Poincaré hyperbolic geodesics in half-plane and disc models including outer branch
Solution 1:
The mistake is after the words "leading to" where the square root in the denominator is missing. It should be $$ \frac{r^2}{1-r^2}\cdot\frac{1}{\sqrt{r^2+r'^2}}=C_1. $$ P.S. Now it is much more fun integrating this guy :-)