How shall I get an estimate of $\int_{1-\frac{c}{log t}-iT}^{1-\frac{c}{log t}+iT}\frac{\zeta(s-1)}{\zeta(s)}\frac{x^s}{s}ds$?

Question

Please help me on the following.

I need to estimate $$\int_{1-\frac{c}{\log t}-iT}^{1-\frac{c}{\log t}+iT}\frac{\zeta(s-1)}{\zeta(s)}\frac{x^s}{s}ds$$ where $c$ is a constant, $T>0$.

What i tried to use is Functional equation $\zeta(s)\Gamma(\frac s2)=\pi^{\frac12-s} \frac{\Gamma(\frac{1-s}{2})}{\Gamma(\frac{s}{2})}\zeta(1-s)$. BUt I have no idea how shall i proceed.