The resp. Pynac-0.7.17 enhancements should be doctested here:
sage: exp(asinh(2))
sqrt(5) + 2
sage: exp(2*asinh(2))
(sqrt(5) + 2)^2
sage: exp(1/2*acosh(3))
sqrt(2*sqrt(2) + 3)
sage: exp(atanh(1/2))
sqrt(3)
sage: exp(atanh(-2, hold=True)) # known bug
sqrt(1/3)
sage: exp(acoth(31))
4*sqrt(1/15)
sage: exp(asech(1/3))
2*sqrt(2) + 3
sage: exp(acsch(13))
1/13*sqrt(170) + 1/13