Suppose that $\Phi'= \phi$. Is it possible to calculate $$ \int \phi^{-1}(c + \phi(x))\,dx $$ directly in terms of $\Phi$ (and possibly $\Phi^{-1}, \phi, \phi^{-1}$ and so on)?
At first I tried per partes, since the substitution of $t = c + \phi(x)$ is impossible at this time. $$ \int \phi^{-1}(c + \phi(x))\,dx = x \phi^{-1}(c + \phi(x)) - \int \frac{x \phi'(x)}{\phi'(\phi^{-1}(c+\phi(x)))}\,dx $$ This doesn't help much.
So I tried to see some special cases to see if a pattern emerges: $$\begin{align} \int e^{c+ \log(x)}\,dx &= e^c \frac{x^2}{2} + K\\ \int \log(c+ e^x)\,dx &= Li_2\left(-\frac{e^x}{c}\right) + x [\log(c + e^x) - \log(c)] + K \\ \int (c + \sqrt{x})^2\,dx &= c^2x + \frac{4}{3}cx^{\frac{3}{2}} + \frac{x^2}{2} + K \end{align}$$
But again, nothing obvious.
Is this a futile quest, or is it worth pursuing more?