If `int(sin x)/(sin^(3)x + cos^(3)x) dx = alpha log_(e)|1 + tan x| + beta log_(e)|1-tan x + tan^(2)x| + gamma tan^(-1)((2 tan x - 1)/(sqrt(3))) + C`, when C is constant of integration, then the value of `18(alpha+ beta + gamma^(2))` is __________.

Answer

To solve the integral

\[ \int \frac{\sin x}{\sin^3 x + \cos^3 x} \, dx, \]

we will follow these steps:

### Step 1: Simplify the Integral
We start by dividing both the numerator and the denominator by \(\cos^3 x\):

\[ \int \frac{\sin x}{\sin^3 x + \cos^3 x} \, dx = \int \frac{\frac{\sin x}{\cos^3 x}}{\frac{\sin^3 x}{\cos^3 x} + 1} \, dx = \int \frac{\tan x \sec^2 x}{\tan^3 x + 1} \, dx. \]

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

\[ \sec^2 x \, dx = dt \quad \text{or} \quad dx = \frac{dt}{\sec^2 x} = \frac{dt}{1 + t^2}. \]

Substituting this into the integral gives:

\[ \int \frac{t}{t^3 + 1} \, dt. \]

### Step 3: Partial Fraction Decomposition
We can decompose \( \frac{t}{t^3 + 1} \):

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

Thus, we can express:

\[ \frac{t}{t^3 + 1} = \frac{A}{t + 1} + \frac{Bt + C}{t^2 - t + 1}. \]

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

### Step 4: Solve for Coefficients
Setting up the equation:

\[ t = A(t^2 - t + 1) + (Bt + C)(t + 1). \]

Expanding and collecting like terms will yield a system of equations to solve for \( A, B, \) and \( C \).

### Step 5: Integrate Each Term
Once we have \( A, B, \) and \( C \), we can integrate each term separately:

1. The integral of \( \frac{A}{t + 1} \) gives \( A \ln |t + 1| \).
2. The integral of \( \frac{Bt + C}{t^2 - t + 1} \) can be handled using substitution or completing the square.

### Step 6: Back Substitute
After integrating, we will substitute back \( t = \tan x \) to express the integral in terms of \( x \).

### Step 7: Identify Constants
From the final expression, we will identify \( \alpha, \beta, \gamma \) based on the logarithmic and arctangent terms in the result.

### Step 8: Calculate \( 18(\alpha + \beta + \gamma^2) \)
Finally, we substitute the values of \( \alpha, \beta, \gamma \) into the expression \( 18(\alpha + \beta + \gamma^2) \) to find the answer.