Integral of product of matrix exponentials

Question

Is there an analytic expression for

$$ \int_{0}^{\infty} e^{-At}e^{-Bt} dt $$ where $A$ and $B$ are non-commuting matrices?

Answer 1

There is no general analytic expression. However, the integral (when it exists) satisfies Sylvester's equation, $$ AX + XB = \text{Id}, $$ and can be computed using specialized algorithms, such as the Bartels–Stewart algorithm.

Surprisingly, neither of these links mentions the connection to exponential integrals.