Optimal. Leaf size=82 \[ -\frac {\log \left (x^3+1\right )}{6 \sqrt [3]{2}}+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}+\frac {\tan ^{-1}\left (\frac {2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}} \]
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Rubi [A] time = 0.06, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {444, 55, 617, 204, 31} \[ -\frac {\log \left (x^3+1\right )}{6 \sqrt [3]{2}}+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}+\frac {\tan ^{-1}\left (\frac {2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 55
Rule 204
Rule 444
Rule 617
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} (1+x)} \, dx,x,x^3\right )\\ &=-\frac {\log \left (1+x^3\right )}{6 \sqrt [3]{2}}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{2^{2/3}+\sqrt [3]{2} x+x^2} \, dx,x,\sqrt [3]{1-x^3}\right )-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{2}-x} \, dx,x,\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}\\ &=-\frac {\log \left (1+x^3\right )}{6 \sqrt [3]{2}}+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}-\frac {\operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2^{2/3} \sqrt [3]{1-x^3}\right )}{\sqrt [3]{2}}\\ &=\frac {\tan ^{-1}\left (\frac {1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {\log \left (1+x^3\right )}{6 \sqrt [3]{2}}+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 73, normalized size = 0.89 \[ \frac {-\log \left (x^3+1\right )+3 \log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{6 \sqrt [3]{2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.28, size = 90, normalized size = 1.10 \[ \frac {1}{6} \, \sqrt {6} 2^{\frac {1}{6}} \arctan \left (\frac {1}{6} \cdot 2^{\frac {1}{6}} {\left (\sqrt {6} 2^{\frac {1}{3}} + 2 \, \sqrt {6} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}\right ) - \frac {1}{12} \cdot 2^{\frac {2}{3}} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}\right ) + \frac {1}{6} \cdot 2^{\frac {2}{3}} \log \left (-2^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 87, normalized size = 1.06 \[ \frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} + 2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}\right ) - \frac {1}{12} \cdot 2^{\frac {2}{3}} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}\right ) + \frac {1}{6} \cdot 2^{\frac {2}{3}} \log \left ({\left | -2^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}} \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 5.59, size = 476, normalized size = 5.80 \[ \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}\right ) \ln \left (\frac {72 x^{3} \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+15 x^{3} \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{3}+72 x^{3} \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}\right )+15 x^{3} \RootOf \left (\textit {\_Z}^{3}-4\right )+126 \left (-x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )-168 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}\right )-35 \RootOf \left (\textit {\_Z}^{3}-4\right )+42 \left (-x^{3}+1\right )^{\frac {2}{3}}}{\left (x +1\right ) \left (x^{2}-x +1\right )}\right )+\frac {\RootOf \left (\textit {\_Z}^{3}-4\right ) \ln \left (-\frac {45 x^{3} \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 x^{3} \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{3}-15 x^{3} \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}\right )-2 x^{3} \RootOf \left (\textit {\_Z}^{3}-4\right )-63 \left (-x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )+105 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}\right )+14 \RootOf \left (\textit {\_Z}^{3}-4\right )-21 \left (-x^{3}+1\right )^{\frac {2}{3}}}{\left (x +1\right ) \left (x^{2}-x +1\right )}\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.30, size = 86, normalized size = 1.05 \[ \frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} + 2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}\right ) - \frac {1}{12} \cdot 2^{\frac {2}{3}} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}\right ) + \frac {1}{6} \cdot 2^{\frac {2}{3}} \log \left (-2^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.89, size = 100, normalized size = 1.22 \[ \frac {2^{2/3}\,\ln \left ({\left (1-x^3\right )}^{1/3}-2^{1/3}\right )}{6}+\frac {2^{2/3}\,\ln \left ({\left (1-x^3\right )}^{1/3}-\frac {2^{1/3}\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{4}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{12}-\frac {2^{2/3}\,\ln \left ({\left (1-x^3\right )}^{1/3}-\frac {2^{1/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{4}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\sqrt [3]{- \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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