I have been struggling computing the Fisher's information of the Wishart distribution. I'll write what I have gone through. Let's $\Omega$ denote a $p\times p$ Wishart random variate denoted by $\mathcal{W}(k,V)$ where $k$ is the degrees of freedom and $V$ a positive definite scale matrix. If we write $\mathcal{W}(\Omega\,|\,k,V)$ for the density function, $$ \begin{align} \nabla_{\operatorname{vech}(V)}\log\mathcal{W}(\Omega\,|\,k,V) &= \dfrac{1}{2}D_{p}'\left(V^{-1}\otimes V^{-1}\right)D_{p}\operatorname{vech}(\Omega)-\dfrac{k}{2}D_{p}\operatorname{vech}\left(V^{-1}\right)\\ \nabla_{k}\log\mathcal{W}(\Omega\,|\,k,V) &= \dfrac{1}{2}\log|\Omega|-\dfrac{1}{2}\log|V|-\dfrac{p}{2}\log 2-\dfrac{1}{2}\sum_{i=1}^{p}\psi\left(\dfrac{k+1-i}{2}\right) \end{align} $$ where $D_{p}$ is the unique duplication matrix such that $D_{p}\operatorname{vech}(A)=\operatorname{vec}(A)$, $\otimes$ Kronecker product, and $\psi$ digamma function.
Is there a closed-form expression for the Fisher information matrix of the Wishart distribution?
UPDATE I have computed the following two terms: $$ \begin{align}\operatorname{Var}(\operatorname{vec}(\Omega)) &= k\left(\mathbf{I}_{p^{2}}+K_{pp}\right)(V\otimes V)\\ \operatorname{Var}(\log|\Omega|) &= \sum_{i=1}^{p}\psi_{1}\left(\dfrac{k-i+1}{2}\right) \end{align} $$ where $K_{pp}$ is a $p^{2}\times p^{2}$ commutation matrix such that $$ \begin{equation} K_{pp}\operatorname{vec}(C) = \operatorname{vec}(C') \end{equation} $$ for a $p\times p$ matrix $C$ and $\psi_{1}$ is the trigamma function. But I have no idea how to get the following covariance term. $$ \operatorname{Cov}(\operatorname{vec}(\Omega),\log|\Omega|) $$