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

The above Integral can be expressed as \[-\dfrac{\gamma}{a}, \]

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

Find \( a\).

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