Symbolic 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
Exponentialpolynomial 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 \(f(x)\),
\[F(s) = \int_0^\infty x^{s1} f(x) \mathrm{d}x.\] 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)
whereF
is the Mellin transform off
,(a, b)
is the fundamental strip (as above), andcond
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()
. Ifnoconds=False
, then only \(F\) will be returned (i.e. notcond
, 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)

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(x + 1)/2  Heaviside(x  1)/(2*x) >>> inverse_mellin_transform(f, s, x, (1, oo)) (x/2 + 1/(2*x))*Heaviside(x + 1)

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)
whereF
is the Laplace transform off
, \(\operatorname{Re}(s) > a\) is the halfplane of convergence, andcond
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()
. Ifnoconds=True
, only \(F\) will be returned (i.e. notcond
, and also not the planea
).>>> 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)

sympy.integrals.transforms.
inverse_laplace_transform
(F, s, t, plane=None, **hints)[source]¶ Compute the inverse Laplace transform of \(F(s)\), defined as
\[f(t) = \frac{1}{2\pi i} \int_{ci\infty}^{c+i\infty} e^{st} F(s) \mathrm{d}s,\]for \(c\) so large that \(F(s)\) has no singularites in the halfplane \(\operatorname{Re}(s) > c\epsilon\).
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 nonnegative \(t\), 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)

sympy.integrals.transforms.
fourier_transform
(f, x, k, **hints)[source]¶ Compute the unitary, ordinaryfrequency 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 defaultnoconds=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.
inverse_fourier_transform
(F, k, x, **hints)[source]¶ Compute the unitary, ordinaryfrequency 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 defaultnoconds=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)

sympy.integrals.transforms.
sine_transform
(f, x, k, **hints)[source]¶ Compute the unitary, ordinaryfrequency 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 defaultnoconds=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**(a + 1/2)*k**(a  1)*gamma(a/2 + 1)/gamma(a/2 + 1/2)

sympy.integrals.transforms.
inverse_sine_transform
(F, k, x, **hints)[source]¶ Compute the unitary, ordinaryfrequency 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 defaultnoconds=True
.>>> from sympy import inverse_sine_transform, exp, sqrt, gamma, pi >>> from sympy.abc import x, k, a >>> inverse_sine_transform(2**((12*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)

sympy.integrals.transforms.
cosine_transform
(f, x, k, **hints)[source]¶ Compute the unitary, ordinaryfrequency 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 defaultnoconds=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))

sympy.integrals.transforms.
inverse_cosine_transform
(F, k, x, **hints)[source]¶ Compute the unitary, ordinaryfrequency 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 defaultnoconds=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)

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 defaultnoconds=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)

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 defaultnoconds=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)
Internals¶
There is a general method for calculating antiderivatives of elementary functions, called the Risch algorithm. 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.
SymPy currently uses a simplified version of the Risch algorithm, called the RischNorman algorithm. This algorithm is much faster, but may fail to find an antiderivative, although it is still very powerful. SymPy also uses pattern matching and heuristics to speed up evaluation of some types of integrals, e.g. polynomials.
For nonelementary definite integrals, SymPy uses socalled Meijer Gfunctions. Details are described here:
API reference¶

sympy.integrals.
integrate
(f, var, ...)[source]¶ Compute definite or indefinite integral of one or more variables using RischNorman 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 nonelementary. 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 rewriting the integrand in terms of socalled Meijer Gfunctions. Indefinite integrals of a single Gfunction can always be computed, and the definite integral of a product of two Gfunctions can be computed from zero to infinity. Various strategies are implemented to rewrite integrands as Gfunctions, 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 Gfunction 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 Gfunction 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 Gfunctions second to last, and heuristic Risch last).
 If still not successful, try Gfunctions irrespective of the limits.
The option meijerg=True, False, None can be used to, respectively: always use Gfunction methods and no others, never use Gfunction methods, or use all available methods (in order as described above). It defaults to None.
See also
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.
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.
See also
Integral.doit
 Perform the integration using any hints
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 lefthand 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 righthand 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 righthand 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
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].

doit
(**hints)[source]¶ Perform the integration using any hints given.
See also
sympy.integrals.trigonometry.trigintegrate
,sympy.integrals.risch.heurisch
,sympy.integrals.rationaltools.ratint
as_sum
 Approximate the integral using a sum
Examples
>>> from sympy import Integral >>> from sympy.abc import x, i >>> Integral(x**i, (i, 1, 3)).doit() Piecewise((x**3/log(x)  x/log(x), (x > 1)  ((x >= 0) & (x < 1))), (2, True))

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.
See also
function
,limits
,variables
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.
References
[R412] https://en.wikipedia.org/wiki/Cauchy_principal_value [R413] http://mathworld.wolfram.com/CauchyPrincipalValue.html 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

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:
 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).
 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 usubstitution.
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.
See also
variables
 Lists the integration variables
as_dummy
 Replace integration variables with dummy ones
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 usubstitution 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 “usubstitution”:
>>> 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, a + 1)) >>> Integral(x, (x, 0, 1)).transform(x, (u + a, a)) Integral(a + u, (a, u, u + 1))

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 GaussLegendre quadrature [R414] points and weights.
The GaussLegendre 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:
\[w_i = \frac{2}{\left(1x_i^2\right) \left(P'_n(x_i)\right)^2}\]Parameters: n : the order of quadrature
n_digits : number of significant digits of the points and weights to return
Returns: (x, w) : the
x
andw
are lists of points and weights as Floats.The points \(x_i\) and weights \(w_i\) are returned as
(x, w)
tuple of lists.See also
gauss_laguerre
,gauss_gen_laguerre
,gauss_hermite
,gauss_chebyshev_t
,gauss_chebyshev_u
,gauss_jacobi
,gauss_lobatto
References
[R414] (1, 2) https://en.wikipedia.org/wiki/Gaussian_quadrature [R415] http://people.sc.fsu.edu/~jburkardt/cpp_src/legendre_rule/legendre_rule.html 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]

sympy.integrals.quadrature.
gauss_laguerre
(n, n_digits)[source]¶ Computes the GaussLaguerre quadrature [R416] points and weights.
The GaussLaguerre 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
andw
are lists of points and weights as Floats.The points \(x_i\) and weights \(w_i\) are returned as
(x, w)
tuple of lists.See also
gauss_legendre
,gauss_gen_laguerre
,gauss_hermite
,gauss_chebyshev_t
,gauss_chebyshev_u
,gauss_jacobi
,gauss_lobatto
References
[R416] (1, 2) https://en.wikipedia.org/wiki/Gauss%E2%80%93Laguerre_quadrature [R417] http://people.sc.fsu.edu/~jburkardt/cpp_src/laguerre_rule/laguerre_rule.html 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, 8.9855e7]

sympy.integrals.quadrature.
gauss_hermite
(n, n_digits)[source]¶ Computes the GaussHermite quadrature [R418] points and weights.
The GaussHermite 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:
\[w_i = \frac{2^{n1} n! \sqrt{\pi}}{n^2 \left(H_{n1}(x_i)\right)^2}\]Parameters: n : the order of quadrature
n_digits : number of significant digits of the points and weights to return
Returns: (x, w) : the
x
andw
are lists of points and weights as Floats.The points \(x_i\) and weights \(w_i\) are returned as
(x, w)
tuple of lists.See also
gauss_legendre
,gauss_laguerre
,gauss_gen_laguerre
,gauss_chebyshev_t
,gauss_chebyshev_u
,gauss_jacobi
,gauss_lobatto
References
[R418] (1, 2) https://en.wikipedia.org/wiki/GaussHermite_Quadrature [R419] http://people.sc.fsu.edu/~jburkardt/cpp_src/hermite_rule/hermite_rule.html [R420] http://people.sc.fsu.edu/~jburkardt/cpp_src/gen_hermite_rule/gen_hermite_rule.html 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]

sympy.integrals.quadrature.
gauss_gen_laguerre
(n, alpha, n_digits)[source]¶ Computes the generalized GaussLaguerre quadrature [R421] points and weights.
The generalized GaussLaguerre 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:
\[w_i = \frac{\Gamma(\alpha+n)} {n \Gamma(n) L^{\alpha}_{n1}(x_i) L^{\alpha+1}_{n1}(x_i)}\]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
andw
are lists of points and weights as Floats.The points \(x_i\) and weights \(w_i\) are returned as
(x, w)
tuple of lists.See also
gauss_legendre
,gauss_laguerre
,gauss_hermite
,gauss_chebyshev_t
,gauss_chebyshev_u
,gauss_jacobi
,gauss_lobatto
References
[R421] (1, 2) https://en.wikipedia.org/wiki/Gauss%E2%80%93Laguerre_quadrature [R422] http://people.sc.fsu.edu/~jburkardt/cpp_src/gen_laguerre_rule/gen_laguerre_rule.html 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]

sympy.integrals.quadrature.
gauss_chebyshev_t
(n, n_digits)[source]¶ Computes the GaussChebyshev quadrature [R423] points and weights of the first kind.
The GaussChebyshev quadrature of the first kind approximates the integral:
\[\int_{1}^{1} \frac{1}{\sqrt{1x^2}} f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i)\]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
andw
are lists of points and weights as Floats.The points \(x_i\) and weights \(w_i\) are returned as
(x, w)
tuple of lists.See also
gauss_legendre
,gauss_laguerre
,gauss_hermite
,gauss_gen_laguerre
,gauss_chebyshev_u
,gauss_jacobi
,gauss_lobatto
References
[R423] (1, 2) https://en.wikipedia.org/wiki/Chebyshev%E2%80%93Gauss_quadrature [R424] http://people.sc.fsu.edu/~jburkardt/cpp_src/chebyshev1_rule/chebyshev1_rule.html 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]

sympy.integrals.quadrature.
gauss_chebyshev_u
(n, n_digits)[source]¶ Computes the GaussChebyshev quadrature [R425] points and weights of the second kind.
The GaussChebyshev quadrature of the second kind approximates the integral:
\[\int_{1}^{1} \sqrt{1x^2} f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i)\]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
andw
are lists of points and weights as Floats.The points \(x_i\) and weights \(w_i\) are returned as
(x, w)
tuple of lists.See also
gauss_legendre
,gauss_laguerre
,gauss_hermite
,gauss_gen_laguerre
,gauss_chebyshev_t
,gauss_jacobi
,gauss_lobatto
References
[R425] (1, 2) https://en.wikipedia.org/wiki/Chebyshev%E2%80%93Gauss_quadrature [R426] http://people.sc.fsu.edu/~jburkardt/cpp_src/chebyshev2_rule/chebyshev2_rule.html 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]

sympy.integrals.quadrature.
gauss_jacobi
(n, alpha, beta, n_digits)[source]¶ Computes the GaussJacobi quadrature [R427] points and weights.
The GaussJacobi quadrature of the first kind approximates the integral:
\[\int_{1}^1 (1x)^\alpha (1+x)^\beta f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i)\]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
andw
are lists of points and weights as Floats.The points \(x_i\) and weights \(w_i\) are returned as
(x, w)
tuple of lists.See also
gauss_legendre
,gauss_laguerre
,gauss_hermite
,gauss_gen_laguerre
,gauss_chebyshev_t
,gauss_chebyshev_u
,gauss_lobatto
References
[R427] (1, 2) https://en.wikipedia.org/wiki/Gauss%E2%80%93Jacobi_quadrature [R428] http://people.sc.fsu.edu/~jburkardt/cpp_src/jacobi_rule/jacobi_rule.html [R429] http://people.sc.fsu.edu/~jburkardt/cpp_src/gegenbauer_rule/gegenbauer_rule.html 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]
Integration over Polytopes¶
The intpoly
module in SymPy implements methods to calculate the integral of a polynomial over 2/3Polytopes.
Uses evaluation techniques as described in Chin et al. (2015) [1].
The input for 2Polytope or Polygon uses the already existing Polygon
data structure in SymPy. See
sympy.geometry.polygon
for how to create a polygon.
For the 3Polytope 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): 967981
PDF link : http://dilbert.engr.ucdavis.edu/~suku/quadrature/clsintegration.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 3Polytopes/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 2/3Polytopes.
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 2Polytope 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}