I agree that "Inverse hyperbolic *" is better than "Hyperbolic arc*"
(when I was taught hyperbolic functions and their inverses some 45 years
ago, they were denoted ch, sh, and th and the inverses argch, argsh,
and argth, I have no idea about the today fashion!-).
However the inverse hyperbolic functions appear in many places and
not in non-Euclidian geometry only, so I am not very fond
(to say the least) of "hyperbolic area". Why not
"computes the inverse @var{X} of the hyperbolic * (@code{*(X)})."
(or any suitable translation from Frenglish to native English!).
well, I'm starting to think that there may be regional differences in
the naming of the inverse hyperbolics (though I'm not sure about
this)?
The common naming convention in Germany, I think, is that the inverse
trigonometrics are called ARCTAN etc, while in contrast the inverse
hyperbolics are labeled ATANH or ARTANH, where the 'A' or 'AR' in
front stands for 'area' (and not for 'arc'). This is confirmed by my
copy of the Mathematical Handbook by Bronstein et al. (which I think
is Russian by origin), and which even goes into some depth to explain
where this naming comes from.