TITLE: Closed form of $\int_0^{\infty} \frac{\log(x)}{\cosh(x) \sec(x)- \tan(x)} \ dx$
QUESTION [16 upvotes]: What real analysis tools would you recommend me for getting the closed form of the integral below?
$$\int_0^{\infty} \frac{\log(x)}{\cosh(x) \sec(x)- \tan(x)} \ dx$$

REPLY [5 votes]: $$\int_0^{\infty} \frac{\log(x)}{\cosh(x) \sec(x)- \tan(x)} \ dx=\frac{\pi}{4}\log\left(\displaystyle \frac{4 \pi^3}{\Gamma\left(\displaystyle \frac{1}{4}\right)^4}\right)-\left(\frac{\pi}{4}\right)^2-\log^2(2)$$