I encountered the following integral in the physics literature
$$ \int_{-1}^{1}P_{\ell}^m(x)^2P_{n}^\prime(x){\rm d}x $$
where $P_{\ell}^m(x)$ is an associated Legendre polynomial of degree $\ell$ and order $m$, $P_n(x)$ is a Legendre polynomial of degree $n$ and $^\prime$ denotes derivatives with respect to $x$. Any idea how to proceed?
One thing I tried was to decompose $P_{\ell}^m(x)^2$ as a sum of Legendre polynomials (using Gaunt's formula), and the resulting integral only involves the sum of products of Legendre polynomials and derivatives of Legendre polynomials. The final expression can then be written as a sum of products of 3$-J$ symbols. However, I don't find this very elegant, as I was expecting the integral to be expressed as a product of 3$-J$ symbols, and not as a sum of products of 3$-J$ symbols.
