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.