Definite Integration

If \(\int_\limits{\frac{\pi}{6}}^{\frac{\pi}{3}} \sqrt{1-\sin 2 x} d x=\alpha+\beta \sqrt{2}+\gamma \sqrt{3}\), where \(\alpha, \beta\) and \(\gamma\) are rational numbers, then
\(3 \alpha+4 \beta-\gamma\) is equal to _________.