"I am Back" says Integration Part 7

Calculus Level 5

\[\large \int_{-\infty}^{\infty} {xe^{3x}e^{-{e}^{2x}} \, dx}\]

If the above Integral can be expressed as \[\dfrac{{\pi}^{A/2}}{B} \left( C - \gamma - D\ln(E)\right),\]

where \(A,B,C,D, E\) are positive integers, with \(E\) prime, find \(A+B+C+D+E\).

Notation: \(\gamma\) denotes the Euler-Mascheroni constant, \( \displaystyle \gamma = \lim_{n\to\infty} \left( - \ln n + \sum_{k=1}^n \dfrac1k \right) \approx 0.5772 \).

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