The two integrals in the attached notebook are algebraically identical but produce different results. I’d like to understand why, and whether there is something I can do to consistently obtain the nonzero result.
As far as I can tell, that’s a bug.
I’ve encountered the situation before where the order of integration greatly effects the results. You might find this post to be interesting reading
Note that there is a singularity for theta=phi=0. This can cause trouble.