Indefinite integral of fraction?

Question

How to integrate $$\int\frac{1}{x^{14}+1}dx$$ I've tried to use partial fraction decomposition but it doesn't work

the indefinite integral doesn't have a closed form, however if you are integrating in the whole $\mathbb{R}$ then you can use the residue theorem to find a closed form for it value — Masacroso, Apr 23 '20 at 13:40
@Masacroso So, any "simple" methods (like from algebra) will not work? — Limenal, Apr 23 '20 at 13:44

score 0 · Answer 1 · answered Apr 23 '20 at 13:45

The integrand is the sum of a geometric series with common ratio $-x^{14}$, so your integral equals

$$\int 1 - x^{14} + x^{28} - x^{42} + \cdots \; dx $$

$$= x - \frac{x^{15}}{15} + \frac{x^{29}}{29} - \cdots.$$

That's probably as good as you're going to do. (Plus constant.)

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