Not another composed function

Algebra Level 4

Define the function $f(x)=\frac{2x}{1-x^2}$ . Find the number of distinct real solutions of the equation $f^{(5)} (x) =x.$

Details and assumptions

$f^{(n)} (x)$ denotes the function $f$ applied $n$ times. In particular, $f^{(5)} (x) = f(f(f(f(f(x)))))$ .