$\int\frac{1}{x\left(x^3+1\right)^2}dx$

$\ln\left|x\right|+\frac{1}{9\left(x+1\right)}+\frac{-4\sqrt{3}}{27}\left(\frac{\sqrt{\left(3\right)^{3}}}{-8\left(\left(x-\frac{1}{2}\right)^2+\frac{3}{4}\right)}+\frac{1}{2}\arctan\left(\frac{-1+2x}{\sqrt{3}}\right)+\frac{\sqrt{3}\left(x-\frac{1}{2}\right)}{4\left(\left(x-\frac{1}{2}\right)^2+\frac{3}{4}\right)}\right)-\frac{1}{3}\ln\left|x+1\right|+\frac{2\sqrt{3}\arctan\left(\frac{2\left(x-\frac{1}{2}\right)}{\sqrt{3}}\right)}{27}-\frac{2}{3}\ln\left|\sqrt{\left(x-\frac{1}{2}\right)^2+\frac{3}{4}}\right|+C_2$