The integral $\int \frac{\left(1-\frac{1}{\sqrt{3}}\right)(\cos \mathrm{x}-\sin \mathrm{x})}{\left(1+\frac{2}{\sqrt{3}}\sin 2\mathrm{x}\right)}\mathrm{dx}$ is equal to

$\begin{array}{l}\mathrm{I}=\int \frac{\left(1-\frac{1}{\sqrt{3}}\right)(\cos \mathrm{x}-\sin \mathrm{x})}{\left(1+\frac{2}{\sqrt{3}}\sin 2\mathrm{x}\right)}\mathrm{dx}\\ \frac{\sqrt{3}}{2}\int \frac{\left(1-\frac{1}{\sqrt{3}}\right)(\cos \mathrm{x}-\sin \mathrm{x})}{\left(\frac{\sqrt{3}}{2}+\sin 2\mathrm{x}\right)}\mathrm{dx}\\ \int \frac{\left(\frac{\sqrt{3}}{2}-\frac{1}{2}\right)(\cos \mathrm{x}-\sin \mathrm{x})}{\sin {60}^{\circ }+\sin 2\mathrm{x}}\mathrm{dx}\\ \int \frac{\left(\frac{\sqrt{3}}{2}\cos \mathrm{x}-\frac{1}{2}\cos \mathrm{x}-\frac{\sqrt{3}}{2}\sin \mathrm{x}+\frac{1}{2}\sin \mathrm{x}\right)}{2\sin \left(\mathrm{x}+\frac{\mathrm{\pi }}{6}\right)\cos \left(\mathrm{x}-\frac{\mathrm{\pi }}{6}\right)}\mathrm{dx}\\ \int \frac{\left(\cos \left(\mathrm{x}-\frac{\mathrm{\pi }}{6}\right)-\sin \left(\mathrm{x}+\frac{\mathrm{\pi }}{6}\right)\right)}{2\sin \left(\mathrm{x}+\frac{\mathrm{\pi }}{6}\right)\cos \left(\mathrm{x}-\frac{\mathrm{\pi }}{6}\right)}\mathrm{dx}\end{array}$

$\begin{array}{l}\frac{1}{2}\left(\int \frac{\mathrm{dx}}{\sin \left(\mathrm{x}+\frac{\mathrm{\pi }}{6}\right)}-\int \frac{\mathrm{dx}}{\cos \left(\mathrm{x}-\frac{\mathrm{\pi }}{6}\right)}\right)\\ \frac{1}{2}\ln \left|\frac{\tan \left(\frac{\mathrm{x}}{2}+\frac{\mathrm{\pi }}{12}\right)}{\tan \left(\frac{\mathrm{x}}{2}+\frac{\mathrm{\pi }}{6}\right)}\right|\end{array}$