The following integral cannot be evaluated by maxima:

sage: integrate(1/sqrt(1+x^3),x)
integrate(1/sqrt(x^3 + 1), x)

With 'sympy' algorithm the computation fails:

sage: integrate(1/sqrt(1+x^3),x,algorithm='sympy')
...
AttributeError: 'gamma' object has no attribute '_sage_'

However, SymPy can compute the integral and gives the result in terms of gamma and hypergeometric functions:

sage: import sympy
sage: sympy.integrate(1/sqrt(1+x**3))
x*gamma(1/3)*hyper((1/3, 1/2), (4/3,), x**3*exp_polar(I*pi))/(3*gamma(4/3))

It can be seen that not only gamma is a problem (already fixed in sympy master) but also exp_polar which Sage does not know.

This ticket should track the status of the Sympy pull request fixing the exp_polar issue, and it should implement a skeleton exp_polar on the Sage side.

Depends on #20185

CC: @videlec @staroste @kcrisman

Component: symbolics

Keywords: sd66

Author: Tomáš Kalvoda

Branch/Commit: 6d3aeaa

Reviewer: Ralf Stephan

Issue created by migration from https://trac.sagemath.org/ticket/18085