Integration failure

[tverticalis]

The following integral should evaluate to 1 (for beta > 0). The function is a probability density function (PDF) of the angular elevation of points distributed on a unit sphere. If beta = 1, the points are uniformly distributed on the sphere. In the general case, Python 3.6.3 does not produce an answer in a reasonable amount of time (< 30 min):

from sympy import Symbol, integrate, Rational, Pow, exp, sin, cos, pi
beta = Symbol('beta', positive = True)
theta = Symbol('theta')
res = integrate(beta*sin(theta)/(2*Pow(1+(beta**2-1)*(cos(theta))**2, Rational(3,2))), (theta,0,pi))

(I am not sure why line breaks are not formatted correctly in the Preview.)

It works for beta = 1 (which greatly simplifies the expression) but again fails for beta = 2.

For comparison, Mathematica 11.2 produces the expected result (1) in less than 0.3 s on the same system:

Assuming[beta > 0, Integrate[beta*Sin[theta]/(2*(1 + (beta^2 - 1)*Cos[theta]^2)^(3/2)), {theta, 0, Pi}]] // Timing

In all other respects, Python did as well as Mathematica.

Thank you.

[ylemkimon]

You can use triple backticks(```) for multi-line codes.