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.