# Is $\int{\sqrt[3]{x+\sin(x)}}\space\mathrm{d}{x}$ elementary?

I found this strange one while watching a mit integration bee video, $\int_{\log 1/2}^{\log2}{\sqrt[3]{x+sin(x)}}\space\mathrm{d}{x}$.

Although it is quite clear that the definite integral is zero, the antiderivative itself seemingly lies outside modern CASes' grasp: Axiom admits that it can neither compute it nor prove it being non-elementary, Mathematica fails similarily.

Notably, replacing cubic root with square root does not make the problem simplier, as both CASes fail there as well...

• Is there some reason for believing that $\int\sqrt[3]{x+\sin x}\,dx$ or $\int\sqrt[3]{\text{random increasing function}(x)}\,dx$ are elementary? – Jack D'Aurizio May 2 '18 at 19:37
• This define integral is not zero. – Mariusz Iwaniuk May 2 '18 at 21:53
• or is it? We're assuming reals... – Thehx May 3 '18 at 18:58
• Jack, yes, there are ways to prove that function $f$ does not have an elementary antiderivative. I was hoping that someone who knows exactly how Risch's algorithm (which does more than that, it either obtains the answer, or proves that no elementary answer exist) works would at least point out why computer algebra systems fail at this case. – Thehx May 3 '18 at 19:02