I am looking for families of invariant integrals $\int_G dg f(g)$ (where $dg$ is a Haar measure) over a semisimple Lie group that can be evaluated in closed form, together with references where I can find proofs.

I don't know yet what I'll need, but $f$ can be quite involved, so I want to study the techniques that are available for deriving such formulas. My primary interest is in $SU(n)$ for general $n$, but results on other groups are likely to be instructive, too.