Indefinite integral of x^x

Question

When I searched for a solution to this in the internet, I always read that it can be expressed in closed form, but I haven't found the proof, is there one ? Or we concluded that it is so, just because we can't find one in closed form ?

Answer 1

We can write $$x^x=e^{x\log x}=\sum_{k=0}^{\infty}\frac{x^k(\log x)^k}{k!}$$

Then since the power series converges uniformly, $$\int x^x\,dx=\sum_{k=0}^{\infty}\int\frac{x^k(\log x)^k}{k!}\,dx$$

You can then attempt to integrate $\dfrac{x^k(\log x)^k}{k!}$, but there is no closed form: the answer will involve the Gamma function and/or this integral.