1/1-x + 1/1+x + 2/1+x^2 + 4/1+x^4

Let's simplify the expression step by step. The expression appears to be:

\[
\frac{1}{1 - x} + \frac{1}{1 + x} + \frac{2}{1 + x^2} + \frac{4}{1 + x^4} + \frac{1}{1 - x} + \frac{1}{1 + x} + \frac{2}{1 + x^2} + \frac{4}{1 + x^4}.
\]

We can combine like terms. Notice that the first four terms and the last four terms are identical. Therefore, we can simplify it as:

\[
2\left(\frac{1}{1 - x} + \frac{1}{1 + x} + \frac{2}{1 + x^2} + \frac{4}{1 + x^4}\right).
\]

Now, let's analyze each component separately:

1. **Combine \(\frac{1}{1 - x}\) and \(\frac{1}{1 + x}\)**:

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

2. **Compose \(\frac{2}{1 + x^2} + \frac{4}{1 + x^4}\)**:

We can write \(1+x^4 = (1 + x^2)(1 - x^2 + x^4)\). Thus, we can combine these fractions.

The common denominator would be \((1 + x^2)(1 + x^4)\):

- First, we need to express \(\frac{2}{1 + x^2}\) with the common denominator:

\[
\frac{2(1 + x^4)}{(1 + x^2)(1 + x^4)} = \frac{2 + 2x^4}{(1 + x^2)(1 + x^4)}.
\]

- The next one is:

\[
\frac{4}{1 + x^4} = \frac{4(1 + x^2)}{(1 + x^2)(1 + x^4)} = \frac{4 + 4x^2}{(1 + x^2)(1 + x^4)}.
\]

Now, we can add these two results:

\[
\frac{(2 + 2x^4) + (4 + 4x^2)}{(1 + x^2)(1 + x^4)} = \frac{6 + 4x^2 + 2x^4}{(1 + x^2)(1 + x^4)}.
\]

3. **Putting everything back together**:

Now plug the combined fractions back into the expression:

\[
2\left(\frac{2}{1 - x^2} + \frac{6 + 4x^2 + 2x^4}{(1 + x^2)(1 + x^4)}\right).
\]

Thus, the expression simplifies to:

\[
2\left(\frac{2}{1 - x^2} + \frac{6 + 4x^2 + 2x^4}{(1 + x^2)(1 + x^4)}\right).
\]

This is a clearer way to represent the original expression. Further simplifications, if desired, will depend on specific values of \(x\) or additional context.