calculus - Evaluating $intlimits_0^infty{frac{1}{1+x^2+x^alpha}dx}$

I'm trying to evaluate$$f(\alpha)=\int\limits_0^\infty{\frac{1}{1+x^2+x^\alpha}dx}$$
I proved:
$f(\alpha)$ converges when $\alpha\in\mathbb{R}$
$f(2-\alpha)=f(\alpha)$
$f(0)=f(2)=\frac{\pi}{2\sqrt{2}}$
$f(1)=\frac{2\pi}{3\sqrt{3}}$
$f(-\infty)=f(\infty)=\frac{\pi}{4}$
Similar question:$$\int\limits_0^\infty{\frac{1}{1+x^\alpha}dx}=\frac{\pi}{\alpha}\csc\frac{\pi}{\alpha}$$
I tried all of the techniques can be used in evaluating this integral, but I still cannot get the answer.
When I was using complex analysis, I found that the poles of $\frac{1}{1+x^2+x^\alpha}$ is hard to be found.