Integration of invariant matrices and moments of inverses of Ginibre and Wishart matrices

We consider random matrices that have invariance properties under the action of unitary groups (either a left-right invariance, or a conjugacy invariance), and we give formulas for moments in terms of functions of eigenvalues. Our main tool is the Weingarten calculus. As an application, we obtain new formulas for the pseudoinverse of Gaussian matrices and for the inverse of compound Wishart matrices.