I am wondering whether the following integral over Haar measure （edit: say U is 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.