$$ \begin{align*}
\color{red}{\int \frac {dx}{x^n+1}} &= \int(1+x^n)^{-1}dx \\
& = \int\left(1 - x^n + x^{2n} - x^{3n} + x^{4n} - x^{5n}...\right)dx \\
& = x\left(1 - \frac {x^n}{n + 1} + \frac {x^{2n}}{2n + 1} - \frac {x^{3n}}{3n + 1} + \frac {x^{4n}}{4n + 1} - \frac {x^{5n}}{5n + 1}...\right) + C \\
& = x\left(1 + \frac {1.\frac 1n}{1 + \frac 1n}\frac {(-x^n)^1 }{1!} + \frac{1.2.\left(\frac 1n (\frac 1n + 1)\right)}{\left(\frac 1n + 1\right)\left(\frac 1n + 2\right)}\frac {(-x^n)^2}{2!}
+ \right) + C \\
& = \color{red}{x\,{}_2F_1\left(1, \frac1n;1+\frac1n;-x^n\right) + C}
\end{align*} $$
Here, I used Gaussian Hypergeometric Function as I believed $b$ and $c$ have telescopic ratio :) and $(a)_k$ is simply $k!$.
