integration - Calculus Question: Improper integral $displaystyleint_{-infty}^{infty} x^{2}e^{x-e^{2x}}dx$

I am curious about evaluation of the following integral

$$\int_{-\infty}^{\infty} x^{2}e^{x-e^{2x}}dx$$
Is it possible to evaluate it? This not my homework but I will share my attempt. I tried standard technique, integration by part but without any success. I also couldn't find any suitable substitution. The integral seems as if it were evaluating the expected value or moment generating function of a certain distribution but I couldn't find any pdf like the integrand in my textbook table.

Answer

Hint: Show that $\displaystyle\int_{-\infty}^\infty e^{ax-e^x}~dx=\Gamma(a).~$ Then, after substituting $x=2t$, differentiate twice with

regard to a, and let $a=\dfrac12$.