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?

Well...maybe the division isn't that bad. Do you know about geometric series? What's 
$$
1 + r + r^2 + \ldots + r^{32}?
$$
It's 
$$
\frac{r^{33} - 1}{r-1}
$$
If you apply this to $r = -x^3$, you get
$$
\frac{-x^{99} - 1}{-(x^3)-1}
= \frac{x^{99} + 1}{x^3 + 1}
$$
I know that's not what you wanted, but if you do a little algebra, you can do this:
\begin{align}
\frac{x^{100} + 1}{x^3 + 1} 
&= \frac{x^{100} + x - x + 1}{x^3 + 1}\\
&= \frac{x^{100} + x}{x^3 + 1} + \frac{-x + 1}{x^3 + 1}\\
&= x\frac{x^{99} + 1}{x^3 + 1} + \frac{-x + 1}{x^3 + 1}.
\end{align}
Now integrating the first fraction is easy because of the geometric series, and all you have to deal with is the last one. 
