General formula of fraction integral Consider the sequences of integrals:
$\int \frac{x}{1+x} dx, \int \frac{x}{1+x^2}dx, \int \frac{x}{1+x^3}dx,\ldots,\int \frac{x}{1+x^n}dx$.
Question: What is the general formula of $\int \frac{x}{1+x^n} dx$  for $n>0$?

It's not a complete answer, but too long for a comment. It gives the answer for $n=4m$, $m$ is positive integer. Denote $I_n:=\int\frac{x}{1+x^n}\,dx$.  If $x<0$ then substituting $x:=-z$ we can reduce the problem for $x>0$. So we calculate $I_{4m}:=\int\frac{x}{1+x^{4m}}\,dx$. Substitute $x:=\sqrt{z},\,z>0$. Then $I_{4m}=\frac{1}{2}\int\frac{1}{1+z^{2m}}\,dz$. The result of the integral is (Demidovich, Mathematical Analysis, problem 1925) 
$$
-\frac{1}{2m}\sum_{k=1}^{m}\cos\frac{\pi(2k-1)}{2m}\ln\left(1-2x\cos\frac{2k-1}{2m}\pi+x^2 \right)$$
$$+\frac{1}{m}\sum_{k=1}^m\sin\frac{\pi(2k-1)}{2m}\arctan\frac{x-\cos\frac{2k-1}{2m}\pi}{\sin\frac{2k-1}{2m}\pi}.
$$
