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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?

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    @Angada: there's no reason to expect a rule for integrals of this kind. The only general rules with wide applicability that I can really think of are $u$-substitution and integration by parts.2011-09-26

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I suggest integration by parts where $u=f(x)$ and $dv=f(f(x))dx$

du=f'(x)dx and $v=\int f(f(x))dx$ , this integral we may evaluate as $v=\int f(x)dx \int f(x)dx$

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No, but there is a rule for integrating f'(x)\,f'(f(x)).

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    The derivative of $f(f(x))$ is what I wrote. So, the integral of what I wrote is $f(f(x)) +C$.2019-01-07
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Not in general. If, say, $f(x)=\ln\,x$, one requires a nonelementary function to represent the integral...