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

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
$$
\int_0^\infty \frac{d\tau}{\tau}\tau^{s+1}e^{-\tau}=\Gamma(s+1),\qquad\mbox{for}\quad s>-1\ .
$$
