If $I=\int {\sec }^{2}x{\text{\hspace{0.17em}cosec}}^{4}xdx=A{\cot }^{3}x+B\tan x+Ccotx+D$ , where D is integration constant, then which of the following is/are true?

$$\begin{gathered} I=\int {\sec }^{2}x{\text{\hspace{0.17em}\hspace{0.17em}cosec}}^{4}xdx=\int \frac{{({\sin }^{2}x+{\cos }^{2}x)}^2}{{\cos }^{2}x{\sin }^{4}x}dx \\ =\int \frac{{\sin }^{4}x+{\cos }^{4}x+2{\sin }^{2}x{\cos }^{2}x}{{\cos }^{2}x{\sin }^{4}x}dx \\ =\int ({\sec }^{2}x+2{\text{cosec}}^{2}x+\frac{{\cos }^{2}x}{{\sin }^{4}x})dx \\ =\tan x-2\cot x+\int {\cot }^{2}x{\text{\hspace{0.17em}cosec}}^{2}xdx \\ =\tan x-2cotx-\frac{{\cot }^{3}x}{3}+D \end{gathered}$$