One thing to realize is that Simplify[] and FullSimplify[] are discrete optimization functions that try to minimize the "complexity" of an expression according to a ComplexityFunction. With respect to the default complexity function, the Coth expression is the simplest. To get rid of it, construct a ComplexityFunction that counts Coth as more complex. With trig, you tend to have to penalize all the trig functions, and Log if there is inverse trig, since there are many transformations from one to the other.
Here is the default ComplexityFunction values on three possible answers:
(1 + 2 3^(1 + 2^(-1 + n)))/(3 (-1 + 3^2^(-1 + n))) // Simplify`SimplifyCount
(6*3^2^(-1 + n) + 1)/(3*3^2^(-1 + n) - 3) // Simplify`SimplifyCount
1/6 (5 + 7 Coth[2^(-1 + n) ArcCoth[2]]) // Simplify`SimplifyCount
(*
32
31
18
*)
Here is a way to get rid of the trig functions:
1/6 (5 + 7 Coth[2^(-1 + n) ArcCoth[2]]) //
FullSimplify[#, n \[Element] Integers,
ComplexityFunction -> (LeafCount[#] +
20 Count[#, Coth | Tanh | Cosh | Sinh | Log, Infinity,
Heads -> True] &)
] &
(* 2 + 7/(3 (-1 + 3^2^(-1 + n))) *)
% // Simplify`SimplifyCount
(* 20 *)
This answer is slightly more complex, by the default complexity function, than the Coth answer.
