The configuration of the following problem/integral is based on the fig 1 below. The area of the wedge with thickness $\Delta$ is from $-\infty$ to $-a$, which is non-analytical. From $-a$ to $a$ along the x-axis is the crack (void) area, which is also non-analytical. So, the entire red area is non-analytical. Outside of this red area is analytical.
When I read this problem, I found they gave the solution of the following integral directly (arrowed). I have no idea how they solved it explicitly.
My question is how to get this:
$$\int_{-a}^{a} \frac{1}{z-t} \sqrt{\frac{a-t}{a+t}}dt=\pi(1-\sqrt{\frac{z-a}{z+a}})$$
where $a>0$, $z=x+iy$ is a complex variable.
This integral/problem is on page 154 of Tada, H., Ernst, H. & Paris, P. Westergaard stress functions for displacement-prescribed cracks-II. Extension to mixed problems. Int J Fract 67, 151–167 (1994). https://doi.org/10.1007/BF00019601.