It is known that for $SU(N)$ $$ \int \chi_{\mu_1}(UV_1)\chi_{\mu_2}(U^{-1}V_2)\, dU = \delta_{\mu_1\mu_2}\frac{\chi_{\mu_1}(V_1V_2)}{\dim(\mu_1)} $$ where $dU$ is Haar measure on $SU(N)$ normalized such that with respect to it $\operatorname{Vol}(SU(N))=1$; and $\chi_{\mu}(U)$ means the trace of $U$ in the irreducible representation $\mu$.

I came across this integral in "On quantum gauge theories in two dimensions", Edward Witten, Commun. Math. Phys. 141, 153-209 (1991).

My question is simple. What else is known about integrals on $SU(N)$ of the form $$ \int \chi_{\mu_1}(U^{\pm 1}V_1)\dots\chi_{\mu_n}(U^{\pm 1}V_n)\, dU \ ? $$