TITLE: Integral of $\ln(\tanh(x))$
QUESTION [8 upvotes]: I'd like a hint toward how I could evaluate this definite integral. I'm aware it's likely to be non elementary and I haven't found a way to evaluate it yet:$$\int_0^\infty \ln(\tanh(x))\,\,\mathrm{d}x$$
If you're curious where this came from, I was looking at an integral involving $\ln(\sin(x))$ and I thought of this one.
Thanks.

REPLY [2 votes]: Another option: $$\ln\tanh x=-2\operatorname{artanh}\exp -2x=-2\sum_{k\ge 0}\frac{\exp -(4k+2)x}{2k+1},$$so$$\int_0^\infty\ln\tanh x dx=-\sum_{k\ge 0}\frac{1}{(2k+1)^2}=-\frac{\pi^2}{8}.$$