Basically the title explains most of my question here. Purely out of curiosity (I have no real application), I am wondering if there are any "interesting" functions $f$ where we know of a function $F$ (perhaps $f$'s antiderivative, but not necessarily) such that

$$\int f(x + f(x + f(x + \dotsb)))dx = F(x + F(x + F(x + \dotsb))) + C.$$

One obvious case where this is true: $f(x) = 0$, $F(x) = 0$.

Are there any other examples where $f$ is doing any real transformation? Or is such a pair $f$ and $F$ impossible?

Basically the title explains most of my question here. Purely out of curiosity (I have no real application), I am wondering if there are any "interesting" functions $f$ where we know of a function $F$ (perhaps $f$'s antiderivative, but not necessarily) such that

$$\int f(x + f(x + f(x + \dotsb)))\,dx = F(x + F(x + F(x + \dotsb))) + C.$$

One obvious case where this is true: $f(x) = 0$, $F(x) = 0$.

Are there any other examples where $f$ is doing any real transformation? Or is such a pair $f$ and $F$ impossible?