Evaluate the integral:\(\int \frac{x^2\left(x \sec ^2 x+\tan x\right)}{(x \tan x+1)^2} d x\)

\(\int {x}^{2}\cdot \frac{\left(x{\sec }^{2}x+\tan x\right)}{(x\tan x+1{)}^{2}}dx\\ III\)

Using integral by parts

\(=\frac{-{x}^{2}}{(x\tan x+1)}+\int \frac{2x}{x\tan x+1}dx\\ letI=\int \frac{2x}{x\tan x+1}dx\)

\(I=2\int \frac{x}{x\tan x+1}dx\)

\(=2\int \frac{x\cos x}{x\sin x+\cos x}dx\\ \text{ Let }x\sin x+\cos x=t\\ (x\cos x+\sin x-\sin x)dx=dt\)

\(=2\int \frac{dt}{t}=2\log t+{c}^{'}\\ =2\log |x\sin x+\cos x|+{c}^{'}\\ ∴\int \frac{{x}^{2}\left(x{\sec }^{2}x+\tan x\right)}{(x\tan x+1{)}^{2}}dx\\ =\frac{-{x}^{2}}{x\tan x+1}+2\log |x\sin x+\cos x|+c\)