Number of equivalence classes of rational functions over finite fields

Two rational functions $f,g\in \mathbb F_q(X)$ are said to be <em>equivalent</em> if there exist $\phi ,\psi \in \mathbb F_q(X)$ of degree $1$ such that $g=\phi \circ f\circ \psi $. We give an explicit formula for the number of equivalence classes of rati