I was wondering if there is a way to solve for the overlapping volume between a cone and sphere where the cone's axis isn't necessarily aligned with the sphere's. Here's an image showing a possible case of overlapping volumes (source). I know that the overlap becomes simpler in the case of an aligned axis (e.g. Finding the volume between a cone and a sphere) but I'm interested in the more general case.
Let the cone have an opening angle $\alpha$, height $h$, and the sphere's radius is $R$. I know we could work in spherical coordinates and try to integrate over the overlapping region between the two shapes to solve for the volume. But the curve of intersection is quite complicated so I'd expect that such an integral can only be solved numerically for fixed values of $\alpha$, $h$, and $R$. I was wondering if such an integral could be solved without resorting to numerics or if there is an alternative way to solve this problem?