Have you tried the method of section 8.7, i.e., solving Abel's equation $\phi(P(x))-\phi(x)=1$? Here we expect $\phi(t)=t^{-1}+\sum_{n=1}^{\infty}c_n t^n$, and you can find the coefficients of $\phi$ one by one. For example, I took $P(x)=x-x^2+x^3+x^4$ and immediately found $c_1=-2$, $c_2=-5/2$, $c_3=-7/2$, $c_4=-17/4$... Not a general formula, but you can get as many terms as you want for a given $P$.