$\int \frac{1}{{\cos }^{2}x{\sin }^{4}x}dx=$

$I=\int \frac{1}{{\cos }^{2}x{\sin }^{4}x}dx$

$=\int \frac{{\left({\sin }^{2}x+{\cos }^{2}x\right)}^2}{{\cos }^{2}x{\sin }^{4}x}dx$

$\begin{array}{l}=\int \frac{{\sin }^{4}x+2{\sin }^{2}x{\cos }^{2}x+{\cos }^{4}x}{{\cos }^{2}x{\sin }^{4}x}dx\\ =\int {\sec }^{2}x+2{\mathrm{cosec}}^{2}x+{\cot }^{2}x{\mathrm{cosec}}^{2}x\\ =\tan x-2\cot x-\frac{{\cot }^{3}x}{3}+c\end{array}$