integration of .......?

Question

integrate: (x^2 + 1) / (x^4 + 1) wrt dx.

hint: the integral can be also wirten as (1 + 1/x^2) / (x^2 + 1/x^2 + 2 - 2).......

Answer 1

x^4+1=(x^2-√2 x +1)·(x^2+√2 x +1)
(x^2 + 1) / (x^4 + 1) = 1/2 · [ 1/(x^2-√2 x +1) + 1/(x^2-√2 x +1) ]

x^2-√2 x +1=(x-√2/2)^2 +1/2 = (√2/√2 x- 1/√2)^2 +1/2 =
1/2 · [ (√2x-1)^2+1]

S (x^2 + 1) / (x^4 + 1) dx=
S 1/[ (√2x-1)^2+1] ·dx + S 1/[ (√2x+1)^2+1] ·dx =
√2/2 · [ arctan (√2x-1) +arctan (√2x+1)] +K

Saludos.