Answer
$$\sec x + x + C$$
Work Step by Step
$$\eqalign{
& \int {\sec x\left( {\tan x + \cos x} \right)} dx \cr
& \int {\left( {\sec x\tan x + \sec x\cos x} \right)} dx \cr
& {\text{identity }}\sec x = \frac{1}{{\cos x}} \cr
& \int {\left( {\sec x\tan x + \frac{1}{{\cos x}}\cos x} \right)} dx \cr
& \int {\left( {\sec x\tan x + 1} \right)} dx \cr
& {\text{sum rule}} \cr
& \int {\sec x\tan x} dx + \int {dx} \cr
& {\text{find antiderivatives}} \cr
& = \sec x + x + C \cr} $$
