The point is what exactly do you mean by "solving". The integral cannot be given in terms of elementary functions for arbitrary n. The integral is found in terms of (Gauss's) Hypergeometric series 2F1. A symbolic evaluation gives
\int\dfrac{\mathrm dx}{1+x^n}=x\ _2F_1\left(\dfrac{1}{n},1,1+\dfrac{1}{n};-x^n\right)
You can look up Wikipedia for hypergeometric series.