Primitive of Reciprocal of One plus Fourth Power of x

(or just invoke that page as a second proof of this, based on the fact that this is a special case of that)

Theorem

 * $\displaystyle \int \frac 1 {1 + x^4} \rd x = \frac 1 {2 \sqrt 2} \paren {\map \arctan {\frac 1 {\sqrt 2} \paren {x - \frac 1 x} } + \frac 1 2 \ln \size {\frac {x^2 + \sqrt 2 x + 1} {x^2 - \sqrt 2 x + 1} } } + C$

Proof
Note that, by Derivative of Power:


 * $\dfrac \d {\d x} \paren {x - \dfrac 1 x} = 1 + \dfrac 1 {x^2}$

and:


 * $\dfrac \d {\d x} \paren {x + \dfrac 1 x} = 1 - \dfrac 1 {x^2}$

We have:

We also have:

Then: