiderivative, using all available methods, ordered
by performance (that is try fastest method first, slowest last; in
particular polynomial integration is tried first, Meijer
G-functions second to last, and heuristic Risch last).
- If still not successful, try G-functions irrespective of the
limits.
The option meijerg=True, False, None can be used to, respectively:
always use G-function methods and no others, never use G-function
methods, or use all available methods (in order as described above).
It defaults to None.
Examples
========
>>> from sympy import integrate, log, exp, oo
>>> from sympy.abc import a, x, y
>>> integrate(x*y, x)
x**2*y/2
>>> integrate(log(x), x)
x*log(x) - x
>>> integrate(log(x), (x, 1, a))
a*log(a) - a + 1
>>> integrate(x)
x**2/2
Terms that are independent of x are dropped by indefinite integration:
>>> from sympy import sqrt
>>> integrate(sqrt(1 + x), (x, 0, x))
2*(x + 1)**(3/2)/3 - 2/3
>>> integrate(sqrt(1 + x), x)
2*(x + 1)**(3/2)/3
>>> integrate(x*y)
Traceback (most recent call last):
...
ValueError: specify integration variables to integrate x*y
Note that ``integrate(x)`` syntax is meant only for convenience
in interactive sessions and should be avoided in library code.
>>> integrate(x**a*exp(-x), (x, 0, oo)) # same as conds='piecewise'
Piecewise((gamma(a + 1), re(a) > -1),
(Integral(x**a*exp(-x), (x, 0, oo)), True))
>>> integrate(x**a*exp(-x), (x, 0, oo), conds='none')
gamma(a + 1)
>>> integrate(x**a*exp(-x), (x, 0, oo), conds='separate')
(gamma(a + 1), re(a) > -1)
See Also
========
Integral, Integral.doit
"""
doit_flags = {
'deep': False,
'meijerg': meijerg,
'conds': conds,
'risch': risch,
'heurisch': heurisch,
'manual': manual
}
integral = Integral(*args, **kwargs)
if isinstance(integral, Integral):
return integral.doit(**doit_flags)
else:
new_args = [a.doit(**doit_flags) if isinstance(a, Integral) else a
for a in integral.args]
return integral.func(*new_args)
def line_integrate(field, curve, vars):
"""line_integrate(field, Curve, variables)
Compute the line integral.
Examples
========
>>> from sympy import Curve, line_integrate, E, ln
>>> from sympy.abc import x, y, t
>>> C = Curve([E**t + 1, E**t - 1], (t, 0, ln(2)))
>>> line_integrate(x + y, C, [x, y])
3*sqrt(2)
See Also
========
sympy.integrals.integrals.integrate, Integral
"""
from sympy.geometry import Curve
F = sympify(field)
if not F:
raise ValueError(
"Expecting function specifying field as first argument.")
if not isinstance(curve, Curve):
raise ValueError("Expecting Curve entity as second argument.")
if not is_sequence(vars):
raise ValueError("Expecting ordered iterable for variables.")
if len(curve.functions) != len(vars):
raise ValueError("Field variable size does not match curve dimension.")
if curve.parameter in vars:
raise ValueError("Curve parameter clashes with field parameters.")
# Calculate derivatives for line parameter functions
# F(r) -> F(r(t)) and finally F(r(t)*r'(t))
Ft = F
dldt = 0
for i, var in enumerate(vars):
_f = curve.functions[i]
_dn = diff(_f, curve.parameter)
# ...arc length
dldt = dldt + (_dn * _dn)
Ft = Ft.subs(var, _f)
Ft = Ft * sqrt(dldt)
integral = Integral(Ft, curve.limits).doit(deep=False)
return integral
### Property function dispatching ###
@shape.register(Integral)
def _(expr):
return shape(expr.function)
# Delayed imports
from .deltafunctions import deltaintegrate
from .meijerint import meijerint_definite, meijerint_indefinite, _debug
from .trigonometry import trigintegrate