Evaluate :
`int(sqrt(x+1)-sqrt(x-1))/(sqrt(x+1)+sqrt(x-1))dx`

Answer

To evaluate the integral

\[ \int \frac{\sqrt{x+1} - \sqrt{x-1}}{\sqrt{x+1} + \sqrt{x-1}} \, dx, \]

we will follow these steps:

### Step 1: Rationalize the Numerator
We start by rationalizing the numerator. We multiply and divide by the conjugate of the numerator:

\[ \int \frac{(\sqrt{x+1} - \sqrt{x-1})(\sqrt{x+1} - \sqrt{x-1})}{(\sqrt{x+1} + \sqrt{x-1})(\sqrt{x+1} - \sqrt{x-1})} \, dx. \]

This gives us:

\[ \int \frac{(\sqrt{x+1} - \sqrt{x-1})^2}{(\sqrt{x+1})^2 - (\sqrt{x-1})^2} \, dx. \]

### Step 2: Simplify the Denominator
The denominator simplifies as follows:

\[ (\sqrt{x+1})^2 - (\sqrt{x-1})^2 = (x+1) - (x-1) = 2. \]

So we rewrite the integral:

\[ \int \frac{(\sqrt{x+1} - \sqrt{x-1})^2}{2} \, dx = \frac{1}{2} \int (\sqrt{x+1} - \sqrt{x-1})^2 \, dx. \]

### Step 3: Expand the Numerator
Next, we expand the numerator:

\[ (\sqrt{x+1} - \sqrt{x-1})^2 = (\sqrt{x+1})^2 - 2\sqrt{x+1}\sqrt{x-1} + (\sqrt{x-1})^2 = (x+1) - 2\sqrt{(x+1)(x-1)} + (x-1). \]

This simplifies to:

\[ 2x - 2\sqrt{(x+1)(x-1)} = 2x - 2\sqrt{x^2 - 1}. \]

### Step 4: Substitute Back into the Integral
Substituting this back into the integral, we have:

\[ \frac{1}{2} \int (2x - 2\sqrt{x^2 - 1}) \, dx = \int (x - \sqrt{x^2 - 1}) \, dx. \]

### Step 5: Split the Integral
We can split the integral into two parts:

\[ \int x \, dx - \int \sqrt{x^2 - 1} \, dx. \]

### Step 6: Evaluate Each Integral
1. The first integral:

\[ \int x \, dx = \frac{x^2}{2}. \]

2. The second integral can be evaluated using the formula:

\[ \int \sqrt{x^2 - a^2} \, dx = \frac{x}{2} \sqrt{x^2 - a^2} - \frac{a^2}{2} \log(x + \sqrt{x^2 - a^2}) + C. \]

For \(a = 1\):

\[ \int \sqrt{x^2 - 1} \, dx = \frac{x}{2} \sqrt{x^2 - 1} - \frac{1}{2} \log(x + \sqrt{x^2 - 1}). \]

### Step 7: Combine the Results
Putting it all together, we have:

\[ \int \frac{\sqrt{x+1} - \sqrt{x-1}}{\sqrt{x+1} + \sqrt{x-1}} \, dx = \frac{x^2}{2} - \left( \frac{x}{2} \sqrt{x^2 - 1} - \frac{1}{2} \log(x + \sqrt{x^2 - 1}) \right) + C. \]

### Final Answer
Thus, the final answer is:

\[ \frac{x^2}{2} - \frac{x}{2} \sqrt{x^2 - 1} + \frac{1}{2} \log(x + \sqrt{x^2 - 1}) + C. \]