Nice problem of parametric integral Evaluate
$$I(a)=\int\limits_0^{\infty}\frac{1}{{\cosh}({\pi}x)(1+ax^2)}dx\;,\;\;\text{with}\;\;a>0$$
For example, if $a=4$,then$I(4)=\frac{1}{2}\log(2)$.

The integrand is even. Switch the integral to $\left(-\infty,\infty\right)$. Use complex integration $\left(~\mbox{Residue}\ \mbox{Theorem}~\right)$. Poles of $\cosh\left(\pi x\right)$ are at
${\rm i}\left(n + 1/2\right)$. $n \in {\mathbb Z}$. The "other piece" has poles at $z = \pm{\rm i}a^{-1/2}$. That's all we need.
