Fourier Cos coefficient vs Integrate vs common sense

Question

I was trying to find out what the (discrete) Fourier transformation of the following function is $$ f(\nu, u) = \frac{\sinh \nu}{\cosh \nu - \cos u} $$

So at first, I tried to evaluate a fancy MMA function FourierCoefficient

FourierCoefficient[Sinh[ν]/(Cosh[ν] - Cos[u]), u, n,
    Assumptions -> {ν > 0, Element[n, Integers]}]

to no avail, as the computation never ended.

Then I realized that the function has only cos components, as sin are trivially zero. So I tried to evaluate

FourierCosCoefficient[Sinh[ν]/(Cosh[ν] - Cos[u]), u, n,
    Assumptions -> {ν > 0, Element[n, Integers]}]

which led to some pretty nasty expression

1/((-1 + E^(2 ν)) (-1 + n) π) 2 I (1 + (-1)^n - E^(
   2 ν) + (-1)^(1 + n) E^(2 ν) + (-1)^(1 + n)
     Hypergeometric2F1[1, 1 - n, 2 - n, -E^-ν] - 
   Hypergeometric2F1[1, 1 - n, 2 - n, E^-ν] + (-1)^n E^(2 ν)
     Hypergeometric2F1[1, 1 - n, 2 - n, -E^ν] + 
   E^(2 ν) Hypergeometric2F1[1, 1 - n, 2 - n, E^ν] + (-1)^
    n E^(2 ν) Hypergeometric2F1[1, -1 + n, n, -E^-ν] + 
   E^(2 ν) Hypergeometric2F1[1, -1 + n, n, E^-ν] + (-1)^(
    1 + n) Hypergeometric2F1[1, -1 + n, n, -E^ν] - 
   Hypergeometric2F1[1, -1 + n, n, E^ν]) Sinh[ν]

Moreover, this mess never evaluated if n was integer

%/.n->1
Power::infy: Infinite expression 1/0 encountered.
%/.n->2
Infinity::indet: Indeterminate expression 2-E^(2 ν)-E^(2 ν)+ComplexInfinity+ComplexInfinity+ComplexInfinity+ComplexInfinity-E^(3 ν) Log[1-E^-ν]+E^(3 ν) Log[1+E^-ν]+E^-ν Log[1-E^ν]-E^-ν Log[1+E^ν] encountered.

So I got mad and rewrote the integral from the definition

1/(2 π)
  Integrate[(Sinh[ν])/(Cosh[ν] - Cos[u]) Cos[n u], {u, 0, 
   2 π}, Assumptions -> {ν > 0, Element[n, Integers]}]

which returned the same result as the cos coefficient function.

So I got really mad and just did it "by hand" (by constructing the list for some values of n to see some pattern):

(-(KroneckerDelta[0, #]/(2 π)) + 1/π) Integrate[
     Sinh[ν]/(Cosh[ν] - Cos[u]) Cos[# u], {u, 0, 2 π}, 
     Assumptions -> {ν > 0}] & /@ Range[0, 5] // FullSimplify

{1, 2 E^-ν, 2 E^(-2 ν), 2 E^(-3 ν), 2 E^(-4 ν), 2 E^(-5 ν)}

and the pattern is pretty clear: the zeroeth coefficient is one and the rest can be written as $2 \exp (- n \nu)$, so the formula is: $$ \frac{\sinh \nu}{\cosh \nu - \cos u} = 1 + 2 \sum_{n = 1}^\infty e^{- n \nu} \cos n u $$

The question is: why neither of the FourierCoefficient (with assumptions), FourierCosCoefficient (with assumptions), Integrate (with assumptions) spotted this pattern, and instead yielded some function which cannot even be evaluated in integer values (despite $n$ being integer as was explicitly stated in assumptions)?