Integration over matrix spaces with unique invariant measures

We present a method to calculate integrals over monomials of matrix elements with invariant measures in terms of Wick contractions. The method gives exact results for monomials of low order. For higher-order monomials, it leads to an error of order 1/N, where N is the dimension of the matrix and where α is independent of the degree of the monomial. We give a lower bound on the integer and show how α can be increased systematically. The method is particularly suited for symbolic computer calculation. Explicit results are given for O(N), U(N), and for the circular orthogonal ensemble