# Can it be decided whether $\int\root 3 \of{\cos^2(t)}\,dt$ is expressible by elementary functions? [closed]

I would like to decide by methods of Differential Algebra whether the integral $\int\root 3 \of{\cos^2(t)}\,dt$ contrary to the output of CAS Mathematica Online Integrator might be expressible by elementary functions and if not - why.
Alas I have not the deep enough 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 about Symbolic Integration but presently dont understand the hard stuff.

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## closed as off-topic by Felipe Voloch, Alexandre Eremenko, Chris Godsil, Stefan Kohl, S. Carnahan♦Jun 1 at 22:36

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The substitution $u=\tan(t/2)$ will turn this integral into the integral of an algebraic function of $u$ to which you can apply Liouville's theorem. –  Felipe Voloch Jun 1 at 15:07
My problem is to see the details how one could use/apply Liouville's Theorem. I tried $\tan(t)$ but then have no idea what to do next. –  Wolfgang Tintemann Jun 1 at 17:13
Make the substitution that Felipe suggested, and then you'll probably need an additional substitution $u=v^3$ to get the integral of an integrand involving something like $\sqrt[3]{1+v^6-\sqrt{1+v^6}}$. (I'll let you work it out exactly.) Now it comes down to checking that the algebraic curve $w^3=1+v^6-\sqrt{1+v^6}$ has genus $g\ge1$, which can be done by various CAS. –  Joe Silverman Jun 1 at 17:32
If I understand correctly what you talk about is not using the theories of Differential Algebra. <br/>Do you have a literature source for the theory/theorem(s) by which a connection between the genus and integration by elemenatry functions of algebraic curves is established. <br/>Does e.g. $g=0$ mean integration by elementary functions possible ? <br/>Thank for advice. –  Wolfgang Tintemann Jun 2 at 12:53
Mathematica can explicitly integrate the functions $f_d(t)=(\cos(dt)^{1/d}$ (Your function is, up to a simple change of variables, the case $d=\frac 32$). The result uses hypergeometric functions so it depends on whether you regard them as being elementary. By the way the class of curves with parametrisations $(F_d(t),f_d(t))$ where $F_d$ is a primitive of $f_d$ have remarkable geometrical and mechanical properties due to the fact that for them $f^2+f'^2$ is proportional to a power of $f$---details arXiv 1102:1579. –  blackburne Jun 27 at 8:12