The value of integral $\int \frac{dx}{{\cos }^{3}x\sqrt{\sin 2x}}$ can be expressed as  

write the given integral as 

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

put $\tan x=t$ to obtain 

$\begin{array}{l}I=\frac{1}{\sqrt{2}}\int \frac{1+t^2}{\sqrt{t}}dt=\frac{1}{\sqrt{2}}\int \left[t^{-1/2}+t^{3/2}\right]dt\\ =\frac{1}{\sqrt{2}}\left[2\sqrt{t}+\frac{2}{5}{t}^{5/2}\right]+C\\ =\sqrt{2}\left[\sqrt{\tan x}+\frac{1}{5}(\tan x{)}^{5/2}\right]+C.\end{array}$

Thus, $I$ is an irrational function of $\tan x$