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

${\displaystyle \int \frac{1}{{\sin }^{2}x\sqrt{{\cot }^{2}x-1}}}dx$

             $={\displaystyle \int \frac{\cos e{c}^{2}x}{\sqrt{{\cot }^{2}x-1}}}dx$

$\cot x=t\Rightarrow -\cos e{c}^{2}xdx=dt$

            $=-{\displaystyle \int \frac{1}{\sqrt{t^2-1}}}dt$

            $=-{\cosh }^{-1}\left(\cot x\right)+C$