Chapter 8 - Techniques of Integration - 8.2 Trigonometric Integrals - Exercises - Page 403: 20

$$\frac{1}{2} \tan x \sec x+\frac{1}{2}\ln |\sec x+\tan x|+c$$

Since $$\int \sec ^{m} x d x=\frac{\tan x \sec ^{m-2} x}{m-1}+\frac{m-2}{m-1} \int \sec ^{m-2} x d x$$ Then \begin{align*} \int \sec ^{3} x d x&=\frac{1}{2} \tan x \sec x+\frac{1}{2} \int \sec x d x\\ &=\frac{1}{2} \tan x \sec x+\frac{1}{2}\ln |\sec x+\tan x|+c \end{align*}