The sum of distinct solutions of the equation $\left(4\tan x+{\tan }^{2}x+1\right)=2\sqrt{2}\sin \left(x+\frac{\pi }{4}\right)\left(1+{\tan }^{2}x\right)$ in the interval  $\left[0,\pi \right]$  is:

$\begin{array}{c}\left(4\tan x+{\sec }^{2}x\right)=2\sqrt{2}\sin \left(x+\frac{\pi }{4}\right){\sec }^{2}x\\ \Rightarrow 4\sin x\cos x+1=2\left(\sin x+\cos x\right)\end{array}$

                           $\Rightarrow 2\sin x\left(2\cos x-1\right)-\left(2\cos x-1\right)=0$

                              $\begin{array}{c}\sin x=\frac{1}{2},\cos x=\frac{1}{2}\\ \Rightarrow x=\frac{\pi }{6},\frac{5\pi }{6},\frac{\pi }{3}\end{array}$