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There was a related question concerning the spectral decomposition of a nested function at [1].

Simply put, do the solutions to the integral of a composite function bear any resemblance to the solutions of each function integrated individually? Mine is the same question as the one linked except, without the complex exponential under the integral, it is more general.

[1] Fourier transform of function composition

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No, the antiderivative of a composition need not have any resemblance to the antiderivatives of individual functions in the composition. For example, $e^x$ integrates to $e^x$ and $x^2$ integrates to $\frac13 x^3$, but the integral of $e^{x^2}$ is something entirely different. For a more explicit example, compare $\int \frac{1}{1-x^2}$, $\int \sqrt{x}$, and $\int \sqrt{\frac{1}{1-x^2}}$: the last one is the inverse sine, but the first two have nothing trigonometric in them.

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