Is there a non-elementary function with an elementary derivative and an elementary inverse? Elementary functions are combinations of powers, exponentials and logarithms, using composition and arithmetic operations. The inverse of an elementary function may not be elementary, and the integral of an elementary function may not be elementary. 
There are a couple of equivalent ways to ask my question: 


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*Is there a function $F(x)$ that is non-elementary, but its derivative $F'(x)$ and inverse $F^{-1}(x)$ are both elementary? 

*Is there an elementary function $f(x)$ whose integral $F(x)$ is non-elementary, but can be expressed as the inverse of some elementary function? (i.e. $F^{-1}(x)$ is elementary)
Here are some non-examples: 
The Lambert W function is the inverse of $x e^x$. It is not elementary, but its derivative is not elementary either:
$$W'(x) = \frac{W(x)}{x(1 + W(x))}$$
The "exponential integral" $Ei(x)$ is the integral of $\int \frac{e^x}{x} dx$, which is non-elementary. Its inverse $Ei^{-1}(x)$ is not elementary either, so this is not what I'm looking for. 
 A: My answer shows only one of the possibly suitable function classes.
Let $c,c_1,c_2$ be constants. A constant function is an elementary function.
Let $\Phi$ denote the inverse of $F$: $\Phi=F^{-1}$.
According to the question, we have $\Phi$ is elementary, and
$$F(x)=\int F'(x)dx+c_1,$$
wherein $F$ is non-elementary, and $F'$ and $c_1$ are elementary.
$F(x)$ is a non-elementary integral.
The elementary functions are differentiable, and their derivatives are also elementary.
Because $\Phi$ is elementary, $\Phi(x)=\int \Phi'(x)dx+c_2$, wherein $\Phi'$ is an elementary function.
Assume $F$ is integrable.
Applying the Integral of inverse functions to $\int F(x)dx$ gives
$$\int F(x)dx=xF(x)-\int \Phi(F(x))dF(x)+c.$$
$$\int F(x)dx=xF(x)-\int xdF(x)+c$$
$$\int F(x)dx=xF(x)-\int xF'(x)dx+c$$
Because $F$ is non-elementary, $\int F(x)dx$ and $xF(x)$ are non-elementary, and therefore $\int xF'(x)dx$ can be elementary or non-elementary.
Because $F'$ is elementary, $xF'$ is also elementary. If we, for example, assume that $\int xF'(x)dx$ is non-elementary, $\int xF'(x)dx$ must be a non-elementary integral.
See e.g. Wikipedia: Nonelementary integral and Yadav, D. K.: A Study of Indefinite Nonintegrable Functions. PhD thesis, Vinoba Bhave University, India, 2012 for some very simple kinds of non-elementary integrals.
I don't know if such functions $F$ you asked for with a non-elementary integral $\int xF'(x)dx$ where $F^{-1}$ is elementary actually exist.
Verification of a found $F$ could be difficult: Take a non-elementary integral $\int xF'(x)dx$ and calculate $F$ from that. Calculate the inverse $\Phi$ of $F$. But because $F$ is non-elementary, the closed-form expression of $F(x)$ will, and the closed-form expression of $\Phi(x)$ possibly will contain non-elementary function symbols. Therefore it could possibly be impossible to prove that $\Phi(x)$ can be expressed as an elementary expression.
Is this a method to prove if the inverses of some given non-elementary functions are elementary and if some expressions which contain non-elementary function symbols are elementary?
