Hyperbolic distance in the upper half-space

Question

For the upper half-plane $\mathbb{H}=\{z:\mathrm{Im} z>0\}$, the hyperbolic metric is $\frac{|dz|}{y}$, the hyperbolic geodesics are semicircles and vertical ray and the hyperbolic distance is $$d_{h}(z,w)= \mathrm{Arccosh}\left(1+ \frac{|z-w|^2}{2 \mathrm{Im}(z) \mathrm{Im}(w)}\right).$$ Now for the upper half-space $\mathbb{H}^{n+1}=\{x=(x_1,x_2,\cdots,x_{n+1}):x_{n+1}>0, x_i\in \mathbb{R}, 1\leq i \leq n\}$. I know its hyperbolic metric is $\frac{|dx|}{x_{n+1}}$. What I would like to know is its hyperbolic distance and hyperbolic geodesics. I searched Google and some books of hyperbolic geometry, but I found nothing. Any help or references will be appreciated!