Let $X$ be a standard Wishart matrix, i.e., $$ X = \sum_{j=1}^n g_j \otimes g_j \quad \mbox{where} \quad g_j \sim N(0, I_d). $$ Above, $g_j$ are independent samples from the standard multivariate Gaussian.
Is it possible to calculate the expected value of $$ \mathrm{tr}((X + D)^{-1}), $$ where $D$ is a diagonal matrix with positive diagonal entries?
