If it may also be a graphic solution?

    ((a^2 - 1)*((a^2 - 1)*ArcTanh[a] - a))/(2*Sqrt[2*(a^2 - 1)^4]) == c*x;
    x[a_] = 1/c*((a^2 - 1)*((a^2 - 1)*ArcTanh[a] - a))/(2*Sqrt[2*(a^2 - 1)^4]);

With c as parameter

    Plot[x[a] /. c -> 1, {a, -1, 1}, AxesLabel -> Automatic, GridLines -> Automatic]
[![enter image description here][1]][1]


another way:

    c = 1;
    f = ((a^2 - 1)*((a^2 - 1)*ArcTanh[a] - a))/(2*Sqrt[2*(a^2 - 1)^4]) - c*x;
    p = ContourPlot[f, {a, -1, 1}, {x, -1, 1}, Contours -> 1, 
       ContourShading -> None, PlotPoints -> 100];

    p1 = Join @@ Cases[Normal@p, Line[x1__] :> x1, Infinity];
    a = Interpolation@Thread@{Last /@ p1, First /@ p1}

    Plot[a[x], {x, -1, 1}, GridLines -> Automatic]
[![enter image description here][2]][2]


  [1]: https://i.sstatic.net/M3BeB.gif
  [2]: https://i.sstatic.net/pWkfs.gif