While I was playing with Wolfram Alpha calculator I wondered if it is known a closed-form for $$\int_0^\infty\arctan\left(\frac{1}{\sinh^2 x}\right)dx.\tag{1}$$
Wolfram Alpha provide me the coressponding indefinite integral using this code
int arctan(1/sinh^2(x))dx
but it seems like as science fiction that I can to understand what did this CAS since the integral is very difficult.
Question. Can you provide me an idea to get such indefinite integral $$\int\arctan\left(\frac{1}{\sinh^2 x}\right)dx?$$ Of course, if it is a known integral and closed-form $(1)$ answer as a reference request, refering the literature and I try to find and read such exercise from the literature. Thanks you in advance.