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.



explicit integrals over a Lie group
[Integration with respect to the Haar measure on unitary, orthogonal and symplectic group](http://arxiv.org/abs/math-ph/0402073) will tell you pretty much all we know for integrals of polynomial functions. (For a more recent paper, see [Elementary derivation of Weingarten functions of classical Lie groups](http://arxiv.org/abs/1406.2182).)


There is not much more. Regrettably, even the integral over U(*N*) of a rational function has no closed-form expression.


