Answer
$$\sec x-\ln |\csc x+\cot x|+C $$
Work Step by Step
We integrate as follows:
\begin{align*}
\int \frac{\sec ^{3} x}{\tan x} d x&=\int \frac{\sec ^{2} x \sec x}{\tan x} d x\\
&=\int \frac{\left(\tan ^{2} x+1\right) \sec x}{\tan x} d x\\
&=\int \frac{\tan ^{2} x \sec x}{\tan x} d x+\int \frac{\sec x}{\tan x} d x\\
&=\int \tan x \sec x d x+\int \csc x d x \\
&=\sec x-\ln |\csc x+\cot x|+C
\end{align*}
