How do I integrate
$$ \int_{1}^{2}\int_0^{\sqrt{2x-x^{2}}}\frac{1}{\sqrt{x^2+y^2}}\ dy\ dx $$
using polar coordinates? The base is a quarter circle of radius $1$ centered at $(1,0)$, so my first instinct was to translate it to the origin, giving me
$$ \int_{0}^{\frac{\pi}{2}}\int_0^{1}\frac{1}{\sqrt{r^2+2r\cos\theta+1}}r\ dr\ d\theta $$
but that gives me a pretty ugly integral to deal with. Is there a better conversion?