The integral $$\displaystyle\int \dfrac{1 - \dfrac{1}{\sqrt{3}}(\cos x - \sin x)}{1 + \dfrac{2}{\sqrt{3}}\sin 2x} dx$$ is equal to

$$\dfrac{1}{2}\log_e \dfrac{\tan\left(\frac{x}{2} + \frac{\pi}{12}\right)}{\left(\frac{x}{2} + \frac{\pi}{6}\right)}   + C$$

$$\log_e \dfrac{\tan\left(\frac{x}{2} + \frac{\pi}{6}\right)}{\left(\frac{x}{2} + \frac{\pi}{3}\right)} + C$$

$$\dfrac{1}{2}\log_e \dfrac{\tan\left(\frac{x}{2} + \frac{\pi}{6}\right)}{\left(\frac{x}{2} + \frac{\pi}{3}\right)} + C$$

$$\dfrac{1}{2}\log_e \dfrac{\tan\left(\frac{x}{2} - \frac{\pi}{12}\right)}{\left(\frac{x}{2} - \frac{\pi}{6}\right)} + C$$