More generally it's not at all clear what 'simplify' means.

Is x*x simpler than x^2? Probably? Is sqrt(5)^3 simpler than 5^(3/2)? I don't know.

It entirely depends on what you're going to be doing with the expression later.

I think "simplify" is pretty clear here. For trigonometric functions you would expect a trig function and an inverse trig function to be simplified. We all know what we'd expect if we saw sin(arcsin(x)) (ie x). If we saw cos(arcsin(x)) I'll spoil it for you: it simplifies to sqrt(1-x^2).

Hyperbolic functions aren't used as much but the same principle applies. Here the core identity is cosh^2(x) = sinh^2(x) = 1 so:

      sinh(arccosh(x))
    = sqrt(1 + cosh^2(arccosh(x))
    = sqrt(1 + x^2)