real analysis - An arctan exponential integral

Prove the following for $\Re(z) >0$

$$\log(\Gamma(z)) = 2\int_{0}^{\infty} \tan^{-1} \left(\dfrac{t}{z}\right)\dfrac{\mathrm{d}t}{e^{2\pi t} - 1} + \dfrac{\log(2z)}{2} + \left( z - \dfrac{1}{2} \right)\log (z) -z$$

I found this identity on Wolfram Functions here. I can't see where to start solving this problem.

I'm looking for elementary solution, I haven't learned about Complex Analysis or Abel Plana formula yet.

Please help.
Thanks.