$\int \frac{1}{x(x^n+1)}$$dx$$= \int \frac{(x^n+1)-x^n}{x(x^n+1)}$$dx$$=\int \frac{1}{x} dx - \int \frac{x^{n-1}}{x^n+1} dx$$=log|x|-\frac{1}{n} \int \frac{(x^n+1)'}{x^n+1} dx$$=log|x|-\frac{1}{n}log|x^n+1|+C$