<s>[INST] How to calculate $ \int \frac{x^3+x^2}{x^3-1} \mathrm{d}x$ I was trying to solve this integral:
$$ \int \frac{x^3+x^2}{x^3-1} \mathrm{d}x$$
I made the following steps:


*

*Polynomial division: $$ \frac{x^3+x^2}{x^3-1} = \left(1+\frac{x^2+1}{x^3-1}\right)$$

*Hermite polynomials: $$ \frac{x^2+1}{x^3-1} = \frac{2}{3}\frac{1}{x-1}+\frac{\frac{x}{3}-\frac{1}{3}}{x^2+x+1}$$


So, now the integral becomes: $$x+\frac{2}{3}\log{|x-1|}+\int \dfrac{\frac{x}{3}-\frac{1}{3}}{x^2+x+1} \mathrm{d}x$$
But i think there's something wrong, i can not continue. Any ideas?
 [/INST] What you have done so far is correct, so
$$
\int\frac{x^3+x^2}{x^3-1}dx=
x+\frac{2}{3}\log\left|x-1\right|+\frac{1}{3}\int\frac{x-1}{x^2+x+1}dx.
$$
Now, to determine the last integral, we want to have the derivative of the denominator in the numerator, so we write
$$
\int\frac{x-1}{x^2+x+1}dx=
\int\frac{\frac{1}{2}(2x+1)-\frac{3}{2}}{x^2+x+1}dx=
\frac{1}{2}\log\left|x^2+x+1\right|-\frac{3}{2}\int\frac{dx}{x^2+x+1}.
$$
Can you take it from here?
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