Non-elementary antiderivative of rational function: $f(x)= \frac{1}{a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0}$

Question

Is there any function of the form $$f(x)= \frac{1}{a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0}$$ for $n\in\mathbb{N},a_i\in\mathbb{R}$, that has no elementary antiderivative?

If there is none, is there a proof of that?

Answer 1

Every such function has an antiderivative expressible with elementary functions. You factor the denominator into a product of linear and quadratic factors and decompose using partial fractions.

Of course this is theoretical and neither practical nor closed form unless you have a closed form solution for the roots of the denominator.

See https://en.wikipedia.org/wiki/List_of_integrals_of_rational_functions