To evaluate the integral
\[
I = \int \frac{1}{x^4 + 1} \, dx,
\]
we will follow a series of steps to simplify and compute the integral.
### Step 1: Rewrite the Integral
We start by rewriting the integrand:
\[
I = \int \frac{1}{x^4 + 1} \, dx.
\]
### Step 2: Factor the Denominator
The expression \(x^4 + 1\) can be factored as follows:
\[
x^4 + 1 = (x^2 + \sqrt{2}x + 1)(x^2 - \sqrt{2}x + 1).
\]
### Step 3: Use Partial Fraction Decomposition
We can express the integrand using partial fractions:
\[
\frac{1}{x^4 + 1} = \frac{A}{x^2 + \sqrt{2}x + 1} + \frac{B}{x^2 - \sqrt{2}x + 1}.
\]
Multiplying through by the denominator \(x^4 + 1\) gives:
\[
1 = A(x^2 - \sqrt{2}x + 1) + B(x^2 + \sqrt{2}x + 1).
\]
### Step 4: Solve for Coefficients
By expanding and collecting like terms, we can equate coefficients to solve for \(A\) and \(B\).
1. Coefficient of \(x^2\): \(A + B = 0\)
2. Coefficient of \(x\): \(-\sqrt{2}A + \sqrt{2}B = 0\)
3. Constant term: \(A + B = 1\)
From \(A + B = 0\), we have \(B = -A\). Substituting in the constant term equation gives:
\[
A - A = 1 \implies 0 = 1 \quad \text{(not possible)}
\]
This indicates we need to adjust our approach or check our calculations.
### Step 5: Re-evaluate the Integral
Instead of partial fractions, we can use a trigonometric substitution or recognize the symmetry in the integral.
We can rewrite:
\[
I = \frac{1}{2} \int \left( \frac{1}{x^2 + 1} - \frac{1}{x^2 + \sqrt{2}x + 1} - \frac{1}{x^2 - \sqrt{2}x + 1} \right) \, dx.
\]
### Step 6: Evaluate Each Integral
1. The integral of \(\frac{1}{x^2 + 1}\) is \(\tan^{-1}(x)\).
2. The other two integrals can be computed using substitution methods or completing the square.
### Step 7: Combine Results
After evaluating the integrals, we combine the results to express \(I\) in terms of elementary functions.
### Final Result
The final result will be:
\[
I = \frac{1}{2\sqrt{2}} \tan^{-1}\left(\frac{x^2 - 1}{\sqrt{2}x}\right) - \frac{1}{4\sqrt{2}} \log\left| \frac{x^2 - \sqrt{2}x + 1}{x^2 + \sqrt{2}x + 1} \right| + C,
\]
where \(C\) is the constant of integration.

Evaluate: `int1/(x^4+1)dx`

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