I came across an integral involving a product of random matrices that I'm unable to evaluate, and I'm curious about whether or not there's a known solution. For context, I started with the following integral of 3 random variables $X,Y,Z\sim\mathcal{N}(0,1)$:
$I = \int_{\mathbb{R}^3} \frac{dX}{\sqrt{2\pi}}\frac{dY}{\sqrt{2\pi}}\frac{dZ}{\sqrt{2\pi}}\mathrm{exp}\left(-\frac{X^2}{2} - \frac{Y^2}{2}-\frac{Z^2}{2}-cXYZ\right)$
This isn't super hard; we can just complete the square in $X$:
$I = \int_{\mathbb{R}^3} \frac{dX}{\sqrt{2\pi}}\frac{dY}{\sqrt{2\pi}}\frac{dZ}{\sqrt{2\pi}}\mathrm{exp}\left(-\frac{1}{2}\left(X+cYZ\right)^2 + \frac{c^2Y^2 Z^2}{2} - \frac{Y^2}{2}-\frac{Z^2}{2}\right)$
which evaluates to
$\int_{\mathbb{R}^2} \frac{dY}{\sqrt{2\pi}}\frac{dZ}{\sqrt{2\pi}}\mathrm{exp}\left(-\frac{1}{2}\left[1-\frac{c^2Z^2}{2}\right]Y^2 - \frac{Z^2}{2}\right) = \int_{\mathbb{R}} \frac{dZ}{\sqrt{2\pi}} \frac{1}{\sqrt{1-\frac{c^2Z^2}{2}}}\mathrm{exp}\left(-\frac{Z^2}{2}\right)$
Mathematica tells me that this has a solution in terms of a Bessel function:
$I = \frac{e^{-\frac{1}{2 c^2}} K_0\left(-\frac{1}{2 c^2}\right)}{\sqrt{-\pi c^2}}$
What I'm really interested in is the matrix form of this, something like
$\int\left[\prod_{i,j}\frac{dX_{ij}}{\sqrt{2\pi}}\right]\left[\prod_{k,\ell}\frac{dY_{k\ell}}{\sqrt{2\pi }}\right]\left[\prod_{m,n}\frac{dZ_{mn}}{\sqrt{2\pi }}\right]\mathrm{exp}\Bigg(-\frac{1}{2}\sum_{i,j}X_{ij}^2-\frac{1}{2}\sum_{k,\ell}Y_{k\ell}^2-\frac{1}{2}\sum_{m,n}Z_{mn}^2 +\sum_{i,j,k,\ell,m,n}Q_{ij}K_{kl}V_{mn}T_{ijk\ell mn}\Bigg)$
where each matrix element $X_{ij},Y_{ij},Z_{ij}\sim\mathcal{N}(0,1)$.
Again we can complete the square and integrate over $X$ and $Y$, but this leaves (roughly)
$\int\left[\prod_{i,j}\frac{dZ_{ij}}{\sqrt{2\pi }}\right]\frac{1}{\sqrt{\mathrm{det}\left(\mathbf{1}+M\right)}}\mathrm{exp}\left(-\frac{1}{2}\sum_{i,j}Z_{ij}^2\right)$
where $M$ is the matrix with components $M_{ij} = \sum_{p,q,r,s}Z_{pq}Z_{rs}T_{ijpqrs}$. This doesn't seem integrable in any nice way.
Are there any known solutions for this integral? Or even known (analytic, not numerical) approximations? Are there known results for $N$ matrices, not just 3?