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The Gudermannian function is built into Mathematica, but does not reduce to simpler terms in some simple cases. In particular, the Gudermannian is an odd function of real arguments but Mathematica does not seem to know or use this. For example:
FullSimplify[Gudermannian[2x] + Gudermannian[-2x],Assumptions->{x>0}]
Gudermannian[-2 x] + Gudermannian[2 x]
Am I missing some subtlety that invalidates simplifying to 0 or is Mathematica missing something?
Mathematica is having hard time knowing that ArcTan[E^(-2 x)] + ArcTan[E^(2 x) == Pi/2 and I could not figure how to tell so other than by brute force telling it so in the Assuming.
Once it knew this relation, then the simplification of the FunctionExpand applied to your input, gives zero as expected.
ClearAll[x]
inp = Gudermannian[2 x] + Gudermannian[-2 x];
Assuming[(ArcTan[E^(-2 x)] == Pi/2 - ArcTan[E^(2 x)]),Simplify@FunctionExpand@inp]
From help, it says Use FunctionExpand to expand Gudermannian in terms of elementary functions and that is how the above relation came out. So, without using FunctionExpand it is even harder. So, I would say use ``FunctionExpandonGudermannian` since then there is more chance of doing more simplification on it when it is in that form (i.e. using elementary functions)
