Integrals

The integrals module in SymPy implements methods to calculate definite and indefinite integrals of expressions.

Principal method in this module is integrate()

  • integrate(f, x) returns the indefinite integral \(\int f\,dx\)

  • integrate(f, (x, a, b)) returns the definite integral \(\int_{a}^{b} f\,dx\)

Examples

SymPy can integrate a vast array of functions. It can integrate polynomial functions:

>>> from sympy import *
>>> init_printing(use_unicode=False, wrap_line=False)
>>> x = Symbol('x')
>>> integrate(x**2 + x + 1, x)
 3    2
x    x
-- + -- + x
3    2

Rational functions:

>>> integrate(x/(x**2+2*x+1), x)
1
log(x + 1) + -----
x + 1

Exponential-polynomial functions. These multiplicative combinations of polynomials and the functions exp, cos and sin can be integrated by hand using repeated integration by parts, which is an extremely tedious process. Happily, SymPy will deal with these integrals.

>>> integrate(x**2 * exp(x) * cos(x), x)
 2  x           2  x                         x           x
x *e *sin(x)   x *e *cos(x)      x          e *sin(x)   e *cos(x)
------------ + ------------ - x*e *sin(x) + --------- - ---------
     2              2                           2           2

even a few nonelementary integrals (in particular, some integrals involving the error function) can be evaluated:

>>> integrate(exp(-x**2)*erf(x), x)
  ____    2
\/ pi *erf (x)
--------------
      4

Integral Transforms

SymPy has special support for definite integrals, and integral transforms.

sympy.integrals.transforms.mellin_transform(f, x, s, **hints)[source]

Compute the Mellin transform \(F(s)\) of \(f(x)\),

\[F(s) = \int_0^\infty x^{s-1} f(x) \mathrm{d}x.\]
For all “sensible” functions, this converges absolutely in a strip

\(a < \operatorname{Re}(s) < b\).

The Mellin transform is related via change of variables to the Fourier transform, and also to the (bilateral) Laplace transform.

This function returns (F, (a, b), cond) where F is the Mellin transform of f, (a, b) is the fundamental strip (as above), and cond are auxiliary convergence conditions.

If the integral cannot be computed in closed form, this function returns an unevaluated MellinTransform object.

For a description of possible hints, refer to the docstring of sympy.integrals.transforms.IntegralTransform.doit(). If noconds=False, then only \(F\) will be returned (i.e. not cond, and also not the strip (a, b)).

>>> from sympy.integrals.transforms import mellin_transform
>>> from sympy import exp
>>> from sympy.abc import x, s
>>> mellin_transform(exp(-x), x, s)
(gamma(s), (0, oo), True)
class sympy.integrals.transforms.MellinTransform[source]

Class representing unevaluated Mellin transforms.

For usage of this class, see the IntegralTransform docstring.

For how to compute Mellin transforms, see the mellin_transform() docstring.

sympy.integrals.transforms.inverse_mellin_transform(F, s, x, strip, **hints)[source]

Compute the inverse Mellin transform of \(F(s)\) over the fundamental strip given by strip=(a, b).

This can be defined as

\[f(x) = \frac{1}{2\pi i} \int_{c - i\infty}^{c + i\infty} x^{-s} F(s) \mathrm{d}s,\]

for any \(c\) in the fundamental strip. Under certain regularity conditions on \(F\) and/or \(f\), this recovers \(f\) from its Mellin transform \(F\) (and vice versa), for positive real \(x\).

One of \(a\) or \(b\) may be passed as None; a suitable \(c\) will be inferred.

If the integral cannot be computed in closed form, this function returns an unevaluated InverseMellinTransform object.

Note that this function will assume x to be positive and real, regardless of the sympy assumptions!

For a description of possible hints, refer to the docstring of sympy.integrals.transforms.IntegralTransform.doit().

>>> from sympy.integrals.transforms import inverse_mellin_transform
>>> from sympy import oo, gamma
>>> from sympy.abc import x, s
>>> inverse_mellin_transform(gamma(s), s, x, (0, oo))
exp(-x)

The fundamental strip matters:

>>> f = 1/(s**2 - 1)
>>> inverse_mellin_transform(f, s, x, (-oo, -1))
(x/2 - 1/(2*x))*Heaviside(x - 1)
>>> inverse_mellin_transform(f, s, x, (-1, 1))
-x*Heaviside(1 - x)/2 - Heaviside(x - 1)/(2*x)
>>> inverse_mellin_transform(f, s, x, (1, oo))
(-x/2 + 1/(2*x))*Heaviside(1 - x)
class sympy.integrals.transforms.InverseMellinTransform[source]

Class representing unevaluated inverse Mellin transforms.

For usage of this class, see the IntegralTransform docstring.

For how to compute inverse Mellin transforms, see the inverse_mellin_transform() docstring.

sympy.integrals.transforms.laplace_transform(f, t, s, **hints)[source]

Compute the Laplace Transform \(F(s)\) of \(f(t)\),

\[F(s) = \int_0^\infty e^{-st} f(t) \mathrm{d}t.\]

For all “sensible” functions, this converges absolutely in a half plane \(a < \operatorname{Re}(s)\).

This function returns (F, a, cond) where F is the Laplace transform of f, \(\operatorname{Re}(s) > a\) is the half-plane of convergence, and cond are auxiliary convergence conditions.

If the integral cannot be computed in closed form, this function returns an unevaluated LaplaceTransform object.

For a description of possible hints, refer to the docstring of sympy.integrals.transforms.IntegralTransform.doit(). If noconds=True, only \(F\) will be returned (i.e. not cond, and also not the plane a).

>>> from sympy.integrals import laplace_transform
>>> from sympy.abc import t, s, a
>>> laplace_transform(t**a, t, s)
(s**(-a)*gamma(a + 1)/s, 0, re(a) > -1)
class sympy.integrals.transforms.LaplaceTransform[source]

Class representing unevaluated Laplace transforms.

For usage of this class, see the IntegralTransform docstring.

For how to compute Laplace transforms, see the laplace_transform() docstring.

sympy.integrals.transforms.inverse_laplace_transform(F, s, t, plane=None, **hints)[source]

Compute the inverse Laplace transform of \(F(s)\), defined as

\[f(t) = \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} e^{st} F(s) \mathrm{d}s,\]

for \(c\) so large that \(F(s)\) has no singularites in the half-plane \(\operatorname{Re}(s) > c-\epsilon\).

The plane can be specified by argument plane, but will be inferred if passed as None.

Under certain regularity conditions, this recovers \(f(t)\) from its Laplace Transform \(F(s)\), for non-negative \(t\), and vice versa.

If the integral cannot be computed in closed form, this function returns an unevaluated InverseLaplaceTransform object.

Note that this function will always assume \(t\) to be real, regardless of the sympy assumption on \(t\).

For a description of possible hints, refer to the docstring of sympy.integrals.transforms.IntegralTransform.doit().

>>> from sympy.integrals.transforms import inverse_laplace_transform
>>> from sympy import exp, Symbol
>>> from sympy.abc import s, t
>>> a = Symbol('a', positive=True)
>>> inverse_laplace_transform(exp(-a*s)/s, s, t)
Heaviside(-a + t)
class sympy.integrals.transforms.InverseLaplaceTransform[source]

Class representing unevaluated inverse Laplace transforms.

For usage of this class, see the IntegralTransform docstring.

For how to compute inverse Laplace transforms, see the inverse_laplace_transform() docstring.

sympy.integrals.transforms.fourier_transform(f, x, k, **hints)[source]

Compute the unitary, ordinary-frequency Fourier transform of \(f\), defined as

\[F(k) = \int_{-\infty}^\infty f(x) e^{-2\pi i x k} \mathrm{d} x.\]

If the transform cannot be computed in closed form, this function returns an unevaluated FourierTransform object.

For other Fourier transform conventions, see the function sympy.integrals.transforms._fourier_transform().

For a description of possible hints, refer to the docstring of sympy.integrals.transforms.IntegralTransform.doit(). Note that for this transform, by default noconds=True.

>>> from sympy import fourier_transform, exp
>>> from sympy.abc import x, k
>>> fourier_transform(exp(-x**2), x, k)
sqrt(pi)*exp(-pi**2*k**2)
>>> fourier_transform(exp(-x**2), x, k, noconds=False)
(sqrt(pi)*exp(-pi**2*k**2), True)
sympy.integrals.transforms._fourier_transform(f, x, k, a, b, name, simplify=True)[source]

Compute a general Fourier-type transform

\[F(k) = a \int_{-\infty}^{\infty} e^{bixk} f(x)\, dx.\]

For suitable choice of a and b, this reduces to the standard Fourier and inverse Fourier transforms.

class sympy.integrals.transforms.FourierTransform[source]

Class representing unevaluated Fourier transforms.

For usage of this class, see the IntegralTransform docstring.

For how to compute Fourier transforms, see the fourier_transform() docstring.

sympy.integrals.transforms.inverse_fourier_transform(F, k, x, **hints)[source]

Compute the unitary, ordinary-frequency inverse Fourier transform of \(F\), defined as

\[f(x) = \int_{-\infty}^\infty F(k) e^{2\pi i x k} \mathrm{d} k.\]

If the transform cannot be computed in closed form, this function returns an unevaluated InverseFourierTransform object.

For other Fourier transform conventions, see the function sympy.integrals.transforms._fourier_transform().

For a description of possible hints, refer to the docstring of sympy.integrals.transforms.IntegralTransform.doit(). Note that for this transform, by default noconds=True.

>>> from sympy import inverse_fourier_transform, exp, sqrt, pi
>>> from sympy.abc import x, k
>>> inverse_fourier_transform(sqrt(pi)*exp(-(pi*k)**2), k, x)
exp(-x**2)
>>> inverse_fourier_transform(sqrt(pi)*exp(-(pi*k)**2), k, x, noconds=False)
(exp(-x**2), True)
class sympy.integrals.transforms.InverseFourierTransform[source]

Class representing unevaluated inverse Fourier transforms.

For usage of this class, see the IntegralTransform docstring.

For how to compute inverse Fourier transforms, see the inverse_fourier_transform() docstring.

sympy.integrals.transforms.sine_transform(f, x, k, **hints)[source]

Compute the unitary, ordinary-frequency sine transform of \(f\), defined as

\[F(k) = \sqrt{\frac{2}{\pi}} \int_{0}^\infty f(x) \sin(2\pi x k) \mathrm{d} x.\]

If the transform cannot be computed in closed form, this function returns an unevaluated SineTransform object.

For a description of possible hints, refer to the docstring of sympy.integrals.transforms.IntegralTransform.doit(). Note that for this transform, by default noconds=True.

>>> from sympy import sine_transform, exp
>>> from sympy.abc import x, k, a
>>> sine_transform(x*exp(-a*x**2), x, k)
sqrt(2)*k*exp(-k**2/(4*a))/(4*a**(3/2))
>>> sine_transform(x**(-a), x, k)
2**(1/2 - a)*k**(a - 1)*gamma(1 - a/2)/gamma(a/2 + 1/2)
class sympy.integrals.transforms.SineTransform[source]

Class representing unevaluated sine transforms.

For usage of this class, see the IntegralTransform docstring.

For how to compute sine transforms, see the sine_transform() docstring.

sympy.integrals.transforms.inverse_sine_transform(F, k, x, **hints)[source]

Compute the unitary, ordinary-frequency inverse sine transform of \(F\), defined as

\[f(x) = \sqrt{\frac{2}{\pi}} \int_{0}^\infty F(k) \sin(2\pi x k) \mathrm{d} k.\]

If the transform cannot be computed in closed form, this function returns an unevaluated InverseSineTransform object.

For a description of possible hints, refer to the docstring of sympy.integrals.transforms.IntegralTransform.doit(). Note that for this transform, by default noconds=True.

>>> from sympy import inverse_sine_transform, exp, sqrt, gamma, pi
>>> from sympy.abc import x, k, a
>>> inverse_sine_transform(2**((1-2*a)/2)*k**(a - 1)*
...     gamma(-a/2 + 1)/gamma((a+1)/2), k, x)
x**(-a)
>>> inverse_sine_transform(sqrt(2)*k*exp(-k**2/(4*a))/(4*sqrt(a)**3), k, x)
x*exp(-a*x**2)
class sympy.integrals.transforms.InverseSineTransform[source]

Class representing unevaluated inverse sine transforms.

For usage of this class, see the IntegralTransform docstring.

For how to compute inverse sine transforms, see the inverse_sine_transform() docstring.

sympy.integrals.transforms.cosine_transform(f, x, k, **hints)[source]

Compute the unitary, ordinary-frequency cosine transform of \(f\), defined as

\[F(k) = \sqrt{\frac{2}{\pi}} \int_{0}^\infty f(x) \cos(2\pi x k) \mathrm{d} x.\]

If the transform cannot be computed in closed form, this function returns an unevaluated CosineTransform object.

For a description of possible hints, refer to the docstring of sympy.integrals.transforms.IntegralTransform.doit(). Note that for this transform, by default noconds=True.

>>> from sympy import cosine_transform, exp, sqrt, cos
>>> from sympy.abc import x, k, a
>>> cosine_transform(exp(-a*x), x, k)
sqrt(2)*a/(sqrt(pi)*(a**2 + k**2))
>>> cosine_transform(exp(-a*sqrt(x))*cos(a*sqrt(x)), x, k)
a*exp(-a**2/(2*k))/(2*k**(3/2))
class sympy.integrals.transforms.CosineTransform[source]

Class representing unevaluated cosine transforms.

For usage of this class, see the IntegralTransform docstring.

For how to compute cosine transforms, see the cosine_transform() docstring.

sympy.integrals.transforms.inverse_cosine_transform(F, k, x, **hints)[source]

Compute the unitary, ordinary-frequency inverse cosine transform of \(F\), defined as

\[f(x) = \sqrt{\frac{2}{\pi}} \int_{0}^\infty F(k) \cos(2\pi x k) \mathrm{d} k.\]

If the transform cannot be computed in closed form, this function returns an unevaluated InverseCosineTransform object.

For a description of possible hints, refer to the docstring of sympy.integrals.transforms.IntegralTransform.doit(). Note that for this transform, by default noconds=True.

>>> from sympy import inverse_cosine_transform, exp, sqrt, pi
>>> from sympy.abc import x, k, a
>>> inverse_cosine_transform(sqrt(2)*a/(sqrt(pi)*(a**2 + k**2)), k, x)
exp(-a*x)
>>> inverse_cosine_transform(1/sqrt(k), k, x)
1/sqrt(x)
class sympy.integrals.transforms.InverseCosineTransform[source]

Class representing unevaluated inverse cosine transforms.

For usage of this class, see the IntegralTransform docstring.

For how to compute inverse cosine transforms, see the inverse_cosine_transform() docstring.

sympy.integrals.transforms.hankel_transform(f, r, k, nu, **hints)[source]

Compute the Hankel transform of \(f\), defined as

\[F_\nu(k) = \int_{0}^\infty f(r) J_\nu(k r) r \mathrm{d} r.\]

If the transform cannot be computed in closed form, this function returns an unevaluated HankelTransform object.

For a description of possible hints, refer to the docstring of sympy.integrals.transforms.IntegralTransform.doit(). Note that for this transform, by default noconds=True.

>>> from sympy import hankel_transform, inverse_hankel_transform
>>> from sympy import gamma, exp, sinh, cosh
>>> from sympy.abc import r, k, m, nu, a
>>> ht = hankel_transform(1/r**m, r, k, nu)
>>> ht
2*2**(-m)*k**(m - 2)*gamma(-m/2 + nu/2 + 1)/gamma(m/2 + nu/2)
>>> inverse_hankel_transform(ht, k, r, nu)
r**(-m)
>>> ht = hankel_transform(exp(-a*r), r, k, 0)
>>> ht
a/(k**3*(a**2/k**2 + 1)**(3/2))
>>> inverse_hankel_transform(ht, k, r, 0)
exp(-a*r)
class sympy.integrals.transforms.HankelTransform[source]

Class representing unevaluated Hankel transforms.

For usage of this class, see the IntegralTransform docstring.

For how to compute Hankel transforms, see the hankel_transform() docstring.

sympy.integrals.transforms.inverse_hankel_transform(F, k, r, nu, **hints)[source]

Compute the inverse Hankel transform of \(F\) defined as

\[f(r) = \int_{0}^\infty F_\nu(k) J_\nu(k r) k \mathrm{d} k.\]

If the transform cannot be computed in closed form, this function returns an unevaluated InverseHankelTransform object.

For a description of possible hints, refer to the docstring of sympy.integrals.transforms.IntegralTransform.doit(). Note that for this transform, by default noconds=True.

>>> from sympy import hankel_transform, inverse_hankel_transform, gamma
>>> from sympy import gamma, exp, sinh, cosh
>>> from sympy.abc import r, k, m, nu, a
>>> ht = hankel_transform(1/r**m, r, k, nu)
>>> ht
2*2**(-m)*k**(m - 2)*gamma(-m/2 + nu/2 + 1)/gamma(m/2 + nu/2)
>>> inverse_hankel_transform(ht, k, r, nu)
r**(-m)
>>> ht = hankel_transform(exp(-a*r), r, k, 0)
>>> ht
a/(k**3*(a**2/k**2 + 1)**(3/2))
>>> inverse_hankel_transform(ht, k, r, 0)
exp(-a*r)
class sympy.integrals.transforms.InverseHankelTransform[source]

Class representing unevaluated inverse Hankel transforms.

For usage of this class, see the IntegralTransform docstring.

For how to compute inverse Hankel transforms, see the inverse_hankel_transform() docstring.

class sympy.integrals.transforms.IntegralTransform[source]

Base class for integral transforms.

This class represents unevaluated transforms.

To implement a concrete transform, derive from this class and implement the _compute_transform(f, x, s, **hints) and _as_integral(f, x, s) functions. If the transform cannot be computed, raise IntegralTransformError.

Also set cls._name. For instance,

>>> from sympy.integrals.transforms import LaplaceTransform
>>> LaplaceTransform._name
'Laplace'

Implement self._collapse_extra if your function returns more than just a number and possibly a convergence condition.

doit(**hints)[source]

Try to evaluate the transform in closed form.

This general function handles linearity, but apart from that leaves pretty much everything to _compute_transform.

Standard hints are the following:

  • simplify: whether or not to simplify the result

  • noconds: if True, don’t return convergence conditions

  • needeval: if True, raise IntegralTransformError instead of

    returning IntegralTransform objects

The default values of these hints depend on the concrete transform, usually the default is (simplify, noconds, needeval) = (True, False, False).

property function

The function to be transformed.

property function_variable

The dependent variable of the function to be transformed.

property transform_variable

The independent transform variable.

exception sympy.integrals.transforms.IntegralTransformError(transform, function, msg)[source]

Exception raised in relation to problems computing transforms.

This class is mostly used internally; if integrals cannot be computed objects representing unevaluated transforms are usually returned.

The hint needeval=True can be used to disable returning transform objects, and instead raise this exception if an integral cannot be computed.

Internals

SymPy uses a number of algorithms to compute integrals. Algorithms are tried in order until one produces an answer. Most of these algorithms can be enabled or disabled manually using various flags to integrate() or doit().

SymPy first applies several heuristic algorithms, as these are the fastest:

  1. If the function is a rational function, there is a complete algorithm for integrating rational functions called the Lizard-Rioboo-Trager and the Horowitz-Ostrogradsky algorithms. They are implemented in ratint().

    sympy.integrals.rationaltools.ratint(f, x, **flags)[source]

    Performs indefinite integration of rational functions.

    Given a field \(K\) and a rational function \(f = p/q\), where \(p\) and \(q\) are polynomials in \(K[x]\), returns a function \(g\) such that \(f = g'\).

    >>> from sympy.integrals.rationaltools import ratint
    >>> from sympy.abc import x
    
    >>> ratint(36/(x**5 - 2*x**4 - 2*x**3 + 4*x**2 + x - 2), x)
    (12*x + 6)/(x**2 - 1) + 4*log(x - 2) - 4*log(x + 1)
    

    References

    Bro05

    M. Bronstein, Symbolic Integration I: Transcendental Functions, Second Edition, Springer-Verlag, 2005, pp. 35-70

    sympy.integrals.rationaltools.ratint_ratpart(f, g, x)[source]

    Horowitz-Ostrogradsky algorithm.

    Given a field K and polynomials f and g in K[x], such that f and g are coprime and deg(f) < deg(g), returns fractions A and B in K(x), such that f/g = A’ + B and B has square-free denominator.

    Examples

    >>> from sympy.integrals.rationaltools import ratint_ratpart
    >>> from sympy.abc import x, y
    >>> from sympy import Poly
    >>> ratint_ratpart(Poly(1, x, domain='ZZ'),
    ... Poly(x + 1, x, domain='ZZ'), x)
    (0, 1/(x + 1))
    >>> ratint_ratpart(Poly(1, x, domain='EX'),
    ... Poly(x**2 + y**2, x, domain='EX'), x)
    (0, 1/(x**2 + y**2))
    >>> ratint_ratpart(Poly(36, x, domain='ZZ'),
    ... Poly(x**5 - 2*x**4 - 2*x**3 + 4*x**2 + x - 2, x, domain='ZZ'), x)
    ((12*x + 6)/(x**2 - 1), 12/(x**2 - x - 2))
    
    sympy.integrals.rationaltools.ratint_logpart(f, g, x, t=None)[source]

    Lazard-Rioboo-Trager algorithm.

    Given a field K and polynomials f and g in K[x], such that f and g are coprime, deg(f) < deg(g) and g is square-free, returns a list of tuples (s_i, q_i) of polynomials, for i = 1..n, such that s_i in K[t, x] and q_i in K[t], and:

              ___    ___
    d  f   d  \  `   \  `
    -- - = --  )      )   a log(s_i(a, x))
    dx g   dx /__,   /__,
             i=1..n a | q_i(a) = 0
    

    Examples

    >>> from sympy.integrals.rationaltools import ratint_logpart
    >>> from sympy.abc import x
    >>> from sympy import Poly
    >>> ratint_logpart(Poly(1, x, domain='ZZ'),
    ... Poly(x**2 + x + 1, x, domain='ZZ'), x)
    [(Poly(x + 3*_t/2 + 1/2, x, domain='QQ[_t]'),
    ...Poly(3*_t**2 + 1, _t, domain='ZZ'))]
    >>> ratint_logpart(Poly(12, x, domain='ZZ'),
    ... Poly(x**2 - x - 2, x, domain='ZZ'), x)
    [(Poly(x - 3*_t/8 - 1/2, x, domain='QQ[_t]'),
    ...Poly(-_t**2 + 16, _t, domain='ZZ'))]
    
  2. trigintegrate() solves integrals of trigonometric functions using pattern matching

    sympy.integrals.trigonometry.trigintegrate(f, x, conds='piecewise')[source]

    Integrate f = Mul(trig) over x

    >>> from sympy import Symbol, sin, cos, tan, sec, csc, cot
    >>> from sympy.integrals.trigonometry import trigintegrate
    >>> from sympy.abc import x
    
    >>> trigintegrate(sin(x)*cos(x), x)
    sin(x)**2/2
    
    >>> trigintegrate(sin(x)**2, x)
    x/2 - sin(x)*cos(x)/2
    
    >>> trigintegrate(tan(x)*sec(x), x)
    1/cos(x)
    
    >>> trigintegrate(sin(x)*tan(x), x)
    -log(sin(x) - 1)/2 + log(sin(x) + 1)/2 - sin(x)
    

    http://en.wikibooks.org/wiki/Calculus/Integration_techniques

  3. deltaintegrate() solves integrals with DiracDelta objects.

    sympy.integrals.deltafunctions.deltaintegrate(f, x)[source]

    The idea for integration is the following:

    • If we are dealing with a DiracDelta expression, i.e. DiracDelta(g(x)), we try to simplify it.

      If we could simplify it, then we integrate the resulting expression. We already know we can integrate a simplified expression, because only simple DiracDelta expressions are involved.

      If we couldn’t simplify it, there are two cases:

      1. The expression is a simple expression: we return the integral, taking care if we are dealing with a Derivative or with a proper DiracDelta.

      2. The expression is not simple (i.e. DiracDelta(cos(x))): we can do nothing at all.

    • If the node is a multiplication node having a DiracDelta term:

      First we expand it.

      If the expansion did work, then we try to integrate the expansion.

      If not, we try to extract a simple DiracDelta term, then we have two cases:

      1. We have a simple DiracDelta term, so we return the integral.

      2. We didn’t have a simple term, but we do have an expression with simplified DiracDelta terms, so we integrate this expression.

    Examples

    >>> from sympy.abc import x, y, z
    >>> from sympy.integrals.deltafunctions import deltaintegrate
    >>> from sympy import sin, cos, DiracDelta, Heaviside
    >>> deltaintegrate(x*sin(x)*cos(x)*DiracDelta(x - 1), x)
    sin(1)*cos(1)*Heaviside(x - 1)
    >>> deltaintegrate(y**2*DiracDelta(x - z)*DiracDelta(y - z), y)
    z**2*DiracDelta(x - z)*Heaviside(y - z)
    
  4. singularityintegrate() is applied if the function contains a SingularityFunction

    sympy.integrals.singularityfunctions.singularityintegrate(f, x)[source]

    This function handles the indefinite integrations of Singularity functions. The integrate function calls this function internally whenever an instance of SingularityFunction is passed as argument.

    The idea for integration is the following:

    • If we are dealing with a SingularityFunction expression, i.e. SingularityFunction(x, a, n), we just return SingularityFunction(x, a, n + 1)/(n + 1) if n >= 0 and SingularityFunction(x, a, n + 1) if n < 0.

    • If the node is a multiplication or power node having a SingularityFunction term we rewrite the whole expression in terms of Heaviside and DiracDelta and then integrate the output. Lastly, we rewrite the output of integration back in terms of SingularityFunction.

    • If none of the above case arises, we return None.

    Examples

    >>> from sympy.integrals.singularityfunctions import singularityintegrate
    >>> from sympy import SingularityFunction, symbols, Function
    >>> x, a, n, y = symbols('x a n y')
    >>> f = Function('f')
    >>> singularityintegrate(SingularityFunction(x, a, 3), x)
    SingularityFunction(x, a, 4)/4
    >>> singularityintegrate(5*SingularityFunction(x, 5, -2), x)
    5*SingularityFunction(x, 5, -1)
    >>> singularityintegrate(6*SingularityFunction(x, 5, -1), x)
    6*SingularityFunction(x, 5, 0)
    >>> singularityintegrate(x*SingularityFunction(x, 0, -1), x)
    0
    >>> singularityintegrate(SingularityFunction(x, 1, -1) * f(x), x)
    f(1)*SingularityFunction(x, 1, 0)
    
  5. If the heuristic algorithms cannot be applied, risch_integrate() is tried next. The Risch algorithm is a general method for calculating antiderivatives of elementary functions. The Risch algorithm is a decision procedure that can determine whether an elementary solution exists, and in that case calculate it. It can be extended to handle many nonelementary functions in addition to the elementary ones. However, the version implemented in SymPy only supports a small subset of the full algorithm, particularly, on part of the transcendental algorithm for exponentials and logarithms is implemented. An advantage of risch_integrate() over other methods is that if it returns an instance of NonElementaryIntegral, the integral is proven to be nonelementary by the algorithm, meaning the integral cannot be represented using a combination of exponentials, logarithms, trig functions, powers, rational functions, algebraic functions, and function composition.

    sympy.integrals.risch.risch_integrate(f, x, extension=None, handle_first='log', separate_integral=False, rewrite_complex=None, conds='piecewise')[source]

    The Risch Integration Algorithm.

    Only transcendental functions are supported. Currently, only exponentials and logarithms are supported, but support for trigonometric functions is forthcoming.

    If this function returns an unevaluated Integral in the result, it means that it has proven that integral to be nonelementary. Any errors will result in raising NotImplementedError. The unevaluated Integral will be an instance of NonElementaryIntegral, a subclass of Integral.

    handle_first may be either ‘exp’ or ‘log’. This changes the order in which the extension is built, and may result in a different (but equivalent) solution (for an example of this, see issue 5109). It is also possible that the integral may be computed with one but not the other, because not all cases have been implemented yet. It defaults to ‘log’ so that the outer extension is exponential when possible, because more of the exponential case has been implemented.

    If separate_integral is True, the result is returned as a tuple (ans, i), where the integral is ans + i, ans is elementary, and i is either a NonElementaryIntegral or 0. This useful if you want to try further integrating the NonElementaryIntegral part using other algorithms to possibly get a solution in terms of special functions. It is False by default.

    Examples

    >>> from sympy.integrals.risch import risch_integrate
    >>> from sympy import exp, log, pprint
    >>> from sympy.abc import x
    

    First, we try integrating exp(-x**2). Except for a constant factor of 2/sqrt(pi), this is the famous error function.

    >>> pprint(risch_integrate(exp(-x**2), x))
      /
     |
     |    2
     |  -x
     | e    dx
     |
    /
    

    The unevaluated Integral in the result means that risch_integrate() has proven that exp(-x**2) does not have an elementary anti-derivative.

    In many cases, risch_integrate() can split out the elementary anti-derivative part from the nonelementary anti-derivative part. For example,

    >>> pprint(risch_integrate((2*log(x)**2 - log(x) - x**2)/(log(x)**3 -
    ... x**2*log(x)), x))
                                             /
                                            |
      log(-x + log(x))   log(x + log(x))    |   1
    - ---------------- + --------------- +  | ------ dx
             2                  2           | log(x)
                                            |
                                           /
    

    This means that it has proven that the integral of 1/log(x) is nonelementary. This function is also known as the logarithmic integral, and is often denoted as Li(x).

    risch_integrate() currently only accepts purely transcendental functions with exponentials and logarithms, though note that this can include nested exponentials and logarithms, as well as exponentials with bases other than E.

    >>> pprint(risch_integrate(exp(x)*exp(exp(x)), x))
     / x\
     \e /
    e
    >>> pprint(risch_integrate(exp(exp(x)), x))
      /
     |
     |  / x\
     |  \e /
     | e     dx
     |
    /
    
    >>> pprint(risch_integrate(x*x**x*log(x) + x**x + x*x**x, x))
       x
    x*x
    >>> pprint(risch_integrate(x**x, x))
      /
     |
     |  x
     | x  dx
     |
    /
    
    >>> pprint(risch_integrate(-1/(x*log(x)*log(log(x))**2), x))
         1
    -----------
    log(log(x))
    
    class sympy.integrals.risch.NonElementaryIntegral[source]

    Represents a nonelementary Integral.

    If the result of integrate() is an instance of this class, it is guaranteed to be nonelementary. Note that integrate() by default will try to find any closed-form solution, even in terms of special functions which may themselves not be elementary. To make integrate() only give elementary solutions, or, in the cases where it can prove the integral to be nonelementary, instances of this class, use integrate(risch=True). In this case, integrate() may raise NotImplementedError if it cannot make such a determination.

    integrate() uses the deterministic Risch algorithm to integrate elementary functions or prove that they have no elementary integral. In some cases, this algorithm can split an integral into an elementary and nonelementary part, so that the result of integrate will be the sum of an elementary expression and a NonElementaryIntegral.

    Examples

    >>> from sympy import integrate, exp, log, Integral
    >>> from sympy.abc import x
    
    >>> a = integrate(exp(-x**2), x, risch=True)
    >>> print(a)
    Integral(exp(-x**2), x)
    >>> type(a)
    <class 'sympy.integrals.risch.NonElementaryIntegral'>
    
    >>> expr = (2*log(x)**2 - log(x) - x**2)/(log(x)**3 - x**2*log(x))
    >>> b = integrate(expr, x, risch=True)
    >>> print(b)
    -log(-x + log(x))/2 + log(x + log(x))/2 + Integral(1/log(x), x)
    >>> type(b.atoms(Integral).pop())
    <class 'sympy.integrals.risch.NonElementaryIntegral'>
    
  6. For non-elementary definite integrals, SymPy uses so-called Meijer G-functions. Details are described in Computing Integrals using Meijer G-Functions.

  7. All the algorithms mentioned thus far are either pattern-matching based heuristic, or solve integrals using algorithms that are much different from the way most people are taught in their calculus courses. SymPy also implements a method that can solve integrals in much the same way you would in calculus. The advantage of this method is that it is possible to extract the integration steps from, so that one can see how to compute the integral “by hand”. This is used by SymPy Gamma. This is implemented in the manualintegrate() function. The steps for an integral can be seen with the integral_steps() function.

    sympy.integrals.manualintegrate.manualintegrate(f, var)[source]

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

    Unlike integrate(), var can only be a single symbol.

    Examples

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

    Returns the steps needed to compute an integral.

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

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

    Returns

    rule : namedtuple

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

    Examples

    >>> from sympy import exp, sin, cos
    >>> from sympy.integrals.manualintegrate import integral_steps
    >>> from sympy.abc import x
    >>> print(repr(integral_steps(exp(x) / (1 + exp(2 * x)), x)))     
    URule(u_var=_u, u_func=exp(x), constant=1,
    substep=PiecewiseRule(subfunctions=[(ArctanRule(a=1, b=1, c=1, context=1/(_u**2 + 1), symbol=_u), True),
        (ArccothRule(a=1, b=1, c=1, context=1/(_u**2 + 1), symbol=_u), False),
        (ArctanhRule(a=1, b=1, c=1, context=1/(_u**2 + 1), symbol=_u), False)],
    context=1/(_u**2 + 1), symbol=_u), context=exp(x)/(exp(2*x) + 1), symbol=x)
    >>> print(repr(integral_steps(sin(x), x)))     
    TrigRule(func='sin', arg=x, context=sin(x), symbol=x)
    >>> print(repr(integral_steps((x**2 + 3)**2 , x)))     
    RewriteRule(rewritten=x**4 + 6*x**2 + 9,
    substep=AddRule(substeps=[PowerRule(base=x, exp=4, context=x**4, symbol=x),
        ConstantTimesRule(constant=6, other=x**2,
            substep=PowerRule(base=x, exp=2, context=x**2, symbol=x),
                context=6*x**2, symbol=x),
        ConstantRule(constant=9, context=9, symbol=x)],
    context=x**4 + 6*x**2 + 9, symbol=x), context=(x**2 + 3)**2, symbol=x)
    
  8. Finally, if all the above fail, SymPy also uses a simplified version of the Risch algorithm, called the Risch-Norman algorithm. This algorithm is tried last because it is often the slowest to compute. This is implemented in heurisch():

    sympy.integrals.heurisch.heurisch(f, x, rewrite=False, hints=None, mappings=None, retries=3, degree_offset=0, unnecessary_permutations=None, _try_heurisch=None)[source]

    Compute indefinite integral using heuristic Risch algorithm.

    This is a heuristic approach to indefinite integration in finite terms using the extended heuristic (parallel) Risch algorithm, based on Manuel Bronstein’s “Poor Man’s Integrator”.

    The algorithm supports various classes of functions including transcendental elementary or special functions like Airy, Bessel, Whittaker and Lambert.

    Note that this algorithm is not a decision procedure. If it isn’t able to compute the antiderivative for a given function, then this is not a proof that such a functions does not exist. One should use recursive Risch algorithm in such case. It’s an open question if this algorithm can be made a full decision procedure.

    This is an internal integrator procedure. You should use toplevel ‘integrate’ function in most cases, as this procedure needs some preprocessing steps and otherwise may fail.

    Specification

    heurisch(f, x, rewrite=False, hints=None)

    where

    f : expression x : symbol

    rewrite -> force rewrite ‘f’ in terms of ‘tan’ and ‘tanh’ hints -> a list of functions that may appear in anti-derivate

    • hints = None –> no suggestions at all

    • hints = [ ] –> try to figure out

    • hints = [f1, …, fn] –> we know better

    Examples

    >>> from sympy import tan
    >>> from sympy.integrals.heurisch import heurisch
    >>> from sympy.abc import x, y
    
    >>> heurisch(y*tan(x), x)
    y*log(tan(x)**2 + 1)/2
    

    See Manuel Bronstein’s “Poor Man’s Integrator”:

    [1] http://www-sop.inria.fr/cafe/Manuel.Bronstein/pmint/index.html

    For more information on the implemented algorithm refer to:

    [2] K. Geddes, L. Stefanus, On the Risch-Norman Integration

    Method and its Implementation in Maple, Proceedings of ISSAC’89, ACM Press, 212-217.

    [3] J. H. Davenport, On the Parallel Risch Algorithm (I),

    Proceedings of EUROCAM’82, LNCS 144, Springer, 144-157.

    [4] J. H. Davenport, On the Parallel Risch Algorithm (III):

    Use of Tangents, SIGSAM Bulletin 16 (1982), 3-6.

    [5] J. H. Davenport, B. M. Trager, On the Parallel Risch

    Algorithm (II), ACM Transactions on Mathematical Software 11 (1985), 356-362.

    sympy.integrals.heurisch.components(f, x)[source]

    Returns a set of all functional components of the given expression which includes symbols, function applications and compositions and non-integer powers. Fractional powers are collected with minimal, positive exponents.

    >>> from sympy import cos, sin
    >>> from sympy.abc import x, y
    >>> from sympy.integrals.heurisch import components
    
    >>> components(sin(x)*cos(x)**2, x)
    {x, sin(x), cos(x)}
    

    See also

    heurisch

API reference

sympy.integrals.integrals.integrate(f, var, ...)[source]

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)
sympy.integrals.line_integrate(field, Curve, variables)[source]

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)

The class Integral represents an unevaluated integral and has some methods that help in the integration of an expression.

class sympy.integrals.integrals.Integral[source]

Represents unevaluated integral.

is_commutative

Returns whether all the free symbols in the integral are commutative.

as_sum(n=None, method='midpoint', evaluate=True)[source]

Approximates a definite integral by a sum.

Arguments

n

The number of subintervals to use, optional.

method

One of: ‘left’, ‘right’, ‘midpoint’, ‘trapezoid’.

evaluate

If False, returns an unevaluated Sum expression. The default is True, evaluate the sum.

These methods of approximate integration are described in [1].

[1] https://en.wikipedia.org/wiki/Riemann_sum#Methods

Examples

>>> from sympy import sin, sqrt
>>> from sympy.abc import x, n
>>> from sympy.integrals import Integral
>>> 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

doit(**hints)[source]

Perform the integration using any hints given.

Examples

>>> from sympy import Integral, 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
property free_symbols

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}
principal_value(**kwargs)[source]

Compute the Cauchy Principal Value of the definite integral of a real function in the given interval on the real axis. 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 Dummy, symbols, integrate, limit, oo
>>> from sympy.integrals.integrals import Integral
>>> from sympy.calculus.singularities import singularities
>>> x = symbols('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

R476

https://en.wikipedia.org/wiki/Cauchy_principal_value

R477

http://mathworld.wolfram.com/CauchyPrincipalValue.html

transform(x, u)[source]

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:

\[\int_a^b x \mathrm{d}x \rightarrow \int_{F(a)}^{F(b)} f(x) \frac{\mathrm{d}}{\mathrm{d}x}\]

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, b, c, d, x, u, y
>>> from sympy import Integral, S, 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

Integral subclasses from ExprWithLimits, which is a common superclass of Integral and Sum.

class sympy.concrete.expr_with_limits.ExprWithLimits[source]
property bound_symbols

Return only variables that are dummy variables.

Examples

>>> from sympy import Integral
>>> from sympy.abc import x, i, j, k
>>> Integral(x**i, (i, 1, 3), (j, 2), k).bound_symbols
[i, j]

See also

function, limits, free_symbols

as_dummy

Rename dummy variables

sympy.integrals.integrals.Integral.transform

Perform mapping on the dummy variable

property free_symbols

This method returns the symbols in the object, excluding those that take on a specific value (i.e. the dummy symbols).

Examples

>>> from sympy import Sum
>>> from sympy.abc import x, y
>>> Sum(x, (x, y, 1)).free_symbols
{y}
property function

Return the function applied across limits.

Examples

>>> from sympy import Integral
>>> from sympy.abc import x
>>> Integral(x**2, (x,)).function
x**2
property has_finite_limits

Returns True if the limits are known to be finite, either by the explicit bounds, assumptions on the bounds, or assumptions on the variables. False if known to be infinite, based on the bounds. None if not enough information is available to determine.

Examples

>>> from sympy import Sum, Integral, Product, oo, Symbol
>>> x = Symbol('x')
>>> Sum(x, (x, 1, 8)).has_finite_limits
True
>>> Integral(x, (x, 1, oo)).has_finite_limits
False
>>> M = Symbol('M')
>>> Sum(x, (x, 1, M)).has_finite_limits
>>> N = Symbol('N', integer=True)
>>> Product(x, (x, 1, N)).has_finite_limits
True
property has_reversed_limits

Returns True if the limits are known to be in reversed order, either by the explicit bounds, assumptions on the bounds, or assumptions on the variables. False if known to be in normal order, based on the bounds. None if not enough information is available to determine.

Examples

>>> from sympy import Sum, Integral, Product, oo, Symbol
>>> x = Symbol('x')
>>> Sum(x, (x, 8, 1)).has_reversed_limits
True
>>> Sum(x, (x, 1, oo)).has_reversed_limits
False
>>> M = Symbol('M')
>>> Integral(x, (x, 1, M)).has_reversed_limits
>>> N = Symbol('N', integer=True, positive=True)
>>> Sum(x, (x, 1, N)).has_reversed_limits
False
>>> Product(x, (x, 2, N)).has_reversed_limits
>>> Product(x, (x, 2, N)).subs(N, N + 2).has_reversed_limits
False
property is_number

Return True if the Sum has no free symbols, else False.

property limits

Return the limits of expression.

Examples

>>> from sympy import Integral
>>> from sympy.abc import x, i
>>> Integral(x**i, (i, 1, 3)).limits
((i, 1, 3),)
property variables

Return a list of the limit variables.

>>> from sympy import Sum
>>> from sympy.abc import x, i
>>> Sum(x**i, (i, 1, 3)).variables
[i]

See also

function, limits, free_symbols

as_dummy

Rename dummy variables

sympy.integrals.integrals.Integral.transform

Perform mapping on the dummy variable

TODO and Bugs

There are still lots of functions that SymPy does not know how to integrate. For bugs related to this module, see https://github.com/sympy/sympy/issues?q=label%3AIntegration

Numeric Integrals

SymPy has functions to calculate points and weights for Gaussian quadrature of any order and any precision:

sympy.integrals.quadrature.gauss_legendre(n, n_digits)[source]

Computes the Gauss-Legendre quadrature [R478] points and weights.

The Gauss-Legendre quadrature approximates the integral:

\[\int_{-1}^1 f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i)\]

The nodes \(x_i\) of an order \(n\) quadrature rule are the roots of \(P_n\) and the weights \(w_i\) are given by:

\[w_i = \frac{2}{\left(1-x_i^2\right) \left(P'_n(x_i)\right)^2}\]
Parameters

n : the order of quadrature

n_digits : number of significant digits of the points and weights to return

Returns

(x, w) : the x and w are lists of points and weights as Floats.

The points \(x_i\) and weights \(w_i\) are returned as (x, w) tuple of lists.

Examples

>>> from sympy.integrals.quadrature import gauss_legendre
>>> x, w = gauss_legendre(3, 5)
>>> x
[-0.7746, 0, 0.7746]
>>> w
[0.55556, 0.88889, 0.55556]
>>> x, w = gauss_legendre(4, 5)
>>> x
[-0.86114, -0.33998, 0.33998, 0.86114]
>>> w
[0.34785, 0.65215, 0.65215, 0.34785]

References

R478(1,2)

https://en.wikipedia.org/wiki/Gaussian_quadrature

R479

http://people.sc.fsu.edu/~jburkardt/cpp_src/legendre_rule/legendre_rule.html

sympy.integrals.quadrature.gauss_laguerre(n, n_digits)[source]

Computes the Gauss-Laguerre quadrature [R480] points and weights.

The Gauss-Laguerre quadrature approximates the integral:

\[\int_0^{\infty} e^{-x} f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i)\]

The nodes \(x_i\) of an order \(n\) quadrature rule are the roots of \(L_n\) and the weights \(w_i\) are given by:

\[w_i = \frac{x_i}{(n+1)^2 \left(L_{n+1}(x_i)\right)^2}\]
Parameters

n : the order of quadrature

n_digits : number of significant digits of the points and weights to return

Returns

(x, w) : the x and w are lists of points and weights as Floats.

The points \(x_i\) and weights \(w_i\) are returned as (x, w) tuple of lists.

Examples

>>> from sympy.integrals.quadrature import gauss_laguerre
>>> x, w = gauss_laguerre(3, 5)
>>> x
[0.41577, 2.2943, 6.2899]
>>> w
[0.71109, 0.27852, 0.010389]
>>> x, w = gauss_laguerre(6, 5)
>>> x
[0.22285, 1.1889, 2.9927, 5.7751, 9.8375, 15.983]
>>> w
[0.45896, 0.417, 0.11337, 0.010399, 0.00026102, 8.9855e-7]

References

R480(1,2)

https://en.wikipedia.org/wiki/Gauss%E2%80%93Laguerre_quadrature

R481

http://people.sc.fsu.edu/~jburkardt/cpp_src/laguerre_rule/laguerre_rule.html

sympy.integrals.quadrature.gauss_hermite(n, n_digits)[source]

Computes the Gauss-Hermite quadrature [R482] points and weights.

The Gauss-Hermite quadrature approximates the integral:

\[\int_{-\infty}^{\infty} e^{-x^2} f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i)\]

The nodes \(x_i\) of an order \(n\) quadrature rule are the roots of \(H_n\) and the weights \(w_i\) are given by:

\[w_i = \frac{2^{n-1} n! \sqrt{\pi}}{n^2 \left(H_{n-1}(x_i)\right)^2}\]
Parameters

n : the order of quadrature

n_digits : number of significant digits of the points and weights to return

Returns

(x, w) : the x and w are lists of points and weights as Floats.

The points \(x_i\) and weights \(w_i\) are returned as (x, w) tuple of lists.

Examples

>>> from sympy.integrals.quadrature import gauss_hermite
>>> x, w = gauss_hermite(3, 5)
>>> x
[-1.2247, 0, 1.2247]
>>> w
[0.29541, 1.1816, 0.29541]
>>> x, w = gauss_hermite(6, 5)
>>> x
[-2.3506, -1.3358, -0.43608, 0.43608, 1.3358, 2.3506]
>>> w
[0.00453, 0.15707, 0.72463, 0.72463, 0.15707, 0.00453]

References

R482(1,2)

https://en.wikipedia.org/wiki/Gauss-Hermite_Quadrature

R483

http://people.sc.fsu.edu/~jburkardt/cpp_src/hermite_rule/hermite_rule.html

R484

http://people.sc.fsu.edu/~jburkardt/cpp_src/gen_hermite_rule/gen_hermite_rule.html

sympy.integrals.quadrature.gauss_gen_laguerre(n, alpha, n_digits)[source]

Computes the generalized Gauss-Laguerre quadrature [R485] points and weights.

The generalized Gauss-Laguerre quadrature approximates the integral:

\[\int_{0}^\infty x^{\alpha} e^{-x} f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i)\]

The nodes \(x_i\) of an order \(n\) quadrature rule are the roots of \(L^{\alpha}_n\) and the weights \(w_i\) are given by:

\[w_i = \frac{\Gamma(\alpha+n)} {n \Gamma(n) L^{\alpha}_{n-1}(x_i) L^{\alpha+1}_{n-1}(x_i)}\]
Parameters

n : the order of quadrature

alpha : the exponent of the singularity, \(\alpha > -1\)

n_digits : number of significant digits of the points and weights to return

Returns

(x, w) : the x and w are lists of points and weights as Floats.

The points \(x_i\) and weights \(w_i\) are returned as (x, w) tuple of lists.

Examples

>>> from sympy import S
>>> from sympy.integrals.quadrature import gauss_gen_laguerre
>>> x, w = gauss_gen_laguerre(3, -S.Half, 5)
>>> x
[0.19016, 1.7845, 5.5253]
>>> w
[1.4493, 0.31413, 0.00906]
>>> x, w = gauss_gen_laguerre(4, 3*S.Half, 5)
>>> x
[0.97851, 2.9904, 6.3193, 11.712]
>>> w
[0.53087, 0.67721, 0.11895, 0.0023152]

References

R485(1,2)

https://en.wikipedia.org/wiki/Gauss%E2%80%93Laguerre_quadrature

R486

http://people.sc.fsu.edu/~jburkardt/cpp_src/gen_laguerre_rule/gen_laguerre_rule.html

sympy.integrals.quadrature.gauss_chebyshev_t(n, n_digits)[source]

Computes the Gauss-Chebyshev quadrature [R487] points and weights of the first kind.

The Gauss-Chebyshev quadrature of the first kind approximates the integral:

\[\int_{-1}^{1} \frac{1}{\sqrt{1-x^2}} f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i)\]

The nodes \(x_i\) of an order \(n\) quadrature rule are the roots of \(T_n\) and the weights \(w_i\) are given by:

\[w_i = \frac{\pi}{n}\]
Parameters

n : the order of quadrature

n_digits : number of significant digits of the points and weights to return

Returns

(x, w) : the x and w are lists of points and weights as Floats.

The points \(x_i\) and weights \(w_i\) are returned as (x, w) tuple of lists.

Examples

>>> from sympy import S
>>> from sympy.integrals.quadrature import gauss_chebyshev_t
>>> x, w = gauss_chebyshev_t(3, 5)
>>> x
[0.86602, 0, -0.86602]
>>> w
[1.0472, 1.0472, 1.0472]
>>> x, w = gauss_chebyshev_t(6, 5)
>>> x
[0.96593, 0.70711, 0.25882, -0.25882, -0.70711, -0.96593]
>>> w
[0.5236, 0.5236, 0.5236, 0.5236, 0.5236, 0.5236]

References

R487(1,2)

https://en.wikipedia.org/wiki/Chebyshev%E2%80%93Gauss_quadrature

R488

http://people.sc.fsu.edu/~jburkardt/cpp_src/chebyshev1_rule/chebyshev1_rule.html

sympy.integrals.quadrature.gauss_chebyshev_u(n, n_digits)[source]

Computes the Gauss-Chebyshev quadrature [R489] points and weights of the second kind.

The Gauss-Chebyshev quadrature of the second kind approximates the integral:

\[\int_{-1}^{1} \sqrt{1-x^2} f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i)\]

The nodes \(x_i\) of an order \(n\) quadrature rule are the roots of \(U_n\) and the weights \(w_i\) are given by:

\[w_i = \frac{\pi}{n+1} \sin^2 \left(\frac{i}{n+1}\pi\right)\]
Parameters

n : the order of quadrature

n_digits : number of significant digits of the points and weights to return

Returns

(x, w) : the x and w are lists of points and weights as Floats.

The points \(x_i\) and weights \(w_i\) are returned as (x, w) tuple of lists.

Examples

>>> from sympy import S
>>> from sympy.integrals.quadrature import gauss_chebyshev_u
>>> x, w = gauss_chebyshev_u(3, 5)
>>> x
[0.70711, 0, -0.70711]
>>> w
[0.3927, 0.7854, 0.3927]
>>> x, w = gauss_chebyshev_u(6, 5)
>>> x
[0.90097, 0.62349, 0.22252, -0.22252, -0.62349, -0.90097]
>>> w
[0.084489, 0.27433, 0.42658, 0.42658, 0.27433, 0.084489]

References

R489(1,2)

https://en.wikipedia.org/wiki/Chebyshev%E2%80%93Gauss_quadrature

R490

http://people.sc.fsu.edu/~jburkardt/cpp_src/chebyshev2_rule/chebyshev2_rule.html

sympy.integrals.quadrature.gauss_jacobi(n, alpha, beta, n_digits)[source]

Computes the Gauss-Jacobi quadrature [R491] points and weights.

The Gauss-Jacobi quadrature of the first kind approximates the integral:

\[\int_{-1}^1 (1-x)^\alpha (1+x)^\beta f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i)\]

The nodes \(x_i\) of an order \(n\) quadrature rule are the roots of \(P^{(\alpha,\beta)}_n\) and the weights \(w_i\) are given by:

\[w_i = -\frac{2n+\alpha+\beta+2}{n+\alpha+\beta+1} \frac{\Gamma(n+\alpha+1)\Gamma(n+\beta+1)} {\Gamma(n+\alpha+\beta+1)(n+1)!} \frac{2^{\alpha+\beta}}{P'_n(x_i) P^{(\alpha,\beta)}_{n+1}(x_i)}\]
Parameters

n : the order of quadrature

alpha : the first parameter of the Jacobi Polynomial, \(\alpha > -1\)

beta : the second parameter of the Jacobi Polynomial, \(\beta > -1\)

n_digits : number of significant digits of the points and weights to return

Returns

(x, w) : the x and w are lists of points and weights as Floats.

The points \(x_i\) and weights \(w_i\) are returned as (x, w) tuple of lists.

Examples

>>> from sympy import S
>>> from sympy.integrals.quadrature import gauss_jacobi
>>> x, w = gauss_jacobi(3, S.Half, -S.Half, 5)
>>> x
[-0.90097, -0.22252, 0.62349]
>>> w
[1.7063, 1.0973, 0.33795]
>>> x, w = gauss_jacobi(6, 1, 1, 5)
>>> x
[-0.87174, -0.5917, -0.2093, 0.2093, 0.5917, 0.87174]
>>> w
[0.050584, 0.22169, 0.39439, 0.39439, 0.22169, 0.050584]

References

R491(1,2)

https://en.wikipedia.org/wiki/Gauss%E2%80%93Jacobi_quadrature

R492

http://people.sc.fsu.edu/~jburkardt/cpp_src/jacobi_rule/jacobi_rule.html

R493

http://people.sc.fsu.edu/~jburkardt/cpp_src/gegenbauer_rule/gegenbauer_rule.html

sympy.integrals.quadrature.gauss_lobatto(n, n_digits)[source]

Computes the Gauss-Lobatto quadrature [R494] points and weights.

The Gauss-Lobatto quadrature approximates the integral:

\[\int_{-1}^1 f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i)\]

The nodes \(x_i\) of an order \(n\) quadrature rule are the roots of \(P'_(n-1)\) and the weights \(w_i\) are given by:

\[\begin{split}&w_i = \frac{2}{n(n-1) \left[P_{n-1}(x_i)\right]^2},\quad x\neq\pm 1\\ &w_i = \frac{2}{n(n-1)},\quad x=\pm 1\end{split}\]
Parameters

n : the order of quadrature

n_digits : number of significant digits of the points and weights to return

Returns

(x, w) : the x and w are lists of points and weights as Floats.

The points \(x_i\) and weights \(w_i\) are returned as (x, w) tuple of lists.

Examples

>>> from sympy.integrals.quadrature import gauss_lobatto
>>> x, w = gauss_lobatto(3, 5)
>>> x
[-1, 0, 1]
>>> w
[0.33333, 1.3333, 0.33333]
>>> x, w = gauss_lobatto(4, 5)
>>> x
[-1, -0.44721, 0.44721, 1]
>>> w
[0.16667, 0.83333, 0.83333, 0.16667]

References

R494(1,2)

https://en.wikipedia.org/wiki/Gaussian_quadrature#Gauss.E2.80.93Lobatto_rules

R495

http://people.math.sfu.ca/~cbm/aands/page_888.htm

Integration over Polytopes

The intpoly module in SymPy implements methods to calculate the integral of a polynomial over 2/3-Polytopes. Uses evaluation techniques as described in Chin et al. (2015) [1].

The input for 2-Polytope or Polygon uses the already existing Polygon data structure in SymPy. See sympy.geometry.polygon for how to create a polygon.

For the 3-Polytope or Polyhedron, the most economical representation is to specify a list of vertices and then to provide each constituting face(Polygon) as a list of vertex indices.

For example, consider the unit cube. Here is how it would be represented.

unit_cube = [[(0, 0, 0), (0, 0, 1), (0, 1, 0), (0, 1, 1), (1, 0, 0),(1, 0, 1), (1, 1, 0), (1, 1, 1)],

[3, 7, 6, 2], [1, 5, 7, 3], [5, 4, 6, 7], [0, 4, 5, 1], [2, 0, 1, 3], [2, 6, 4, 0]]

Here, the first sublist is the list of vertices. The other smaller lists such as [3, 7, 6, 2] represent a 2D face of the polyhedra with vertices having index 3, 7, 6 and 2 in the first sublist(in that order).

Principal method in this module is polytope_integrate()

  • polytope_integrate(Polygon((0, 0), (0, 1), (1, 0)), x) returns the integral of \(x\) over the triangle with vertices (0, 0), (0, 1) and (1, 0)

  • polytope_integrate(unit_cube, x + y + z) returns the integral of \(x + y + z\) over the unit cube.

References

[1] : Chin, Eric B., Jean B. Lasserre, and N. Sukumar. “Numerical integration of homogeneous functions on convex and nonconvex polygons and polyhedra.” Computational Mechanics 56.6 (2015): 967-981

PDF link : http://dilbert.engr.ucdavis.edu/~suku/quadrature/cls-integration.pdf

Examples

For 2D Polygons

Single Polynomial:

>>> from sympy.integrals.intpoly import *
>>> init_printing(use_unicode=False, wrap_line=False)
>>> polytope_integrate(Polygon((0, 0), (0, 1), (1, 0)), x)
1/6
>>> polytope_integrate(Polygon((0, 0), (0, 1), (1, 0)), x + x*y + y**2)
7/24

List of specified polynomials:

>>> polytope_integrate(Polygon((0, 0), (0, 1), (1, 0)), [3, x*y + y**2, x**4], max_degree=4)
          4               2
{3: 3/2, x : 1/30, x*y + y : 1/8}
>>> polytope_integrate(Polygon((0, 0), (0, 1), (1, 0)), [1.125, x, x**2, 6.89*x**3, x*y + y**2, x**4], max_degree=4)
                       2              3  689    4               2
{1.125: 9/16, x: 1/6, x : 1/12, 6.89*x : ----, x : 1/30, x*y + y : 1/8}
                                         2000

Computing all monomials up to a maximum degree:

>>> polytope_integrate(Polygon((0, 0), (0, 1), (1, 0)),max_degree=3)
                        2         3                 2         3                      2         2
{0: 0, 1: 1/2, x: 1/6, x : 1/12, x : 1/20, y: 1/6, y : 1/12, y : 1/20, x*y: 1/24, x*y : 1/60, x *y: 1/60}

For 3-Polytopes/Polyhedra

Single Polynomial:

>>> from sympy.integrals.intpoly import *
>>> cube = [[(0, 0, 0), (0, 0, 5), (0, 5, 0), (0, 5, 5), (5, 0, 0), (5, 0, 5), (5, 5, 0), (5, 5, 5)], [2, 6, 7, 3], [3, 7, 5, 1], [7, 6, 4, 5], [1, 5, 4, 0], [3, 1, 0, 2], [0, 4, 6, 2]]
>>> polytope_integrate(cube, x**2 + y**2 + z**2 + x*y + y*z + x*z)
-21875/4
>>> octahedron = [[(S(-1) / sqrt(2), 0, 0), (0, S(1) / sqrt(2), 0), (0, 0, S(-1) / sqrt(2)), (0, 0, S(1) / sqrt(2)), (0, S(-1) / sqrt(2), 0), (S(1) / sqrt(2), 0, 0)], [3, 4, 5], [3, 5, 1], [3, 1, 0], [3, 0, 4], [4, 0, 2], [4, 2, 5], [2, 0, 1], [5, 2, 1]]
>>> polytope_integrate(octahedron, x**2 + y**2 + z**2 + x*y + y*z + x*z)
  ___
\/ 2
-----
  20

List of specified polynomials:

>>> polytope_integrate(Polygon((0, 0), (0, 1), (1, 0)), [3, x*y + y**2, x**4], max_degree=4)
          4               2
{3: 3/2, x : 1/30, x*y + y : 1/8}
>>> polytope_integrate(Polygon((0, 0), (0, 1), (1, 0)), [1.125, x, x**2, 6.89*x**3, x*y + y**2, x**4], max_degree=4)
                       2              3  689    4               2
{1.125: 9/16, x: 1/6, x : 1/12, 6.89*x : ----, x : 1/30, x*y + y : 1/8}
                                         2000

Computing all monomials up to a maximum degree:

>>> polytope_integrate(Polygon((0, 0), (0, 1), (1, 0)),max_degree=3)
                        2         3                 2         3                      2         2
{0: 0, 1: 1/2, x: 1/6, x : 1/12, x : 1/20, y: 1/6, y : 1/12, y : 1/20, x*y: 1/24, x*y : 1/60, x *y: 1/60}

API reference

sympy.integrals.intpoly.polytope_integrate(poly, expr=None, **kwargs)[source]

Integrates polynomials over 2/3-Polytopes.

This function accepts the polytope in \(poly\) and the function in \(expr\) (uni/bi/trivariate polynomials are implemented) and returns the exact integral of \(expr\) over \(poly\).

Parameters

poly : The input Polygon.

expr : The input polynomial.

clockwise : Binary value to sort input points of 2-Polytope clockwise.(Optional)

max_degree : The maximum degree of any monomial of the input polynomial.(Optional)

Examples

>>> from sympy.abc import x, y
>>> from sympy.geometry.polygon import Polygon
>>> from sympy.geometry.point import Point
>>> from sympy.integrals.intpoly import polytope_integrate
>>> polygon = Polygon(Point(0, 0), Point(0, 1), Point(1, 1), Point(1, 0))
>>> polys = [1, x, y, x*y, x**2*y, x*y**2]
>>> expr = x*y
>>> polytope_integrate(polygon, expr)
1/4
>>> polytope_integrate(polygon, polys, max_degree=3)
{1: 1, x: 1/2, y: 1/2, x*y: 1/4, x*y**2: 1/6, x**2*y: 1/6}