Optimal. Leaf size=70 \[ \frac {1}{2} \left (1-x^3\right )^{2/3}+\frac {1}{2} \log \left (1-\sqrt [3]{1-x^3}\right )+\frac {\tan ^{-1}\left (\frac {2 \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\log (x)}{2} \]
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Rubi [A] time = 0.04, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {266, 50, 55, 618, 204, 31} \[ \frac {1}{2} \left (1-x^3\right )^{2/3}+\frac {1}{2} \log \left (1-\sqrt [3]{1-x^3}\right )+\frac {\tan ^{-1}\left (\frac {2 \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\log (x)}{2} \]
Antiderivative was successfully verified.
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Rule 31
Rule 50
Rule 55
Rule 204
Rule 266
Rule 618
Rubi steps
\begin {align*} \int \frac {\left (1-x^3\right )^{2/3}}{x} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {(1-x)^{2/3}}{x} \, dx,x,x^3\right )\\ &=\frac {1}{2} \left (1-x^3\right )^{2/3}+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} x} \, dx,x,x^3\right )\\ &=\frac {1}{2} \left (1-x^3\right )^{2/3}-\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 )\\ &=\frac {1}{2} \left (1-x^3\right )^{2/3}-\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 )\\ &=\frac {1}{2} \left (1-x^3\right )^{2/3}+\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*}
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Mathematica [A] time = 0.03, size = 65, normalized size = 0.93 \[ \frac {1}{2} \left (\left (1-x^3\right )^{2/3}+\log \left (1-\sqrt [3]{1-x^3}\right )-\log (x)\right )+\frac {\tan ^{-1}\left (\frac {2 \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 75, normalized size = 1.07 \[ \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + \frac {1}{2} \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} - \frac {1}{6} \, \log \left ({\left (-x^{3} + 1\right )}^{\frac {2}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{3} \, \log \left ({\left (-x^{3} + 1\right )}^{\frac {1}{3}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.89, size = 74, normalized size = 1.06 \[ \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {1}{2} \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} - \frac {1}{6} \, \log \left ({\left (-x^{3} + 1\right )}^{\frac {2}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{3} \, \log \left ({\left | {\left (-x^{3} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.10, size = 66, normalized size = 0.94 \[ -\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (\frac {2 \pi \sqrt {3}\, x^{3} \hypergeom \left (\left [\frac {1}{3}, 1, 1\right ], \left [2, 2\right ], x^{3}\right )}{3 \Gamma \left (\frac {2}{3}\right )}-\frac {\left (3 \ln \relax (x )+\frac {3}{2}-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+i \pi \right ) \pi \sqrt {3}}{\Gamma \left (\frac {2}{3}\right )}\right )}{9 \pi } \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.41, size = 73, normalized size = 1.04 \[ \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {1}{2} \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} - \frac {1}{6} \, \log \left ({\left (-x^{3} + 1\right )}^{\frac {2}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{3} \, \log \left ({\left (-x^{3} + 1\right )}^{\frac {1}{3}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.40, size = 91, normalized size = 1.30 \[ \frac {\ln \left ({\left (1-x^3\right )}^{1/3}-1\right )}{3}+\ln \left ({\left (1-x^3\right )}^{1/3}-9\,{\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}^2\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )-\ln \left ({\left (1-x^3\right )}^{1/3}-9\,{\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}^2\right )\,\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )+\frac {{\left (1-x^3\right )}^{2/3}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.02, size = 41, normalized size = 0.59 \[ - \frac {x^{2} e^{\frac {2 i \pi }{3}} \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {2}{3} \\ \frac {1}{3} \end {matrix}\middle | {\frac {1}{x^{3}}} \right )}}{3 \Gamma \left (\frac {1}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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