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We present a method to calculate integrals over monomials of matrix elementswith invariant measures in terms of Wick contractions. The method gives exactresults for monomials of low order. For higher--order monomials, it leads to anerror of order 1/N^alpha where N is the dimension of the matrix and where alphais independent of the degree of the monomial. We give a lower bound on theinteger alpha and show how alpha can be increased systematically. The method isparticularly suited for symbolic computer calculation. Explicit results aregiven for O(N), U(N) and for the circular orthogonal ensemble.Comment: 12 pages in revtex, no figure

Integration over matrix spaces with unique invariant measures