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Does there exist any rational $t$, such that $|t|<1/10$ and the integral $$\frac{1}{\pi^2}\int\limits_0^{\pi t}\arccos{\left(\frac{\sin{s}}{1+2\sin{s}}\right)}ds$$ is irrational?

The integral is from http://arxiv.org/abs/1304.7464 where it is explained why it is desirable to get the answer to this question.

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Well, the author himself says that the problem to decide whether the above integral is rational or not is a difficult problem. Do you believe otherwise ? –  Dietrich Burde May 5 '13 at 15:21
I agree with the author that the problem seems to be difficult. But sometimes difficult problems have simple enough solutions. Examples are mateforum.ro/articole/irationale_faimoase.pdf pracownicy.uksw.edu.pl/mwolf/Poorten_MI_195_0.pdf and arxiv.org/abs/math/0506086 –  Zurab Silagadze May 6 '13 at 7:43
Oh, I see. You are right. There are certainly experts for such a question (and, e.g., Zudilin has a user account for mathoverflow). –  Dietrich Burde May 6 '13 at 10:02

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