Overview
I want to calculate this integral via complex analysis:
$$I(x) = \cfrac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\cfrac{\log\zeta(s)}{s}x^s\mathrm{d}s,$$ where i is the imaginary unit, c > 1 and log is the natural logarithm. I already have a rough idea how I may solve this integral but it also could be wrong. I know the solution, it's $$I(x) = \operatorname{Li}(x)-\sum_\rho\operatorname{Li}(x^\rho)-\log2+\int_{x}^{\infty}\cfrac{\mathrm{d}t}{t(t^2-1)\log t},$$ where \rho are the nontrivial zeros of the Riemann-Zeta function. But how can I derive to that solution step by step?
My Rough idea
We know that $$\log\zeta(s) = \sum_{\rho}\log\left(1-\cfrac{s}{\rho}\right)-\log2-\log\Gamma\left(1+\cfrac{s}{2}\right)+\cfrac{s}{2}\log\pi-\log(s-1).$$ Is it possible to evaluate these terms seperately? If yes, please show how, if no, please explain another method step by step. Thank you!