\(\int \frac{1}{\cos ^3 x \sqrt{\sin 2 x}} \mathrm{~d} x=\)

\( \text {Let } \mathrm{I} =\int \frac{1}{\cos ^3 x \sqrt{\sin 2 x}} \mathrm{~d} x \)
\( =\frac{1}{\sqrt{2}} \int \frac{\sec ^3 x}{\sqrt{\sin x \cos x}} \mathrm{~d} \)
\( =\frac{1}{\sqrt{2}} \int \frac{\sec ^4 x}{\sqrt{\tan x}} \mathrm{~d} x\)
Let \(\tan x=\mathrm{t} \Rightarrow \sec ^2 x=\mathrm{dt}\)
\(\therefore I =\frac{1}{\sqrt{2}} \int \frac{1+t^2}{\sqrt{t}} d t \)
\( =\frac{1}{\sqrt{2}} \int t^{-\frac{1}{2}} d t+\frac{1}{\sqrt{2}} \int t^{\frac{3}{2}} d t \)
\( =\frac{1}{\sqrt{2}}\left(\frac{t^{\frac{1}{2}}}{\frac{1}{2}}\right)+\frac{1}{\sqrt{2}}\left(\frac{t^{\frac{5}{2}}}{\frac{5}{2}}\right)+\mathrm{c} \)
\( =\sqrt{2}\left(\sqrt{\tan x}+\frac{1}{5}(\tan x)^{\frac{5}{2}}\right)+\mathrm{c}\)