Optimal. Leaf size=83 \[ -\frac {\log \left (x^3+1\right )}{6\ 2^{2/3}}+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}-\frac {\tan ^{-1}\left (\frac {2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}} \]
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Rubi [A] time = 0.06, antiderivative size = 83, 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, 57, 617, 204, 31} \begin {gather*} -\frac {\log \left (x^3+1\right )}{6\ 2^{2/3}}+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}-\frac {\tan ^{-1}\left (\frac {2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 57
Rule 204
Rule 444
Rule 617
Rubi steps
\begin {align*} \int \frac {x^2}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{(1-x)^{2/3} (1+x)} \, dx,x,x^3\right )\\ &=-\frac {\log \left (1+x^3\right )}{6\ 2^{2/3}}-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{2}-x} \, dx,x,\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}-\frac {\operatorname {Subst}\left (\int \frac {1}{2^{2/3}+\sqrt [3]{2} x+x^2} \, dx,x,\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}\\ &=-\frac {\log \left (1+x^3\right )}{6\ 2^{2/3}}+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}+\frac {\operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2^{2/3} \sqrt [3]{1-x^3}\right )}{2^{2/3}}\\ &=-\frac {\tan ^{-1}\left (\frac {1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}-\frac {\log \left (1+x^3\right )}{6\ 2^{2/3}}+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 94, normalized size = 1.13 \begin {gather*} -\frac {-2 \log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )+\log \left (\left (1-x^3\right )^{2/3}+\sqrt [3]{2-2 x^3}+2^{2/3}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{6\ 2^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.10, size = 117, normalized size = 1.41 \begin {gather*} \frac {\log \left (2^{2/3} \sqrt [3]{1-x^3}-2\right )}{3\ 2^{2/3}}-\frac {\log \left (\sqrt [3]{2} \left (1-x^3\right )^{2/3}+2^{2/3} \sqrt [3]{1-x^3}+2\right )}{6\ 2^{2/3}}-\frac {\tan ^{-1}\left (\frac {2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 98, normalized size = 1.18 \begin {gather*} -\frac {1}{6} \cdot 4^{\frac {1}{6}} \sqrt {3} \arctan \left (\frac {1}{6} \cdot 4^{\frac {1}{6}} {\left (4^{\frac {2}{3}} \sqrt {3} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 4^{\frac {1}{3}} \sqrt {3}\right )}\right ) - \frac {1}{24} \cdot 4^{\frac {2}{3}} \log \left (4^{\frac {2}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 2 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} + 2 \cdot 4^{\frac {1}{3}}\right ) + \frac {1}{12} \cdot 4^{\frac {2}{3}} \log \left (-4^{\frac {2}{3}} + 2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 87, normalized size = 1.05 \begin {gather*} -\frac {1}{6} \, \sqrt {3} 2^{\frac {1}{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 {1}{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 {1}{3}} \log \left ({\left | -2^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}} \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 6.28, size = 529, normalized size = 6.37 \begin {gather*} \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}\right ) \ln \left (\frac {144 x^{3} \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-2\right )^{3}-6 x^{3} \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{4}-24 x^{3} \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )+x^{3} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+168 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )-7 \RootOf \left (\textit {\_Z}^{3}-2\right )^{2}-252 \left (-x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}\right )-42 \left (-x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-2\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}-2\right ) \ln \left (-\frac {180 x^{3} \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-2\right )^{3}+6 x^{3} \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{4}+90 x^{3} \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )+3 x^{3} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2}-210 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )-7 \RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+252 \left (-x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}\right )+42 \left (-x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-2\right )-42 \left (-x^{3}+1\right )^{\frac {2}{3}}}{\left (x +1\right ) \left (x^{2}-x +1\right )}\right )}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.18, size = 86, normalized size = 1.04 \begin {gather*} -\frac {1}{6} \, \sqrt {3} 2^{\frac {1}{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 {1}{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 {1}{3}} \log \left (-2^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.05, size = 102, normalized size = 1.23 \begin {gather*} \frac {2^{1/3}\,\ln \left (3\,2^{1/3}-3\,{\left (1-x^3\right )}^{1/3}\right )}{6}+\frac {2^{1/3}\,\ln \left (3\,{\left (1-x^3\right )}^{1/3}-\frac {3\,2^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{12}-\frac {2^{1/3}\,\ln \left (\frac {3\,2^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}+3\,{\left (1-x^3\right )}^{1/3}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{12} \end {gather*}
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 \begin {gather*} \int \frac {x^{2}}{\left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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