`intsqrt(1+2 tan x (tan x + sec x ) ) dx`

Answer

To solve the integral \( \int \sqrt{1 + 2 \tan x (\tan x + \sec x)} \, dx \), we will follow these steps:

### Step 1: Simplify the expression under the square root
We start with the expression inside the square root:
\[ 1 + 2 \tan x (\tan x + \sec x) \]
Expanding this, we have:
\[ 1 + 2 \tan^2 x + 2 \tan x \sec x \]

### Step 2: Use the identity \( \sec^2 x - \tan^2 x = 1 \)
We can rewrite \( 1 \) as \( \sec^2 x - \tan^2 x \):
\[ \sec^2 x - \tan^2 x + 2 \tan^2 x + 2 \tan x \sec x \]
This simplifies to:
\[ \sec^2 x + \tan^2 x + 2 \tan x \sec x \]

### Step 3: Recognize the perfect square
Notice that:
\[ \tan^2 x + 2 \tan x \sec x + \sec^2 x = (\tan x + \sec x)^2 \]
Thus, we can rewrite the integral as:
\[ \int \sqrt{(\tan x + \sec x)^2} \, dx \]
Since \( \tan x + \sec x \) is positive in the interval we are considering, we have:
\[ \sqrt{(\tan x + \sec x)^2} = \tan x + \sec x \]

### Step 4: Integrate
Now we can integrate:
\[ \int (\tan x + \sec x) \, dx = \int \tan x \, dx + \int \sec x \, dx \]

### Step 5: Use known integrals
The integrals of \( \tan x \) and \( \sec x \) are:
\[ \int \tan x \, dx = -\log |\cos x| + C_1 \]
\[ \int \sec x \, dx = \log |\sec x + \tan x| + C_2 \]
Combining these, we get:
\[ \int (\tan x + \sec x) \, dx = -\log |\cos x| + \log |\sec x + \tan x| + C \]

### Final Result
Thus, the final result of the integral is:
\[ \int \sqrt{1 + 2 \tan x (\tan x + \sec x)} \, dx = \log |\sec x + \tan x| - \log |\cos x| + C \]