integration - Prove $gamma = frac{1}{2} + 2 cdot int_0^infty frac{sin(arctan(x))}{(e^{2 pi x} - 1) cdot sqrt{1 + x^2} } dx$

I've found the following integral on the Wikipediapage of the Euler-Mascheroni constant and I want to prove it.

$\gamma = \frac{1}{2} + 2 \cdot \int_0^\infty \frac{\sin(\arctan(x))}{(e^{2 \pi x} - 1) \cdot \sqrt{1 + x^2} } dx$

I know that

$\sin(\arctan(x)) = \frac{t}{\sqrt{t^2 + 1}}$. I tried to apply the Abel-Plana- Formula to the first derivate of the digammafunction, but it does not work.

Any help would be appreciated. Thanks in advance.