I need to calculate the following integral:
$$\int_{\theta, \phi \in S^2} \left [ P_\ell(1-2\sin ^2\theta \sin^2\phi) \right ]^2 \sin\theta\, d\theta\, d\phi$$
where $S^2$ represents the unit sphere and $P_\ell$ is the $\ell$th order Legendre polynomial. The most important goal is to try to find an asymptotic behaviour of this expression with large $\ell$.