2-sided Laplace transform of $\exp(-(t + e^{-t}))$

Question

I'm having trouble finding an analytic solution to the 2-sided Laplace transform of;

$$f(t) = \exp(-(t + e^{-t}))$$ Integration by parts doesn't seem to help. Any pointers appreciated. It seems like there should be a solution since the function is well-behaved (approaches $0$ for large $t$ in either direction).

Answer 1

This the double-sided Laplace transform of a Gumbel probability density function (pdf): $$\int_{-\infty}^\infty dt\ e^{-st} e^{-t} e^{-e^{-t}}\ . $$ With the substitution $e^{-t}=\tau$ it becomes $$ \int_0^\infty \frac{d\tau}{\tau}\tau^{s+1}e^{-\tau}=\Gamma(s+1),\qquad\mbox{for}\quad s>-1\ . $$