  We consider the problem of computing the integral $$
  \int_{\mathcal{U}(d)} u_{i_1j_1}\cdots u_{i_nj_n} \bar{u}_{i'_1j'_1} \cdots
\bar{u}_{i'_{n'}j'_{n'}} dU, $$ where the integration takes place with respect
to the probability Haar measure on the unitary group $\mathcal{U}(d)$, and the
$u_{ij}$ denotes the $ij$-th entry of a unitary matrix $U$. We present a
unified approach connecting classical results, the explicit formula for the
integral given by B. Collins and P. Sniady and subsequent works of various
authors providing different points of view. Finally we are able to provide an
explicit formula for the $2n$-th moment of the trace of a unitary Haar random
matrix, generalizing a result of P. Diaconis.
