While looking for an expression of the hyperbolic distance in the Upper Half Plane $\mathbb{H}=\{z=x +iy \in \mathbb{C}| y>0\},$ I came across two different expressions. Both of them in Wikipedia.
In the page Poincaré Half Plane Model it is explicitly stated that the distance of $z,w \in \mathbb{H}$ is: $$d_{hyp}(z,w)= Arccosh(1+ \frac{|z-w|^2}{2 Im(z) Im(w)}).$$ While in the page Poincaré Metric it is stated that the metric on the Upper Half Plane is : $$\rho(z,w)=2 Arctangh(\frac{|z-w|}{|z-\bar w|}).$$
At the beginnning I thought it would have been an easy exercise to prove the equivalence of the two expressions. But first I failed in doing that, and then I found, using Mathematica, a counterexample (i.e. $z=2i$ and $w=i$).
Question: If they are not equal, which of the two expressions is the right one? Then, how is the metric related to the induced distance?
The question is probably silly, but I'm often confused about the relationships between "metric" objects.
Thank you very much for your time!