Integrating a multidimensional integral in scipy

Question

Motivation: I have a multidimensional integral, which for completeness I have reproduced below. It comes from the computation of the second virial coefficie

Answer 1

Jonathan Dursi has made a very good answer. I will just add on to his answer.

Now scipy.integrate has a function named nquad that one can perform a multi-dimensional integral without hassle. See this link for more information. Below we compute the integral using nquad with Jonathan's example:

from scipy import integrate
import math
import numpy as np

def w(r, theta, phi, alpha, beta, gamma):
    return(-math.log(theta * beta))

def integrand(r, theta, phi, alpha, beta, gamma):
    ww = w(r, theta, phi, alpha, beta, gamma)
    k = 1.
    T = 1.
    return (math.exp(-ww/(k*T)) - 1.)*r*r*math.sin(beta)*math.sin(theta)

result, error = integrate.nquad(integrand, [[0, 1],             # r
                                            [0, 2 * math.pi],   # theta
                                            [0, math.pi],       # phi
                                            [0, 2 * math.pi],   # alpha
                                            [0, 2 * math.pi],   # beta
                                            [0, 2 * math.pi]])  # gamma
expected = 16*math.pow(math.pi,5)/3
diff = abs(result - expected)

The result is more accurate than the chained tplquad:

>>> print(diff)
0.0