Optimal. Leaf size=67 \[ \sqrt [3]{1-x^3}+\frac{1}{2} \log \left (1-\sqrt [3]{1-x^3}\right )-\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{\log (x)}{2} \]
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Rubi [A] time = 0.0372041, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {266, 50, 57, 618, 204, 31} \[ \sqrt [3]{1-x^3}+\frac{1}{2} \log \left (1-\sqrt [3]{1-x^3}\right )-\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{\log (x)}{2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 50
Rule 57
Rule 618
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{\sqrt [3]{1-x^3}}{x} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{\sqrt [3]{1-x}}{x} \, dx,x,x^3\right )\\ &=\sqrt [3]{1-x^3}+\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{(1-x)^{2/3} x} \, dx,x,x^3\right )\\ &=\sqrt [3]{1-x^3}-\frac{\log (x)}{2}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1-x} \, dx,x,\sqrt [3]{1-x^3}\right )-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1+x+x^2} \, dx,x,\sqrt [3]{1-x^3}\right )\\ &=\sqrt [3]{1-x^3}-\frac{\log (x)}{2}+\frac{1}{2} \log \left (1-\sqrt [3]{1-x^3}\right )+\operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{1-x^3}\right )\\ &=\sqrt [3]{1-x^3}-\frac{\tan ^{-1}\left (\frac{1+2 \sqrt [3]{1-x^3}}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{\log (x)}{2}+\frac{1}{2} \log \left (1-\sqrt [3]{1-x^3}\right )\\ \end{align*}
Mathematica [A] time = 0.0236412, size = 90, normalized size = 1.34 \[ \sqrt [3]{1-x^3}+\frac{1}{3} \log \left (1-\sqrt [3]{1-x^3}\right )-\frac{1}{6} \log \left (\left (1-x^3\right )^{2/3}+\sqrt [3]{1-x^3}+1\right )-\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.033, size = 49, normalized size = 0.7 \begin{align*} -{\frac{1}{9\,\Gamma \left ( 2/3 \right ) } \left ( -3\, \left ( 3+1/6\,\pi \,\sqrt{3}-3/2\,\ln \left ( 3 \right ) +3\,\ln \left ( x \right ) +i\pi \right ) \Gamma \left ( 2/3 \right ) +\Gamma \left ({\frac{2}{3}} \right ){x}^{3}{\mbox{$_3$F$_2$}({\frac{2}{3}},1,1;\,2,2;\,{x}^{3})} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.46498, size = 96, normalized size = 1.43 \begin{align*} -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \,{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 1\right )}\right ) +{\left (-x^{3} + 1\right )}^{\frac{1}{3}} - \frac{1}{6} \, \log \left ({\left (-x^{3} + 1\right )}^{\frac{2}{3}} +{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 1\right ) + \frac{1}{3} \, \log \left ({\left (-x^{3} + 1\right )}^{\frac{1}{3}} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.95305, size = 225, normalized size = 3.36 \begin{align*} -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{2}{3} \, \sqrt{3}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + \frac{1}{3} \, \sqrt{3}\right ) +{\left (-x^{3} + 1\right )}^{\frac{1}{3}} - \frac{1}{6} \, \log \left ({\left (-x^{3} + 1\right )}^{\frac{2}{3}} +{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 1\right ) + \frac{1}{3} \, \log \left ({\left (-x^{3} + 1\right )}^{\frac{1}{3}} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.944826, size = 37, normalized size = 0.55 \begin{align*} - \frac{x e^{\frac{i \pi }{3}} \Gamma \left (- \frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{3}, - \frac{1}{3} \\ \frac{2}{3} \end{matrix}\middle |{\frac{1}{x^{3}}} \right )}}{3 \Gamma \left (\frac{2}{3}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14603, size = 97, normalized size = 1.45 \begin{align*} -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \,{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 1\right )}\right ) +{\left (-x^{3} + 1\right )}^{\frac{1}{3}} - \frac{1}{6} \, \log \left ({\left (-x^{3} + 1\right )}^{\frac{2}{3}} +{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 1\right ) + \frac{1}{3} \, \log \left ({\left |{\left (-x^{3} + 1\right )}^{\frac{1}{3}} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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