I wish to calculate the mean log determinant of a random Gaussian matrix, defined by the following definite integral (for $a, b, \sigma\in\mathbb{R}$): $$ I(a, b, \sigma) = \int \frac{d^{n \times n}X}{\sqrt{2\pi}^{n^2}} e^{-\sum_{i,j}^nX_{ij}^2/2\sigma^2-\frac{1}{2}\log\det(aX+bI)} $$
Bonus: allow for a linear term, defined by $A\in\mathbb{R}^{n\times n}$: $$ I = \int \frac{d^{n \times n}X}{\sqrt{2\pi}^{n^2}} e^{-\sum_{i,j}^nX_{ij}^2/2\sigma^2+\sum_{i,j}^nA_{ij}X_{ij}-\frac{1}{2}\log\det(aX+bI)} $$
Random matrices: Law of the determinant provides a CLT-like result for the distribution of the log determinant of X when $n\to\infty$ and assuming $\sigma=1, A=0, b=0, a=1$.