Contour integral of $\int_{-\infty}^\infty \frac{e^{ax}}{1+e^x}dx $ with non rectangular contour Is there a way to solve the integral of $$\int_{-\infty}^\infty \frac{e^{ax}}{1+e^x}dx $$ for $$a\in (0,1)$$
without using the rectangular region like in this post but still using a contour integral?
Perhaps using a semicircular region, circular region, or freshnel contour perhaps? I just don't have a lot of experience with the rectangular region problems. 
Thanks.

(Not using contour integration ; Sorry )
$$I=\int_{-\infty}^{\infty} \frac {e^{ax}}{1+e^x} dx=\int_{-\infty}^{\infty} \frac {e^x\cdot e^{ax}}{e^x+e^{2x}} dx$$
Use the substitution $e^x=t$
$$I=\int_{0}^{\infty} \frac {t^{a-1}}{1+t}dt =B(a,1-a)=\Gamma(a)\Gamma(1-a)=\frac {\pi}{\sin (\pi a)}$$
