Let $M$ be a real symmetric positive definite matrix of size $n \times n$, and let $\log M$ denote its (principal) matrix logarithm.

Is it possible to evaluate the following integral in closed form?

$$Y = \displaystyle\int_{0}^{\infty} \left(M + tI\right)^{-1} (\log M) \left(M + tI\right)^{-1}\:\mathrm{d}t.$$

Not sure if it helps in evaluation, but the above integral has two other representations: $$Y = \displaystyle\int_{-\infty}^{\infty}M^{js}M^{-1/2}(\log M) M^{-1/2} M^{-js} \displaystyle\frac{\pi}{2\cosh^{2}(\pi s)}\:\mathrm{d}s,\\ = \displaystyle\int_{0}^{\infty} \displaystyle\int_{0}^{\infty}\exp(-sM)(\log M)\exp(-tM)\displaystyle\:\frac{\mathrm{d}s\:\mathrm{d}t}{s+t},$$ where $j=\sqrt{-1}$. That the three integral representations are equivalent can be found here (see eqns. (5.4.5), (5.4.8) and the last equation before Appendix with $H=K\equiv M$ and $X\equiv\log M$).

Motivation: the above formulas give the solution of the matrix equation: $\int_{0}^{1}M^{t}YM^{1-t}\mathrm{d}t = \log M$.