Is there a rule for integrating $f(x) f(f(x))$? I am looking through rules/tricks for integration, and there are a lot... but I can't seem to find one specifically applicable to the general form
$$\int_a^b f(x)\,f(f(x)) \,\mathrm dx$$

Is there a nice rule or trick that would generally apply in this situation?
No, but there is a rule for integrating $f&#39;(x)\,f&#39;(f(x))$.
I suggest integration by parts where $u=f(x)$ and $dv=f(f(x))dx$ 

$du=f&#39;(x)dx$ and $v=\int f(f(x))dx$ , this integral we may evaluate as $v=\int f(x)dx \int f(x)dx$