The integral of $\frac{1}{1+x^n}$ Motivated by this question:
Integration of $\displaystyle \int\frac{1}{1+x^8}\,dx$
I got curious about finding a general expression for the integral $\int \frac{1}{1+x^n},\,n \geq 1$. By factoring $1+x^n$, we can get an answer for any given $n$ (in terms of logarithms, arctangents, etc), but I was wondering whether a general one-two-liner formula in terms of elementary functions is known/available (WolframAlpha trials for specific $n$ show some structure.).

I showed in THIS ANSWER, that a general solution is given by 
$$\bbox[5px,border:2px solid #C0A000]{\int\frac{1}{x^n+1}dx=-\frac1n\sum_{k=1}^n\left(\frac12 x_{kr}\log(x^2-2x_{kr}x+1)-x_{ki}\arctan\left(\frac{x-x_{kr}}{x_{ki}}\right)\right)+C'}
$$
where $x_{kr}$ and $x_{ki}$ are the real and imaginary parts of $x_k$, respectively, and are given by
$$x_{kr}=\text{Re}\left(x_k\right)=\cos \left(\frac{(2k-1)\pi}{n}\right)$$
$$x_{ki}=\text{Im}\left(x_k\right)=\sin \left(\frac{(2k-1)\pi}{n}\right)$$
