May the integral $\int\root 3 \of{\cos(x)^2}\,dx$ be expressible by elementary functions?

Question

I would like to decide by methods of differential algebra whether the integral $\int\root 3 \of{\cos(x)^2}\,dx$ might be contrary to the output of CAS Mathematica Online Integrator expressible by elementary functions and if not - why.
Alas I have not the deep knowledge of the subject to be able to tackle this question without months of ( maybe fruitless ) study.
I read about the Risch algorithm which might give the answer and already tried to integrate with the CAS Axiom which was said to have implemented the algorithm.
Also I browsed an article of Bronstein but I presently dont understand the hard stuff.

Answer 1

Wolfram-Alpha gives a closed form solution to your indefinite integral:

$$-\frac{3}{5} \cos^{5/3}(x)\space_2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos^2 x\right)$$

Where $_2F_1$ is an example of a hypergeometric function.