Question 1:
I find the following integral formula from "Table of integrals" but I do not
know how to derive it.
<br /> \int^{\infty}_{0} \frac{\ln (x)}{\cosh^2(x)} dx = \psi(\frac{1}{2}) + \ln(\pi),<br />
where \psi is the Digamma function.

Question 2:
How to get the answer for the slightly modifed integral:
<br /> \int^{\infty}_{0} \frac{\ln (x)}{\cosh^2(x+a)} dx = ?<br />