$$\int\frac{x^{6}-2x^{3}}{\left(x^{3}+1\right)^{3}}dx$$
I tried adding and subtracting 1 to bring a square expression with numerator as $(x^3-1)^2 -1$ but always going to partial fraction which obviously is very lengthy considering the cubic factor in denominator.
Let $\displaystyle I = \int\frac{x^6-2x^3}{(x^3+1)^3}dx = \int\frac{x^6-2x^3}{x^6\left(x+\frac{1}{x^2}\right)^3}dx = \int\frac{1-2x^{-3}}{\left(x+\frac{1}{x^2}\right)^3}dx$
Now Put $\displaystyle x+\frac{1}{x^2} = t\;,$ Then $\displaystyle \left(1-\frac{2}{x^3}\right)dx = dt$
So Integral $\displaystyle I = \int\frac{1}{t^3}dt = -\frac{1}{2t^2}+\mathcal{C} = -\frac{1}{2}\cdot \left(\frac{x^2}{x^3+1}\right)^2+\mathcal{C}$