Evaluating $\int \frac{dx}{x^3+x+1}$

Question

$$\int \frac{dx}{x^3+x+1}$$

I have no idea how to solve this. How do I evaluate it?

Any advice, hint or well-thought solution will be appreciated.

Answer 1

Let $\zeta_1,\zeta_2,\zeta_3$ be the roots of $x^3+x+1$. We have:

$$ \text{Res}\left(\frac{1}{z^3+z+1},z=\zeta_i\right) = \frac{1}{3\zeta_i^2+1},\tag{1}$$ hence: $$ \frac{1}{z^3+z+1}=\sum_{i=1}^{3}\frac{1}{3\zeta_i^2+1}\cdot\frac{1}{z-\zeta_i} \tag{2} $$ and: $$ \int \frac{dz}{z^3+z+1} = C+\sum_{i=1}^{3}\frac{\log(z-\zeta_i)}{3\zeta_i^2+1}.\tag{3}$$