While seeing this post, the following integral is just struck me
\begin{equation} \int_0^\infty \frac{dx}{(1+x^2)(1+\tan x)}\tag1 \end{equation}
I have tried like what user @OlivierOloa did in his answer, but no luck with finding its closed-form. Using substitution $x\mapsto\tan x$ didn't make it any easier either.
\begin{equation} \int_0^{\pi/2}\frac{dx}{1+\tan (\tan x)}\tag2 \end{equation}
I reached a dead end in the attempt. How does one evaluate the integral $(1)$ or $(2)$?