π\pi and Gundermann (happy π\pi day, Gauss bell)

Calculus Level 4

Evaluate: eaxex+1dx\displaystyle \int_{-\infty}^{\infty} \frac{e^{ax}}{e^x + 1} \,dx when a=12a = \frac{1}{2}

Bonus.- Find a beautiful closed form for the above integral being 0<a<10 < a < 1

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