Answer
$$\int\frac{\sec^3x}{\tan x}dx=\sec x-\ln|\csc x+\cot x|+C$$
Work Step by Step
$$A=\int\frac{\sec^3x}{\tan x}dx=\int\frac{\sec^2x\sec x}{\tan x}dx$$ $$A=\int\frac{(\tan^2x+1)\sec x}{\tan x}dx$$ $$A=\int\frac{\tan^2x\sec x}{\tan x}dx+\int\frac{\sec x}{\tan x}dx$$ $$A=\int\sec x\tan xdx+\int\frac{\frac{1}{\cos x}}{\frac{\sin x}{\cos x}}dx$$ $$A=\int d(\sec x)+\int\frac{1}{\sin x}dx$$ $$A=\sec x+\int\csc xdx$$ $$A=\sec x-\ln|\csc x+\cot x|+C$$
