I have proved this equality by means of Cauchy’s Theorem
applied to an adequate function. Since my solution is too long to post it
here, I posted it in arXiv:

* Juan Arias de Reyna, _Computation of a Definite Integral_, arXiv:[1402.3830](http://arxiv.org/abs/1402.3830).

The function 
$$G(z)=\frac{\log(1+(1+i)\,f(z)\,)}z$$
where 
$$f(x)=\frac{\operatorname{arctanh}(x)-\arctan(x)}{\pi}$$
extended analytically.