Compute $\int_{\mathbb R} (x^2+a^2)^{-s} dx$

Question

I want to solve the integral $$\int_{\mathbb R} (x^2+a^2)^{-s} dx$$ for $a>0$ whenever it is defined. I guess this sould be the case for $\Re(s)>1/2$. A simple substitution gives me $$\int_{\mathbb R} (x^2+a^2)^{-s} dx = a^{1-2s} \int_{\mathbb R} (x^2+1)^{-s} dx.$$ Using WolframAlpha I tried a lot of particular values for $s$ and came up with the following formular $$\int_{\mathbb R} (x^2+a^2)^{-s} dx = \frac{\sqrt{\pi} \ \Gamma(s-1/2)}{a^{2s-1}\Gamma(s)}.$$ Now I want to know if it is correct and see in this case a proof. As already explained it is enough to prove the formular for $a=1$.