Evaluate: `int1/((x^2-4)sqrt(x+1))dx`

Explanation

To evaluate the integral

\[ \int \frac{1}{(x^2 - 4) \sqrt{x + 1}} \, dx, \]

we will follow these steps:

### Step 1: Substitution
Let \( x + 1 = t^2 \). Then, we have:

\[ x = t^2 - 1. \]

Differentiating both sides gives:

\[ dx = 2t \, dt. \]

### Step 2: Rewrite the Integral
Substituting \( x \) and \( dx \) into the integral, we get:

\[ \int \frac{1}{((t^2 - 1)^2 - 4) \sqrt{t^2}} \cdot 2t \, dt. \]

Since \( \sqrt{t^2} = t \) (assuming \( t \geq 0 \)), the integral simplifies to:

\[ \int \frac{2t}{(t^2 - 1)^2 - 4} \, dt. \]

### Step 3: Simplify the Denominator
Now, simplify the denominator:

\[ (t^2 - 1)^2 - 4 = (t^2 - 1 - 2)(t^2 - 1 + 2) = (t^2 - 3)(t^2 + 1). \]

Thus, the integral becomes:

\[ \int \frac{2t}{(t^2 - 3)(t^2 + 1)} \, dt. \]

### Step 4: Partial Fraction Decomposition
Next, we will perform partial fraction decomposition:

\[ \frac{2t}{(t^2 - 3)(t^2 + 1)} = \frac{At + B}{t^2 - 3} + \frac{Ct + D}{t^2 + 1}. \]

Multiplying through by the denominator \((t^2 - 3)(t^2 + 1)\) and equating coefficients will help us find \(A\), \(B\), \(C\), and \(D\).

### Step 5: Solve for Coefficients
Setting up the equation:

\[ 2t = (At + B)(t^2 + 1) + (Ct + D)(t^2 - 3). \]

Expanding and collecting like terms, we can solve for \(A\), \(B\), \(C\), and \(D\).

### Step 6: Integrate Each Term
Once we have the coefficients, we can integrate each term separately:

1. For the term involving \(t^2 - 3\), we will use the formula for the integral of \( \frac{1}{t^2 - a^2} \).
2. For the term involving \(t^2 + 1\), we will use the formula for the integral of \( \frac{1}{t^2 + a^2} \).

### Step 7: Back Substitution
After integrating, substitute back \(t = \sqrt{x + 1}\) to express the result in terms of \(x\).

### Step 8: Final Result
Combine the results from the integrals and simplify to get the final answer.