Given a $N \times M$ matrix $X$ comprised of standard normal entries ($M > N$), I'm interested in approximating $E[trace((XX^T\frac{\gamma}{M} + I)^{-1}]$ in terms of $N, M$ and $\gamma$. Unfortunately, I can't necessarily assume $\gamma$ is small. I've had no luck in coming up with any kind of approximation. Thanks!

For context, this problem relates to the effective degrees of freedom for ridge regression.