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?