Is it possible to do this integral with contour integration?

Question

I'm wondering if it's possible to do this integral with contour integration: $$\int_0^\infty dr \frac{r\cos(ar)}{(r^2+1)(1+2\sqrt{r^2+1})^2}. $$ Having integrated this numerically, I'm sure this integral converges. I thought of computing this as the real part and changing $\cos(ar)$ to $\text{Re}(e^{iar})$, but then I'm not sure about what to do about the poles and branch points. I was thinking of using the branch cut $(-\infty,-i]\cup[i,\infty)$ and use a semicircle contour in the upper-half plane which avoids the top branch cut. But then since the integrand is odd, I'm wondering how to deal with the $(-\infty,0]$ and $[0,\infty)$ sections along the real line canceling out. Maybe I'm supposed to compute it with one side picking up a phase from the branch cut?