Primitive of One plus x Squared over One plus Fourth Power of x

Theorem

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

Proof
We have:

Note that, by Derivative of Power:


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

So, we have: