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The given problem is:

\[
\text{Berechnen und vereinfachen Sie den folgenden Ausdruck so weit wie möglich:}
\]

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

Let's solve and simplify this expression.

### Step-by-Step Solution:

1. **Identify the terms**:
   \[
   \frac{1}{1+x} + \frac{1}{1-x^2} + \frac{1}{1-x}
   \]

2. **Recognize that \(1 - x^2\) can be rewritten**:
   \[
   1 - x^2 = (1 + x)(1 - x)
   \]
   So, \(\frac{1}{1 - x^2}\) becomes:
   \[
   \frac{1}{(1 + x)(1 - x)}
   \]

3. **Find the common denominator**:
   The common denominator for all terms is \((1 + x)(1 - x)\).

4. **Express each term with the common denominator**:
   - For \(\frac{1}{1 + x}\):
     \[
     \frac{1}{1 + x} = \frac{1 - x}{(1 + x)(1 - x)}
     \]
   - For \(\frac{1}{1 - x}\):
     \[
     \frac{1}{1 - x} = \frac{1 + x}{(1 + x)(1 - x)}
     \]
   - For \(\frac{1}{1 - x^2}\):
     \[
     \frac{1}{1 - x^2} = \frac{1}{(1 + x)(1 - x)}
     \]

5. **Combine all terms**:
   \[
   \frac{1 - x}{(1 + x)(1 - x)} + \frac{1}{(1 + x)(1 - x)} + \frac{1 + x}{(1 + x)(1 - x)}
   \]

6. **Simplify the numerator**:
   Combine the numerators:
   \[
   (1 - x) + 1 + (1 + x) = 3
   \]
   So, the expression becomes:
   \[
   \frac{3}{(1 + x)(1 - x)}
   \]

7. **Simplify the denominator**:
   The denominator simplifies back to \(1 - x^2\):
   \[
   \frac{3}{1 - x^2}
   \]

### Final Answer:
\[
\boxed{\frac{3}{1 - x^2}}
\]

Would you like more details on any part of this solution or have any questions?

### Related Questions:
1. How would the solution change if we had \(\frac{1}{1+x} - \frac{1}{1-x^2} + \frac{1}{1-x}\)?
2. What happens if \(x = 0\) in the simplified expression?
3. How do you find a common denominator for rational expressions with different factors?
4. What are the domain restrictions for \(\frac{3}{1 - x^2}\)?
5. How can this expression be graphed, and what does its graph look like?

### Tip:
When simplifying rational expressions, always look for ways to factor the denominators, which often helps in finding a common denominator and simplifying further.

Berechnen und vereinfachen Sie den folgenden Ausdruck so weit wie möglich: \frac{1}{1+x} + \frac{1}{1-x^2} + \frac{1}{1-x}

Learn how to simplify the rational expression 1/(1+x) + 1/(1-x^2) + 1/(1-x) using algebraic techniques like factoring the difference of squares. This problem helps reinforce understanding of rational expressions and their simplification. Suitable for students in Grades 9-12.

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