Is there a closed form for the integral $$\int_0^{\pi/2}(P_\nu^\mu(\cos\theta))^2\,\mathrm d\theta,\quad\mu>\nu\gt-\frac12$$ where $P_\nu^\mu(x)$ is the associated Legendre function of the first kind?

I encountered this integral while trying to derive explicit solutions for a certain Sturm-Liouville problem. I am primarily interested in $\mu,\nu$ being nonnegative integers, but a result that is valid for real $\mu,\nu$ (subject to the above restriction) is very much welcome.

Neither Maple nor Mathematica seem to be able to make a dent on this integral, but I was able to at least confirm that for $\mu,\nu$ an integer, I get results that are rational multiples of $\pi$, which makes me believe there ought to be a (simple?) closed form, perhaps involving gamma functions.

I wasn't able to find anything in G&R or the DLMF that resembles this integral, so I am really stuck, and would appreciate any ideas on resolving this.