While investigating a problem in acoustic scattering in bounded domains, I encountered the following integral: $$\int_{-1}^{1}\frac{\text{P}_n(x)\text{P}_m(x)}{\sqrt{1-x^2}}\mathrm{d}x$$ Where $\text{P}_n(x)$ is the nth order Legendre polynomial of the first kind. Is there a neat closed-form solution (or approximation) to this integral?
I know one can replace each Legendre polynomial with a finite sum of Chebyshev polynomials using Gegenbauer function, but that still produces a lengthy sum of fractions which is too cumbersome to be used in further calculations.
Thank you for your help!