recently we developed a general framework to compute asymptotic expansions of certain quantities coming from random matrix theory. more precisely if one considers the expectation of the trace of a sufficiently smooth function evaluated in a random matrix, one can compute a taylor expansion (in the dimension of our random matrix) of this quantity. this method relies notably on free stochastic calculus whom i will briefly talk about. in a previous work we studied the case of gue random matrices, in this talk we consider polynomials in independent haar unitary matrices. i will explain the additional difficulties that this model brings then give a few applications of this result to random matrix theory as well as links with weingarten calculus.