"I am Back" says Integration Part 8

Calculus Level 5

\[\large \int_{-\infty}^{\infty}{xe^{nx}e^{-e^{2x}} \, dx }\]

For positive constant \(n\), the above integral can be expressed as \[\dfrac{a}{b} \Gamma{\left( \dfrac{n}{c} \right)} \psi{\left( \dfrac{n}{d} \right)}, \]

where \(a,b,c\) and \(d\) are positive integers with \(a,b\) coprime. Find \(a \times b \times c \times d\).

Notations:

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