Optimal. Leaf size=53 \[ -\frac{1}{2} \log \left (-\sqrt [3]{1-x^3}-x\right )-\frac{\tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt{3}} \]
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Rubi [A] time = 0.0411156, antiderivative size = 87, normalized size of antiderivative = 1.64, number of steps used = 7, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538, Rules used = {331, 292, 31, 634, 618, 204, 628} \[ \frac{1}{6} \log \left (\frac{x^2}{\left (1-x^3\right )^{2/3}}-\frac{x}{\sqrt [3]{1-x^3}}+1\right )-\frac{1}{3} \log \left (\frac{x}{\sqrt [3]{1-x^3}}+1\right )-\frac{\tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 331
Rule 292
Rule 31
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x}{\left (1-x^3\right )^{2/3}} \, dx &=\operatorname{Subst}\left (\int \frac{x}{1+x^3} \, dx,x,\frac{x}{\sqrt [3]{1-x^3}}\right )\\ &=-\left (\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,\frac{x}{\sqrt [3]{1-x^3}}\right )\right )+\frac{1}{3} \operatorname{Subst}\left (\int \frac{1+x}{1-x+x^2} \, dx,x,\frac{x}{\sqrt [3]{1-x^3}}\right )\\ &=-\frac{1}{3} \log \left (1+\frac{x}{\sqrt [3]{1-x^3}}\right )+\frac{1}{6} \operatorname{Subst}\left (\int \frac{-1+2 x}{1-x+x^2} \, dx,x,\frac{x}{\sqrt [3]{1-x^3}}\right )+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1-x+x^2} \, dx,x,\frac{x}{\sqrt [3]{1-x^3}}\right )\\ &=\frac{1}{6} \log \left (1+\frac{x^2}{\left (1-x^3\right )^{2/3}}-\frac{x}{\sqrt [3]{1-x^3}}\right )-\frac{1}{3} \log \left (1+\frac{x}{\sqrt [3]{1-x^3}}\right )-\operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,-1+\frac{2 x}{\sqrt [3]{1-x^3}}\right )\\ &=\frac{\tan ^{-1}\left (\frac{-1+\frac{2 x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt{3}}+\frac{1}{6} \log \left (1+\frac{x^2}{\left (1-x^3\right )^{2/3}}-\frac{x}{\sqrt [3]{1-x^3}}\right )-\frac{1}{3} \log \left (1+\frac{x}{\sqrt [3]{1-x^3}}\right )\\ \end{align*}
Mathematica [C] time = 0.0091055, size = 37, normalized size = 0.7 \[ \frac{x^2 \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{x^3}{x^3-1}\right )}{2 \left (1-x^3\right )^{2/3}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.033, size = 15, normalized size = 0.3 \begin{align*}{\frac{{x}^{2}}{2}{\mbox{$_2$F$_1$}({\frac{2}{3}},{\frac{2}{3}};\,{\frac{5}{3}};\,{x}^{3})}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.58435, size = 105, normalized size = 1.98 \begin{align*} -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (\frac{2 \,{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{x} - 1\right )}\right ) - \frac{1}{3} \, \log \left (\frac{{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{x} + 1\right ) + \frac{1}{6} \, \log \left (-\frac{{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{x} + \frac{{\left (-x^{3} + 1\right )}^{\frac{2}{3}}}{x^{2}} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55684, size = 225, normalized size = 4.25 \begin{align*} -\frac{1}{3} \, \sqrt{3} \arctan \left (-\frac{\sqrt{3} x - 2 \, \sqrt{3}{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{3 \, x}\right ) - \frac{1}{3} \, \log \left (\frac{x +{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{x}\right ) + \frac{1}{6} \, \log \left (\frac{x^{2} -{\left (-x^{3} + 1\right )}^{\frac{1}{3}} x +{\left (-x^{3} + 1\right )}^{\frac{2}{3}}}{x^{2}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.04767, size = 31, normalized size = 0.58 \begin{align*} \frac{x^{2} \Gamma \left (\frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{2}{3}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle |{x^{3} e^{2 i \pi }} \right )}}{3 \Gamma \left (\frac{5}{3}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{{\left (-x^{3} + 1\right )}^{\frac{2}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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