Need reference or proof of an integral identity involving the first Exponential Integral, $E_1 = \int_y^{\infty}\frac{e^{-t}}{t}dt$

Question

Need a reference for the following identity $$ \int_0 ^ {\log x} \left(\frac{e^{-2t}}{2t} - \frac{e^{-2t}}{1-e^{-2t}}\right)dt = -\frac{\gamma}{2}- \frac{E_1(2 \log x)}{2} - \frac{\log (1-e^{-2 \log x})}{2}$$ where $\gamma$ is the Euler Mascheroni constant and $E_1=\int_y^{\infty}\frac{e^{-t}}{t}dt$