`int((1-x^(- 2))/(x^(1//2)-x^(-1//2))-2/(x^(3//2))+(x^(- 2)-x)/(x^(1//2)-x^(-1//2)))dx`

Answer

To solve the integral

\[ \int \left( \frac{1 - x^{-2}}{x^{1/2} - x^{-1/2}} - \frac{2}{x^{3/2}} + \frac{x^{-2} - x}{x^{1/2} - x^{-1/2}} \right) dx, \]

we will simplify the expression step by step.

### Step 1: Simplifying the Expression
First, we notice that the denominator \( x^{1/2} - x^{-1/2} \) is common in the first and third terms. We can combine these fractions:

\[ \frac{1 - x^{-2} + x^{-2} - x}{x^{1/2} - x^{-1/2}} - \frac{2}{x^{3/2}}. \]

This simplifies to:

\[ \frac{1 - x}{x^{1/2} - x^{-1/2}} - \frac{2}{x^{3/2}}. \]

### Step 2: Further Simplification
Now, we can rewrite the denominator:

\[ x^{1/2} - x^{-1/2} = \frac{x - 1}{x^{1/2}}. \]

Thus, the expression becomes:

\[ \frac{1 - x}{\frac{x - 1}{x^{1/2}}} - \frac{2}{x^{3/2}} = \frac{(1 - x) x^{1/2}}{x - 1} - \frac{2}{x^{3/2}}. \]

Notice that \( 1 - x = -(x - 1) \), so we can rewrite the first term:

\[ -\frac{(x - 1) x^{1/2}}{x - 1} - \frac{2}{x^{3/2}} = -x^{1/2} - \frac{2}{x^{3/2}}. \]

### Step 3: Combining Terms
Now we have:

\[ \int \left( -x^{1/2} - \frac{2}{x^{3/2}} \right) dx. \]

### Step 4: Integrating Each Term
We can integrate each term separately:

1. For \(-x^{1/2}\):

\[ \int -x^{1/2} \, dx = -\frac{2}{3} x^{3/2}. \]

2. For \(-\frac{2}{x^{3/2}}\):

\[ \int -\frac{2}{x^{3/2}} \, dx = -2 \cdot \int x^{-3/2} \, dx = -2 \cdot \left(-\frac{2}{\sqrt{x}}\right) = \frac{4}{\sqrt{x}}. \]

### Step 5: Combining the Results
Combining the results of the integrals, we get:

\[ -\frac{2}{3} x^{3/2} + \frac{4}{\sqrt{x}} + C, \]

where \( C \) is the constant of integration.

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

\[ \int \left( \frac{1 - x^{-2}}{x^{1/2} - x^{-1/2}} - \frac{2}{x^{3/2}} + \frac{x^{-2} - x}{x^{1/2} - x^{-1/2}} \right) dx = -\frac{2}{3} x^{3/2} + \frac{4}{\sqrt{x}} + C. \]