<s>[INST] Integrating $\int_0^\infty\frac{1}{1+x^6}dx$ $$I=\int_0^\infty\frac{1}{1+x^6}dx$$
How do I evaluate this?
 [/INST] Without complex analysis nor special functions: 
$$x^6+1=(x^2+1)(x^4-x^2+1)=(x^2+1)(x^2+\sqrt3x +1)(x^2-\sqrt3x+1).$$
($x=\pm i$ are obviously roots and $(x^4-x^2+1)$ is biquadratic)
And now you have the following partial fraction decomposition:
$${1\over x^6+1} =
{Ax+B\over x^2+1} + {Cx+D\over x^2+\sqrt3x +1} + {Ex+F\over x^2 -\sqrt3x + 1}.$$
$\cdots$
</s>