calculus - Prove that $int_0^inftyfrac1{t^x+1}dt = fracpi x csc fracpi x$

I'm stuck on this identity:

$$
\int_0^\infty\frac1{t^x+1}dt = \frac\pi x \csc \frac\pi x
$$

Could someone show me a proof for this?

What I've tried:

I've thrown a bunch of substitutions and integration by parts at this, but they haven't led me to the answer. I did, however, find these identites:
$$
\int_0^\infty\frac1{t^x+1}dt = \int_0^\infty\frac{t^{x-2}}{t^x+1}dt = \int_0^1\left( \frac{1-t}t \right)^{1/x}dt
$$
But none of these seem to lead anywhere helpful.

I also tried introducing another variable to turn it into something similar to the Laplace Transform, but I'm not very familiar with methods like that, so they've led nowhere.