I'm confused as to how we are supposed to integrate
$$\frac{1}{\pi}\int_U\left[\frac{d}{dz}\left( \frac{z-\alpha}{1-\bar\alpha z }\right)\right]^2 \, dm$$
where $U$ is the unit disc, $|\alpha|<1$, and $m$ denotes Lebesgue measure. I've attempted turning this into a double integral with $r\in(0,1)$ and $\theta\in(0,2\pi)$, but I keep getting stuck. Is there a trick I'm missing in this process? Thanks!