Let $r$ be an integer $\geq -1$ and let $\alpha \in (0,1)$, $0 < \epsilon < \delta < 1 < x$ be real numbers. I want to evaluate the integrals $I_r(x):= \int_{\gamma_1} x^s(s-1)^{r+\alpha}s^{-1} \, ds$ and $J_r(x):= \int_{\gamma_2} x^s(s-1)^{r+\alpha}s^{-1}\, ds$ (in terms of $r, \alpha, \delta, \epsilon$ of course) along the paths $\gamma_1$ and $\gamma_2$, where $\gamma_1$ is the path from $1-\delta$ to $1-\epsilon$ traversed along the "bottom" of the $x$-axis (so it is clear which value the multivalued function $x^s$ should take) and $\gamma_2$ is the path from $1-\epsilon$ to $1-\delta$ traversed along the "top" of the $x$-axis.
Attempting to use the definition of complex integral along a piecewise-differentiable path or converting the given integrals to integrals over circles or semicircles (in order to enable use of the Cauchy Integral Formula or the Residue Theorem) didn't work out for me. Any hints or suggestions would be really appreciated. Thanks.
Edit (9th May, 2020): It would be sufficient for my purpose even if i could just estimate the sum $I_r(x)+J_r(x)$. An intuitive guess tells me that I should be able to do so upto an error term of $O \left( \frac{x}{\log^{\alpha+r+1} x} \right)$ (as $x \rightarrow \infty$) [and the principal term will look something like a constant times $\frac{x}{\log^{\alpha+r} x}$], but I'm unable to rigorously establish the same.