Why can't $\mathbb{Li}(x)$ be evaluated?

Question

Why can't the function $\mathbb{Li}(x)$ be directly evaluated?

I tried using the substitution $e^u=x$, giving $\ln(x)=u$, and then $dx=e^udu$, then using the Reverse Product Rule on it infinitely, but it turns out that that sequence can't be evaluated either.

Any ideas?

Answer 1

With your way you don't really get a closed form.

You obtain what is called a Series solution. Like for the integral of $x^x$.

There is no primitive of that integral. And you can prove it via Risch algorithm.

https://en.wikipedia.org/wiki/Risch_algorithm

https://en.wikipedia.org/wiki/Nonelementary_integral