Fransén–Robinson constant

The Fransén–Robinson constant is defined to be the number $F$ given by the formula $$F = \displaystyle\int_0^{\infty} \dfrac{1}{\Gamma(x)} dx,$$ where $\dfrac{1}{\Gamma}$ denotes the reciprocal gamma function.

Proposition (Relation to $e$ and $\pi$): $F=e+\displaystyle\int_0^{\infty} \dfrac{e^{-x}}{\pi^2+\log(x)^2}.$

Proof: proof goes here █