I see somewhere that the value of the integral $$ I(\alpha)=\int_0^\infty \frac{1}{(1+x^2)(1+x^\alpha)} \mathbb dx,\alpha\in \mathbb R^+ $$ does not depend on $\alpha$. So $I=\pi/4$ for all $\alpha$. How can I prove this?
I tackled the special case $\alpha=2$ quite easily by contour integration, but find it quite awkward to deal with general $\alpha$. Contour integration doesn't seem to work for $\alpha=\sqrt 2$, for example.
I tried $dI/d\alpha$ and $I(\alpha)-I(\beta)$, but both lead to very complicated functions that don't lead to anywhere.
