Is there an alternative way to solve this equation?

Any suggestions to obtain $x$ in terms of $n$ in this equation. Again Mathematica gives: "This system cannot be solved with the methods available to Solve."

$\coth (\pi x) \coth (n x)-\frac{x^4-2 x^2+5}{4 \left(x^2-1\right)}=0$

Coth[n x] Coth[\[Pi] x] - (5 - 2 x^2 + x^4)/(4 (-1 + x^2)) == 0

Any comment to solve this equation is welcomed.

Answer

It should be noted that the type of numeric search, based on ContourPlot, that others mention in the comments, has been automated by Wagon, in his book, Mathematica in Action. J.M. gives a version of Wagon's function in this answer.

Using his function, we get the following:

With[{n = 1},
 FindAllCrossings[Coth[n x] Coth[π x] - (5 - 2 x^2 + x^4)/(4 (-1 + x^2)), {x, -5, 5}, WorkingPrecision -> 20]
 ]

{-1.9201894111730777583, -1.5082193798592498308}

Let's check to make sure that the solution are ok:

With[{n = 1},
 Coth[n x] Coth[π x] - (5 - 2 x^2 + x^4)/(4 (-1 + x^2)) /. 
  x -> -1.92018941117307775830055191944881406955`20.
 ]

0.*10^-20

With[{n = 1},
 Coth[n x] Coth[\[Pi] x] - (5 - 2 x^2 + x^4)/(4 (-1 + x^2)) /. 

  x -> -1.50821937985924983076925593990549242151`20.
 ]

0.*10^-19

Yup, it seems to be working.