We consider integrals on unitary groups U d of the form $\int_{{\text{?U}}_d}U_{i_1\text{?}j_1}\cdots U_{i_q\text{?}j_q}U_{{j^{'}}_1{i^{'}}_1}^\ast\cdots U_{{j^{'}}_{q^{'}}{i^{'}}_{q^{'}}}^{\ast} dU$ . We give an explicit formula in terms of characters of symmetric groups and Schur functions, which allows to rederive an asymptotic expansion as d → ∞. Using this, we rederive and strengthen a result of asymptotic freeness due to Voiculescu. We then study large d asymptotics of matrix-model integrals and of the logarithm of Itzykson-Zuber integrals and show that they converge towards a limit when considered as power series. In particular, we give an explicit formula for ${\text{lim}}_{d\rightarrow\infty}{(\partial^n/\partial z^n\text{?})}d^{-2}\text{?log}\int_{{\text{U}}_d}e^{zd\text{?Tr}{(XUYU^\ast)}}dU\vert_{z=0}$ , assuming that the normalized traces d −1 Tr(X k ) and d −1 Tr (Y k ) converge in the large d limit. We consider as well a different scaling and relate its asymptotics to Voiculescu's R-transform.