Cosh[a ArcSinh[z]] // FunctionExpand
Which seems to suggest that there is no simplification that can be done here. However, there is an identity on the wolfram functions site that states
Cosh[a ArcSinh[z]] == ((1/2) (1 + (z + Sqrt[1 + z^2])^(2 a)))/(z + Sqrt[1 + z^2])^a
which is clearly a simplification. Why did
FunctionExpand not recognize this and what should I do in future to make Mathematica see this sort of simplification directly without looking it up on a webpage?