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I am wondering whether the following integral over Haar measure has explicit form(edit: say $U$ is $d\times d$ unitary, orthogonal or symplectic matrix) $$ \int dU [(U\otimes U^*)X(U^{+}\otimes U^T)]^{\otimes n} $$ where $X$ is a given $d^2\times d^2$ Hermitian matrix, $U^{*}$ is the element-by-element complex conjugation of $U$, $U^{+}$ is the Hermitian transpose of $U$, $U^T$ is the transpose of $U$, and $n,d$ are known integers.

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