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$I(4)=\frac{1}{2}\log(2)$.

i think the OP wants to evaluate it 
By using Evaluate the integral $\int_{0}^{\infty} \frac{1}{(1+x^2)\cosh{(ax)}}dx$
$$ I = \int_0^{\infty} \frac{dx}{\cosh(ax)(1+x^2)} $$
$$ u = bx \Rightarrow I = \frac{1}{b}\int_0^{\infty} \frac{dx}{\cosh \left(\frac{a}{b}x \right)\left( 1 + \frac{x^2}{b^2} \right)} $$
put $ \frac{a}{b} = \pi $
