$Li(x)$ Is a non elementary function being the primitive of $\frac{1}{log(x)}$. I know also that the primitive of $e^{x^2}$ Is not elementary and I know that Gauss calculated some definite integral involving that function. Are there other examples of functions which doesnt have an elementary primitive? How to prove that a function has not an elementary primitive?
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The Risch - algorithm can decide whether a given function has an "elementary" antiderivate. In theory , this problem is undecidable , but apart from pathological cases , it always works. – Peter Jan 04 '24 at 14:57