I was looking at a paper by Takao Suyama on GT theory, and I couldn't figure out how he derived his formula (3.59): $$\frac{1}{\pi}\int_a^bdx\frac{1}{z-x}\frac{\sqrt{(z-a)(z-b)}}{\sqrt{|(x-a)(x-b)|}}\frac{\log(e^{-t_1}x)}{2}=\frac{1}{2}\log\left[\frac{e^{-t_1}}{2\sqrt{ab}+a+b}\left(z+\sqrt{ab}-\sqrt{(z-a)(z-b)}\right)^2\right],$$ where $0<a<b$, $[a,b]\subset\mathbb{R}$, $t_1\in\mathbb{R}$, and $z\in\mathbb{C}\setminus[a,b]$. It's a physics paper, but my question is just how would one do the integral?

I tried expanding everything as a power series, and then using the fact that the integral of $x^n\log x$ is known, but then I couldn't figure out how to resum the resulting series, so I'm a bit confused how one would solve this integral.