If   $\int \frac{dx}{\sqrt[3]{{\sin }^{11}x\cos x}}=$ $-\left(\frac{3}{8}f\left(x\right)+\frac{3}{2}g\left(x\right)\right)+c$  then 

$\int \frac{dx}{\sqrt[3]{\frac{{\sin }^{11}x}{{\cos }^{11}x}.{\cos }^{12}x}}\Rightarrow \int \frac{1+t^2}{t^{11/3}}dt$

Put   tanx =t  ${\sec }^{2}xdx=dt$

$\int \left(t^{-11/3}+t^{-5/3}\right)dt$

$\Rightarrow -\left(\frac{3}{8}.\tan x^{-8/3}+\frac{3}{2}{\tan }^{-2/3}\right)+c$

$\therefore f\left(x\right)={\tan }^{-8/3}x.g\left(x\right)={\tan }^{-2/3}x$