How to integrate $\int \frac{x^{13}\ dx}{x^5 + 1}$ We get this problem from our teacher today. I only wish that it was $x^{14}$ in the numerator, so we can use substitution method:
$$\int \dfrac{x^{13}\ dx}{x^5 + 1}$$
I can't find way to integrate this. Please help me out to solve this seeming simple integral.

$$
\int\frac{x^{13}}{x^5+1}\,dx
=\int\left(x^8-x^3+\frac{x^{3}}{x^5+1}\right)\,dx
=\frac{x^9}{9}-\frac{x^4}{4}+\int\frac{x^{3}}{x^5+1}\,dx
$$
Note that 
$$
x^5+1=(x+1)(x^2+\phi x+1)(x^2-(\phi-1) x+1),
$$
where $\phi=\frac{\sqrt{5}+1}{2}$ is the "Golden Ratio". It can be shown that 
$$
\frac{x^3}{x^5+1}=\frac{1}{5}\left(\frac{\phi x+1-\phi}{x^2-(\phi-1)x+1}+
\frac{(1-\phi)x+\phi}{x^2+\phi x+1}-\frac{1}{x+1}\right)
$$
You can take it from here...
