# Closed form for ∫∞0xn1+xmdx \int_0^\infty {\frac{{{x^n}}}{{1 + {x^m}}}dx }

I’ve been looking at

It seems that it always evaluates in terms of $\sin X$ and $\pi$, where $X$ is to be determined. For example:

So I guess there must be a closed form – the use of $\Gamma(x)\Gamma(1-x)$ first comess to my mind because of the $\dfrac{{\pi x}}{{\sin \pi x}}$ appearing. Note that the arguments are always the ratio of the exponents, like $\dfrac{1}{4}$, $\dfrac{1}{3}$ and $\dfrac{2}{5}$. Is there any way of finding it? I’ll work on it and update with any ideas.

UPDATE:

The integral reduces to finding

With $a =\dfrac{n+1}{m}$ which converges only if

Using series I find the solution is

Let us first assume that $0 < \mu < \nu$ so that the integral
converges absolutely. By the substitution $x = \tan^{2/\nu} \theta$, we have