For $s\in (0,1)$, is there are an explicit expression for $$(-\Delta)^s \left(\frac{1}{\left(1+|x|^2\right)^s}\right)?$$

Edit: My goal is to show that for the function $u(x)=\frac{1}{\left(1+|x|^2\right)^s}$ defined on the ball $B(0,R)$ where $R>1$ we have $(-\Delta)^s u(x)\geq c(n,s) u^2(x)$ where $c(n,s)>0$ is constant dependent on $n$ and $s.$ Any references/hints on how to do this will be much appreciated.