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, you can get it at 
http://arxiv.org/abs/1402.3830

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