The denominator depends on $|x|$, so it's even in $x$. The $4x^{11}$ numerator term is odd, contributing 0. Only the $\pi$ part survives:

$I=2\pi {\int }_0^{\pi /6}\frac{dx}{1-\sin (x+\pi /6)}$.

Substitute $t=x+\pi /6$, then use $\frac{1}{1-\sin t}=\frac{1+\sin t}{{\cos }^{2}t}={\sec }^{2}t+\sec t\tan t$:

$I=2\pi {\left[\tan t+\sec t\right]}_{\pi /6}^{\pi /3}=2\pi \left[(\sqrt{3}+2)-(1/\sqrt{3}+2/\sqrt{3})\right]=2\pi \left[\sqrt{3}+2-\sqrt{3}\right]=4\pi$.