4/(x^{4}+1)

4/x^{2} (x^{2}+1/x^{2})

4/x^{2} = (2+2/x^{2})-(2-2/x^{2})

Integral 2(1+1/x^{2}) /(x^{2}+1/x^{2})dx- Integral 2(1-1/x^{2}) /(x^{2}+1/x^{2})dx

In the first integral put x-1/x = t

In the second integral put x+1/x = u

Integral 2 dt /(t^{2}+2)-integral 2 du /(u^{2}-2)

2/√2 tan ^{-1 } t/√2- integral 2/2√2 [1/(u-√2)-1/(u+√2)]

2/√2 tan ^{-1 } t/√2-1/√2 ln [(u-√2)/(u+√2)]+c

2/√2 tan ^{-1 }(x-1/x)/ √2- 1/√2 ln [(x+1/x-√2)/(x+1+√2)]+c

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