In section 6.3 of D'Hoker and Freedman's paper, they give the following result for the geodesic distance in AdS:

$$d(z,w) = \int_{w}^{z}ds = \ln\left(\frac{1+\sqrt{1-\zeta^2}}{\zeta}\right)$$

with

$$\zeta \equiv \frac{2 z_0 w_0}{z_0^2 + w_0^2 + (\vec{z}-\vec{w})^2}$$

How does one derive this?