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.simplify.fu import sincos_to_sum from sympy.simplify.simplify import simplify from sympy.solvers.solvers import solve, posify 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__ = ('is_commutative',) 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 AddWithLimits.free_symbols.fget(self) 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''')) 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)] 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 = dict(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 # continous 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 = dict(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. 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): 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#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] http://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