<s>[INST] How to integrate $\frac{1}{(1 + x^5)(1 + x^7)}$ My cousin who is in high school asked me if it is possible to integrate
$$
\int \frac{1}{(1 + x^5)(1 + x^7)} \, dx
$$
I checked the list of integrals of rational functions on Wikipedia link
and it doesn't seem to be here.
Is this not possible to do using elementary functions?
Wolfram Alpha can do this but the answer is extremely complicated.
 [/INST] Solve:
$$\begin{align}\frac{1}{(x^5+1)(x^7+1)} & =\frac{a_0}{(x+1)^2}+\frac{a_1}{x+1}\\
&\quad+\frac{a_2x+b_2}{x^2-2\cos(\pi/5)x +1}\\&+\frac{a_3x+b_3}{x^2-2\cos(3\pi/5)x+1}\\
&\quad+\frac{a_4x+b_4}{x^2-2\cos(\pi/7)x +1}\\
&\quad+\frac{a_5x+b_5}{x^2-2\cos(3\pi/7)x +1}\\
&\quad+\frac{a_6x+b_6}{x^2-2\cos(5\pi/7)x +1}\\
\end{align}$$
Then you can use basic integrals to solve each:
$$\int \frac{ax+b}{x^2-2cx+1}\,dx$$
You'll get lots of logarithms and arctans, and it will be just horrible. And that doesn't even count how horrible the $a_i$ and $b_i$ are going to be.
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