Answer
$$ \displaystyle\int \dfrac{1}{{x}^{4}+1}\, \mathrm d x = {{\ln \left(x^2+\sqrt{2}\,x+1\right)-\ln \left(x^2-\sqrt{2}\,x+1 \right)+2\,\arctan \left({{2\,x+\sqrt{2}}\over{\sqrt{2}}}\right)+2\, \arctan \left({{2\,x-\sqrt{2}}\over{\sqrt{2}}}\right)}\over{2^{{{5 }\over{2}}}}} $$
Explanation
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