By considering a rectangular contour, show that for $0<a<1$ we have

$$\int_{-\infty}^{\infty} \frac{e^{a x}}{e^{x}+1} d x=\frac{\pi}{\sin \pi a}$$

Hence evaluate

$$\int_{0}^{\infty} \frac{d t}{t^{5 / 6}(1+t)}$$