Integrate $\iint_D \frac1{(x^2+y^2)^2 }dx dy$ over a region bounded by four circles 

I'm thinking of doing the substitution $x = \frac{u}{u^2 + v^2}, y = \frac{v}{u^2 + v^2},$ but I'm not sure how to exactly compute the range of values $u$ and $v$ take. Clearly, $u=\frac{x}{x^2+y^2}$ and $v = \frac{y}{x^2 + y^2}.$ The Jacobian of the result is $-\frac{1}{u^2 + v^2}$, and so the resulting integral equals $\iint_{D'} dudv,$ where $D'$ is the region of possible values for the pairs $(u,v)$. In the circle $x^2 + y^2 - 2x= 0$, for instance, we have the point $(1,0)$, which corresponds to $u=1, v=0$. Also, we have the point $(2,0)$, corresponding to $u= \frac{1}2, v = 0$. I can't seem to generalize what values u and v can take on.


Given that the first equation could be written
$$
\frac{x}{x^2+y^2}=\frac{1}{2} \implies u=\frac{1}{2}
$$
with similar reasoning $u\in\left[\frac{1}{4},\frac{1}{2}\right]$ and $v\in\left[\frac{1}{6},\frac{1}{2}\right]$
