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.