Solve 3/2arctan[frac{2sqrt{3}}{3}(tanx+1/2)]+frac{3sqrt{3}}{4}tanx+1/2/tan^2x+tanx+1

$$\frac { 3 } { 2 } \arctan [ \frac { 2 \sqrt { 3 } } { 3 } ( \tan x + \frac { 1 } { 2 } ) ] + \frac { 3 \sqrt { 3 } } { 4 } \frac { \tan x + \frac { 1 } { 2 } } { \tan ^ { 2 } x + \tan x + 1 }$$

Evaluate

$$\frac{3\left(2\sin(2x)\arctan(\frac{\sqrt{3}\left(\cos(x)+2\sin(x)\right)}{3\cos(x)})+4\arctan(\frac{\sqrt{3}\left(\cos(x)+2\sin(x)\right)}{3\cos(x)})+\sqrt{3}\left(\cos(x)\right)^{2}+\sqrt{3}\sin(2x)\right)}{4\left(\sin(2x)+2\right)}$$

Factor

$$\frac{3\left(2\sin(2x)\arctan(\frac{\sqrt{3}\left(\cos(x)+2\sin(x)\right)}{3\cos(x)})+4\arctan(\frac{\sqrt{3}\left(\cos(x)+2\sin(x)\right)}{3\cos(x)})+\sqrt{3}\left(\cos(x)\right)^{2}+\sqrt{3}\sin(2x)\right)}{8\left(\frac{1}{2}\sin(2x)+1\right)}$$