# An “Itzykson-Zuber”-like integral

I been told that there exists an integration formula, which states (or something of this sort) $$\int_{U(N)} dU \det[(\mathbb I+XUYU^{-1})^{-r}\propto \frac{\det(1+x_iy_j)^{N-r-1}]_{i,j}}{\Delta_N(x)\Delta_N(y)},$$ where $dU$ is the Haar-measure, $X=\text{diag}(x_1,\ldots,x_N)$ is a positive definite diagonal matrix with $x_i\neq x_j$ for $i\neq j$ (similar for $Y$), and $$\Delta_N(x)=\prod_{1\leq i<j\leq N}(x_i-x_j)$$ is a Vandermonde determinant.

This is somewhat similar to a Harish-Chandra-Itzykson-Zuber integral, but I haven't seen this specific kind before. Are anyone familiar with this kind of integrals, and how to solve them? I would be particular interested in references to the literature.

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