不定積分 $\small \displaystyle \int \dfrac{x^{n}}{x^{3}+1} \,dx$ の結果をまとめています。
$n \geqq 0$ の場合
（※$\operatorname{atan}$は正接（タンジェント）の逆関数）
$\small \displaystyle \int \dfrac{1}{x^{3} + 1} \,dx$ $\small = \dfrac{\log{\left(x + 1 \right)}}{3}-\dfrac{\log{\left(x^{2}-x + 1 \right)}}{6} + \dfrac{\sqrt{3} \operatorname{atan}{\left(\dfrac{2 \sqrt{3} x}{3}-\dfrac{\sqrt{3}}{3} \right)}}{3} +C $
$\small \displaystyle \int \dfrac{x}{x^{3} + 1} \,dx$ $\small =-\dfrac{\log{\left(x + 1 \right)}}{3} + \dfrac{\log{\left(x^{2}-x + 1 \right)}}{6} + \dfrac{\sqrt{3} \operatorname{atan}{\left(\dfrac{2 \sqrt{3} x}{3}-\dfrac{\sqrt{3}}{3} \right)}}{3} +C $
$\small \displaystyle \int \dfrac{x^{2}}{x^{3} + 1} \,dx$ $\small = \dfrac{\log{\left(x^{3} + 1 \right)}}{3} +C $
$\small \displaystyle \int \dfrac{x^{3}}{x^{3} + 1} \,dx$ $\small = x-\dfrac{\log{\left(x + 1 \right)}}{3} + \dfrac{\log{\left(x^{2}-x + 1 \right)}}{6}-\dfrac{\sqrt{3} \operatorname{atan}{\left(\dfrac{2 \sqrt{3} x}{3}-\dfrac{\sqrt{3}}{3} \right)}}{3} +C $
$\small \displaystyle \int \dfrac{x^{4}}{x^{3} + 1} \,dx$ $\small = \dfrac{x^{2}}{2} + \dfrac{\log{\left(x + 1 \right)}}{3}-\dfrac{\log{\left(x^{2}-x + 1 \right)}}{6}-\dfrac{\sqrt{3} \operatorname{atan}{\left(\dfrac{2 \sqrt{3} x}{3}-\dfrac{\sqrt{3}}{3} \right)}}{3} +C $
$\small \displaystyle \int \dfrac{x^{5}}{x^{3} + 1} \,dx$ $\small = \dfrac{x^{3}}{3}-\dfrac{\log{\left(x^{3} + 1 \right)}}{3} +C $
$\small \displaystyle \int \dfrac{x^{6}}{x^{3} + 1} \,dx$ $\small = \dfrac{x^{4}}{4}-x + \dfrac{\log{\left(x + 1 \right)}}{3}-\dfrac{\log{\left(x^{2}-x + 1 \right)}}{6} + \dfrac{\sqrt{3} \operatorname{atan}{\left(\dfrac{2 \sqrt{3} x}{3}-\dfrac{\sqrt{3}}{3} \right)}}{3} +C $
$\small \displaystyle \int \dfrac{x^{7}}{x^{3} + 1} \,dx$ $\small = \dfrac{x^{5}}{5}-\dfrac{x^{2}}{2}-\dfrac{\log{\left(x + 1 \right)}}{3} + \dfrac{\log{\left(x^{2}-x + 1 \right)}}{6} + \dfrac{\sqrt{3} \operatorname{atan}{\left(\dfrac{2 \sqrt{3} x}{3}-\dfrac{\sqrt{3}}{3} \right)}}{3} +C $
$\small \displaystyle \int \dfrac{x^{8}}{x^{3} + 1} \,dx$ $\small = \dfrac{x^{6}}{6}-\dfrac{x^{3}}{3} + \dfrac{\log{\left(x^{3} + 1 \right)}}{3} +C $
$\small \displaystyle \int \dfrac{x^{9}}{x^{3} + 1} \,dx$ $\small = \dfrac{x^{7}}{7}-\dfrac{x^{4}}{4} + x-\dfrac{\log{\left(x + 1 \right)}}{3} + \dfrac{\log{\left(x^{2}-x + 1 \right)}}{6}-\dfrac{\sqrt{3} \operatorname{atan}{\left(\dfrac{2 \sqrt{3} x}{3}-\dfrac{\sqrt{3}}{3} \right)}}{3} +C $
$\small \displaystyle \int \dfrac{x^{10}}{x^{3} + 1} \,dx$ $\small = \dfrac{x^{8}}{8}-\dfrac{x^{5}}{5} + \dfrac{x^{2}}{2} + \dfrac{\log{\left(x + 1 \right)}}{3}-\dfrac{\log{\left(x^{2}-x + 1 \right)}}{6}-\dfrac{\sqrt{3} \operatorname{atan}{\left(\dfrac{2 \sqrt{3} x}{3}-\dfrac{\sqrt{3}}{3} \right)}}{3} +C $
$n < 0$ の場合
$\small \displaystyle \int \dfrac{1}{x \left(x^{3} + 1\right)} \,dx$ $\small = \log{\left(x \right)}-\dfrac{\log{\left(x^{3} + 1 \right)}}{3} +C $
$\small \displaystyle \int \dfrac{1}{x^{2} \left(x^{3} + 1\right)} \,dx$ $\small = \dfrac{\log{\left(x + 1 \right)}}{3}-\dfrac{\log{\left(x^{2}-x + 1 \right)}}{6}-\dfrac{\sqrt{3} \operatorname{atan}{\left(\dfrac{2 \sqrt{3} x}{3}-\dfrac{\sqrt{3}}{3} \right)}}{3}-\dfrac{1}{x} +C $
$\small \displaystyle \int \dfrac{1}{x^{3} \left(x^{3} + 1\right)} \,dx$ $\small =-\dfrac{\log{\left(x + 1 \right)}}{3} + \dfrac{\log{\left(x^{2}-x + 1 \right)}}{6}-\dfrac{\sqrt{3} \operatorname{atan}{\left(\dfrac{2 \sqrt{3} x}{3}-\dfrac{\sqrt{3}}{3} \right)}}{3}-\dfrac{1}{2 x^{2}} +C $
$\small \displaystyle \int \dfrac{1}{x^{4} \left(x^{3} + 1\right)} \,dx$ $\small =-\log{\left(x \right)} + \dfrac{\log{\left(x^{3} + 1 \right)}}{3}-\dfrac{1}{3 x^{3}} +C $
$\small \displaystyle \int \dfrac{1}{x^{5} \left(x^{3} + 1\right)} \,dx$ $\small =-\dfrac{\log{\left(x + 1 \right)}}{3} + \dfrac{\log{\left(x^{2}-x + 1 \right)}}{6} + \dfrac{\sqrt{3} \operatorname{atan}{\left(\dfrac{2 \sqrt{3} x}{3}-\dfrac{\sqrt{3}}{3} \right)}}{3} + \dfrac{4 x^{3}-1}{4 x^{4}} +C $
$\small \displaystyle \int \dfrac{1}{x^{6} \left(x^{3} + 1\right)} \,dx$ $\small = \dfrac{\log{\left(x + 1 \right)}}{3}-\dfrac{\log{\left(x^{2}-x + 1 \right)}}{6} + \dfrac{\sqrt{3} \operatorname{atan}{\left(\dfrac{2 \sqrt{3} x}{3}-\dfrac{\sqrt{3}}{3} \right)}}{3} + \dfrac{5 x^{3}-2}{10 x^{5}} +C $
$\small \displaystyle \int \dfrac{1}{x^{7} \left(x^{3} + 1\right)} \,dx$ $\small = \log{\left(x \right)}-\dfrac{\log{\left(x^{3} + 1 \right)}}{3} + \dfrac{2 x^{3}-1}{6 x^{6}} +C $
$\small \displaystyle \int \dfrac{1}{x^{8} \left(x^{3} + 1\right)} \,dx$ $\small = \dfrac{\log{\left(x + 1 \right)}}{3}-\dfrac{\log{\left(x^{2}-x + 1 \right)}}{6}-\dfrac{\sqrt{3} \operatorname{atan}{\left(\dfrac{2 \sqrt{3} x}{3}-\dfrac{\sqrt{3}}{3} \right)}}{3}-\dfrac{28 x^{6}-7 x^{3} + 4}{28 x^{7}} +C $
$\small \displaystyle \int \dfrac{1}{x^{9} \left(x^{3} + 1\right)} \,dx$ $\small =-\dfrac{\log{\left(x + 1 \right)}}{3} + \dfrac{\log{\left(x^{2}-x + 1 \right)}}{6}-\dfrac{\sqrt{3} \operatorname{atan}{\left(\dfrac{2 \sqrt{3} x}{3}-\dfrac{\sqrt{3}}{3} \right)}}{3}-\dfrac{20 x^{6}-8 x^{3} + 5}{40 x^{8}} +C $
$\small \displaystyle \int \dfrac{1}{x^{10} \left(x^{3} + 1\right)} \,dx$ $\small =-\log{\left(x \right)} + \dfrac{\log{\left(x^{3} + 1 \right)}}{3}-\dfrac{6 x^{6}-3 x^{3} + 2}{18 x^{9}} +C $