<s>[INST] Integrate $\int \frac{1}{1+x+x^4}dx$ 
Find $$\int \frac{{\rm d}x}{1+x+x^4}.$$

WA gives a result, which introduces complex numbers. But according to the algebraic fundamental theorem, any rational fraction can be integrated over the real number field. 
How to do this?
 [/INST] \begin{align} 
\int \frac{{\rm d}x}{1+x+x^4}
\end{align} 
It looks like the main problem is factorization
of the quartic, so here it is one more variant:
for
\begin{align}
a&=\sqrt{\frac1{\sqrt3}\,\cos\Big(\tfrac13\,\arctan(\tfrac19\,\sqrt{687})\Big)}
\\
&\approx 0.72713608449
,
\end{align}
\begin{align}
x^4+x+1&=
\Big(x^2-2\,a\,x+2\,a^2+\sqrt{4\,a^4-1}\Big)
\left(x^2+2\,a\,x+\frac1{2\,a^2+\sqrt{4\,a^4-1})}\right)
\\
&\approx
(x^2-1.45427216898\,x+1.40126836794)
(x^2+1.45427216898\,x+0.713639173537)
.
\end{align}
</s>