If $f(x)=\int\left[\tan ^2 x+\cot ^2 x+\frac{4\left(\sin ^3 x+\cos ^3 x\right)}{\sin ^2 2 x}\right] d x$ and $f\left(\frac{\pi}{4}\right)=0$, then $3\left[f\left(\frac{\pi}{6}\right)+2\right]=$