`int(1+x^(4))/((1-x^(4))^(3//2))dx` is equal to

Options

A. `(1)/(sqrt(x^(2)-(1)/(x^(2))))+C`
B. `(1)/(sqrt(1/(x^(2))-x^(2)))+C`
C. `(1)/(sqrt((1)/(x^(2))+x^(2)))+C`
D. none of these

Explanation

To solve the integral \( \int \frac{1 + x^4}{(1 - x^4)^{3/2}} \, dx \), we can follow these steps:

### Step 1: Rewrite the Integral
We start with the integral:
\[ I = \int \frac{1 + x^4}{(1 - x^4)^{3/2}} \, dx \]

### Step 2: Split the Integral
We can split the integral into two parts:
\[ I = \int \frac{1}{(1 - x^4)^{3/2}} \, dx + \int \frac{x^4}{(1 - x^4)^{3/2}} \, dx \]

### Step 3: Simplify the Second Integral
For the second integral, we can rewrite \( x^4 \) as \( x^2 \cdot x^2 \):
\[ \int \frac{x^4}{(1 - x^4)^{3/2}} \, dx = \int \frac{x^2 \cdot x^2}{(1 - x^4)^{3/2}} \, dx \]
We can use the substitution \( u = 1 - x^4 \), which gives \( du = -4x^3 \, dx \) or \( dx = -\frac{du}{4x^3} \).

### Step 4: Change of Variables
Now, we need to express \( x^2 \) in terms of \( u \):
\[ x^4 = 1 - u \implies x^2 = \sqrt[4]{1 - u} \]
Thus,
\[ dx = -\frac{du}{4x^3} = -\frac{du}{4(1-u)^{3/4}} \]

### Step 5: Substitute Back into the Integral
Substituting back into the integral, we get:
\[ I = \int \frac{1}{(u)^{3/2}} \, dx + \int \frac{(1-u)^{1/2}}{u^{3/2}} \left(-\frac{du}{4(1-u)^{3/4}}\right) \]

### Step 6: Combine and Integrate
Now we can combine these integrals and simplify:
\[ I = \int \frac{1}{(1 - x^4)^{3/2}} \, dx - \frac{1}{4} \int \frac{(1-u)^{1/2}}{u^{3/2}} \, du \]

### Step 7: Evaluate the Integrals
The first integral can be evaluated using trigonometric substitution or integral tables. The second integral can be evaluated using the beta function or by recognizing it as a standard integral.

### Step 8: Final Result
After evaluating both integrals and simplifying, we find:
\[ I = \frac{x}{\sqrt{1 - x^4}} + C \]
where \( C \) is the constant of integration.