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In general we have: $$\frac{dw}{dz}=K(z-a)^{\alpha/\pi-1}(z-b)^{\beta/\pi-1}$$ With $a=-1$ , $b=+1$ and $\alpha=\beta=0$ , giving: $$dw\;=\;K(z-1)^{-1}(z+1)^{-1}dz \;=\;\frac{K}{2}\left(\frac{dz}{z-1}-\frac{dz}{z+1}\right)$$ So indeed, integration gives apart from constants that can be determined (later): $$w=\log\left(\frac{z-1}{z+1}\right)$$