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](https://www.jstage.jst.go.jp/article/iis/16/2/16_2_139/_pdf) (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: $\int\_{0}^{1}M^{t}YM^{1-t}\mathrm{d}t = \log M$. 



Closed form for integral of function of a symmetric positive definite matrix
just go to a basis where $M$ is diagonal, evaluate the integral and find $Y=M^{-1}\log M$.


