${\displaystyle \int \frac{dx}{\cos x\sqrt{\cos 2x}}}=$

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

                 $={\displaystyle \int \frac{{\sec }^{2}x}{\sqrt{1-{\tan }^{2}x}}}dx$                        $\tan x=t$

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

                  $={\sin }^{-1}t+c$

                  $={\sin }^{-1}\left(\tan x\right)+c$