<s>[INST] Faster way to get a rational integral $\int_2^3 \frac{x^{100}+1}{x^3+1} \, dx$ So my problem is calculating this integral:
$$
\int_{2}^{3} \frac{x^{100}+1}{x^3+1} \, dx
$$
I know it can be done by polynomial division but that is really tedious I would have to divide $\approx 30$ times.
And i know that Mathematica would give me the answer in the blink of an eye.
Is there a clever way to do it pen&paper?
 [/INST] We have
$$ J=\int_{2}^{3}\frac{x-1}{x^3+1}\,dx = \frac{1}{3}\log\left(\frac{21}{16}\right)\tag{1}$$
by partial fraction decomposition, and that is the only annoying part, since
$$ I = \int_{2}^{3}\frac{x^{100}+1}{x^3+1}\,dx  = -J+\int_{2}^{3}x\cdot\frac{x^{99}+1}{x^3+1}\,dx \tag{2}$$
and 
$$ \frac{x^{99}+1}{x^3+1} = 1-x^3+x^6-\ldots+x^{96} \tag{3} $$
so:
$$ \boxed{\int_{2}^{3}\frac{x^{100}+1}{x^3+1}\,dx} =-J+ \int_{2}^{3}\sum_{k=0}^{32}(-1)^k x^{3k+1}dx=\boxed{-\frac{1}{3}\log\left(\frac{21}{16}\right)+\sum_{k=0}^{32}(-1)^k\frac{3^{3k+2}-2^{3k+2}}{3k+2}}\tag{4} $$
</s>