Answer
$$
\int \sec ^{5} x d x = \frac{1}{4} \tan x \sec ^{3} x+\frac{3}{8} \tan x \sec x+\frac{3}{8} \ln |\sec x+\tan x|+C
$$
Work Step by Step
$$
\int \sec ^{5} x d x
$$
If we look at the Table of Integrals , we see that the closest entry is number
$77$ with $n=5 $, for the second integral we use Formula $77$ with $n=3,$ and for the last integral we use Formula $14$ :
$$
\begin{aligned}
\int \sec ^{5} x d x & \stackrel{77}{=} \frac{1}{4} \tan x \sec ^{3} x+\frac{3}{4} \int \sec ^{3} x d x\\
&=\frac{1}{4} \tan x \sec ^{3} x+\frac{3}{4}\left(\frac{1}{2} \tan x \sec x+\frac{1}{2} \int \sec x d x\right) \\
&\stackrel{14}{=} \frac{1}{4} \tan x \sec ^{3} x+\frac{3}{8} \tan x \sec x+\frac{3}{8} \ln |\sec x+\tan x|+C
\end{aligned}
$$
