We have
$$
\begin{aligned}
& I=\int \sec ^5 x d x \\
& I=\int \sec ^3 \cdot \sec ^2 x d x \\
& I=\sec ^3 x \tan x-\int 3 \sec ^3 x \tan ^2 x d x \\
& I=\sec ^3 x \tan x-3 \int \sec ^3 x\left(1+\sec ^2 x\right) d x \\
& I=\sec ^3 x \tan x-3 \int \sec ^5 x d x-3 \int \sec ^3 x d x \\
& I=\sec ^3 x \tan x-3 I-3 \int \sec ^2 x \sqrt{1+\tan ^2 x} d x \\
& 4 I=\sec ^3 x \tan x-3 \int \sqrt{t^2+1} d t\left[\because \operatorname{put}^2 \tan x=t\right] \\
& 4 I=\sec ^3 x \tan x-\frac{3 t}{2} \sqrt{1+t^2}+\frac{3}{2} \\
& \log \left|t+\sqrt{1+t^2}\right|+c
\end{aligned}
$$

$$
\begin{aligned}
& I=\frac{\sec ^3 x \tan x}{4}-\frac{3}{8} \sec x \tan x+\frac{3}{8} \log \\
& I=\frac{\left(\sec ^2 x\right) \sec x \tan x}{4}+\frac{3}{8} \sec x \tan x+\frac{3}{8} \\
& \log |\sec x+\tan x|+c \\
& I=\frac{\left(1+\tan ^2 x\right) \sec x \tan x}{4}+\frac{3}{8} \sec x \tan x \\
& I=\frac{\sec x \tan ^3 x}{4}+\frac{3}{8} \sec x \tan x+\frac{3}{8} \log |\sec x+\tan x|+c \\
& \log |\sec x+\tan x|+c
\end{aligned}
$$