Does anyone know how to evaluate integrals of them following form: $$\int_{0}^{\pi} \int_{0}^{2 \pi} (\cos{\phi})^{n_1} (\sin{\phi})^{n_2} (\cos{\theta})^{n_3} (\sin{\theta})^{n_4}Y^m_l Y^p_q Y^i_j \sin{\theta}\; d\phi\; d\theta.$$
Where $Y_l^m$ is the spherical harmonic of degree $l$ and order $m$. The symbols $n_1$, $n_2$, $n_3$, $n_4$ $l$, $m$, $q$, $p$, $i$, and $j$ are integers. For any given combination of these integers it is not hard to evaluate the integral, but I need to find all results for a large number of different values of these integers. So many in fact that even tools like Mathematica become to slow.
My current idea is to make some kind of selection criteria for when the integral should be zero, based on symmetry considerations, and then evaluate the non-zero integrals by help of Mathematica. However, if anyone knows any result that would simplify my efforts that would be great.