TITLE: $\int_0^\infty(\log x)^2(\mathrm{sech}\,x)^2\mathrm dx$
QUESTION [18 upvotes]: Is there any closed-form representation for the following integral?
$$\int_0^\infty(\log x)^2(\mathrm{sech}\,x)^2\mathrm dx,$$
where $\mathrm{sech}\,x$ is the hyperbolic secant, $\mathrm{sech}\,x=\dfrac{2}{e^x+e^{-x}}$.

REPLY [12 votes]: Hint: Use the integral:
$$\int_0^\infty x^n\,\text{sech}^2x\,\mathrm dx=(2^{1-n}-4^{1-n})\,\Gamma(n+1)\,\zeta(n)$$
and take the 2nd derivative with respect to $n$.