I want to prove the matrix logarithmic mean in differential form. $ L(A, B)^{-1}=\int_{0}^{\infty} \frac{(t A+B)^{-1}}{t+1} d t $ for positive definite matrices $A, B$.
I know $L(A, B)^{-1}=\frac{\log A-\log B}{A-B}=\int_{0}^{\infty} \frac{1}{(A+t)(B+t)} d t$
and use the property $(A+t T)^{-1}=A^{-1 / 2}\left(I+t A^{-1 / 2} T A^{-1 / 2}\right)^{-1} A^{-1 / 2}$ to expand the two equations.
However, I cannot prove the equation. Could everyone give some other hints to solve the problem?
Thanks!!!