I would like to integrate:

$$ \int_{0}^{\infty} \left[{\tanh\left(x\right) \over x^{3}}-{\operatorname{sech}\left(x\right) \over x^{2}}\right] \mathrm{d}x $$ I'm not sure where I found this integral, but I have a feeling I wrote it down because its solution. I want to say it's related to the Zeta Function, but I'm not sure. I've managed to rewrite it as: $$\small \sum_{n = 1}^{\infty}\!\!\left(-1\right)^{n + 1}\!\! \left[\!2\ln\left(\!2n - 1 \over n - 1\!\right)\! +\! 4n\ln\left(\!n - 1 \over 2n - 1\!\right)\! +\! 2n^{2}\ln\left(\!n \over n - 1\!\right)\! +\! 4n\ln\left(2\right)\! -\! 2n\! -\! 2\ln\left(2\right)\! +\! 1\!\right] $$

Above follows by writing the hyperbolic functions in terms of exponential functions and then using series. Then I used differentiating under the integral.

This makes me think otherwise about the Zeta Function/having a closed form for the original integral. I would appreciate any help in solving this.