calculus - How to know whether the solution of an indefinite integral can be written in the form of elementary functions or not?

Yesterday, I saw a definite integral here. That question is deleted now. I am writing it as indefinite integral $\int \left ( \frac{2log_{e}x}{x+1} - \frac{log_{e}x}{4-x} \right )\; dx$. I tried to solve this integral using substitution and integration by parts and thought more about it but didn't get the solution. I checked on WolframAlpha where it is showing the solution as $2 Li_{2} (-x) + Li_{2}\left ( \frac{x}{4} \right ) + log_{e}x\left ( log\left ( 1-\frac{x}{4} \right ) + 2log(x+1) \right ) + \; constant$, where $Li_{n}(x)$ is the polylogairthm function.

Can anyone please explain how to know whether the solution of an indefinite integral can be written as elementary functions like trigonometric, exponential, logarithmic etc or not ? Is there any method for it ? Thank you.