It is well known that there exists no elementary function $f$ with
$$\int x^x\,dx \quad = \quad f$$
Is there an elementary function $g$ such that
$$\int x^x\,dx \quad \tilde{} \quad g$$
in the sense that the quotient of both sides converges to $1$ as the argument goes to infinity? Results like Liouville's theorem do not apply to such expressions as there is no rigid algebraic structure to work with.
(For the purposes of this question, Wikipedia's definition of "elementary function" is used.)
The answer can be found in the paper : https://fr.scribd.com/doc/34977341/Sophomore-s-Dream-Function
A special function, called "Sophomore's dream" function is defined as : $$\text{Sphd}(\alpha,x)=\int_0^x t^{\alpha t}dt$$ From the properties of this function, an asymptotic expansion is derived Eq.(6:4). The first term, Eq.(9.2) is : $$\text{Sphd}(\alpha,x)\sim \frac{x^{\alpha x}}{\alpha\big(1+\ln(x)\big)}$$ So, for $x$ large $$\int x^x dx \sim \frac{x^x}{1+\ln(x)}$$