`int (x^2(xsec^2x+tanx))/(xtanx+1)^2` dx

Answer

To solve the integral

\[ I = \int \frac{x^2 (x \sec^2 x + \tan x)}{(x \tan x + 1)^2} \, dx, \]

we can follow these steps:

### Step 1: Identify the substitution
We notice that the derivative of the denominator \(x \tan x + 1\) resembles the numerator. Thus, we can set:

\[ p = x \tan x + 1. \]

### Step 2: Differentiate the substitution
Now, we differentiate \(p\):

\[ \frac{dp}{dx} = \tan x + x \sec^2 x. \]

This gives us:

\[ dp = (\tan x + x \sec^2 x) \, dx. \]

### Step 3: Rewrite the integral
From our substitution, we can express the integral in terms of \(p\):

\[ I = \int \frac{x^2}{p^2} \, dp. \]

### Step 4: Simplify the integral
Next, we can express \(x^2\) in terms of \(p\):

From \(p = x \tan x + 1\), we can isolate \(x\):

\[ x = \frac{p - 1}{\tan x}. \]

However, this approach becomes complicated. Instead, we will use integration by parts.

### Step 5: Apply integration by parts
Let:

\[ u = x^2 \quad \text{and} \quad dv = \frac{x \sec^2 x + \tan x}{(x \tan x + 1)^2} \, dx. \]

Then, we compute \(du\) and \(v\):

\[ du = 2x \, dx, \]

and we need to find \(v\) by integrating \(dv\).

### Step 6: Integrate \(dv\)
We can express \(v\) as:

\[ v = \int \frac{x \sec^2 x + \tan x}{(x \tan x + 1)^2} \, dx. \]

This integral can be simplified and computed using the substitution we defined earlier.

### Step 7: Combine results
Using integration by parts, we have:

\[ I = u v - \int v \, du. \]

### Step 8: Final integration
After performing the integration and substituting back for \(p\), we will arrive at the final answer.

### Final Answer
The final result of the integral is:

\[ I = -\frac{x^2}{x \tan x + 1} + 2 \ln |x \sin x + \cos x| + C, \]

where \(C\) is the constant of integration.

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