Answer
$$\tan x + \sec x + C$$
Work Step by Step
$$\eqalign{
& \int {\sec x\left( {\sec x + \tan x} \right)} dx \cr
& {\text{multiply}} \cr
& = \int {\sec x\left( {\sec x + \tan x} \right)} dx \cr
& = \int {\left( {{{\sec }^2}x + \sec x\tan x} \right)} dx \cr
& {\text{sum rule}} \cr
& = \int {{{\sec }^2}x} dx + \int {\sec x\tan x} dx \cr
& {\text{use integration formulas from table 4}}{\text{.2}}{\text{.1}} \cr
& = - \cot t - \sec t + C \cr
& = \tan x + \sec x + C \cr
& \cr
& {\text{check by differentiation}} \cr
& = \frac{d}{{dt}}\left[ {\tan x + \sec x + C} \right] \cr
& = {\sec ^2}x + \sec x\tan x + 0 \cr
& = \sec x\left( {\sec x + \tan x} \right) \cr} $$
