I would like to derive the exact value of the following integral
$$ I_s= \int_1^\infty r| r^{-2s}- (r^2-1)^{-s}|d r\qquad \text{with}\qquad 0<s<1.$$
The existence of $I_s$ is warranted since around $r=\infty$ one should observe that
$$ r| r^{-2s}- (r^2-1)^{-s}|= \frac{1}{r^{2s-1}} \Big|1-(1-\frac{1}{r^2})^{-s}\Big|\sim \frac{s}{r^{2s+1}}$$
and around $r=1$ we have
$$ r| r^{-2s}- (r^2-1)^{-s}] \sim \frac{2^{-s}}{(r-1)^{s}}.$$
$$I_s=\int_1^\infty\left(\frac{r}{(r^2-1)^s}-\frac{1}{r^{2s-1}}\right)dr=\frac{1}{2-2s}\Big((r^2-1)^{1-s}-r^{2-2s}\Big)\Bigg|_{r\to 1}^{r\to\infty}=\frac{1}{2-2s}.$$