Optimal. Leaf size=97 \[ \frac {1}{5} \left (1-x^3\right )^{5/3}+\frac {\tan ^{-1}\left (\frac {1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {\log \left (1+x^3\right )}{6 \sqrt [3]{2}}+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}} \]
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Rubi [A]
time = 0.05, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {457, 90, 57,
631, 210, 31} \begin {gather*} \frac {\text {ArcTan}\left (\frac {2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {1}{5} \left (1-x^3\right )^{5/3}-\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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 57
Rule 90
Rule 210
Rule 457
Rule 631
Rubi steps
\begin {align*} \int \frac {x^8}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {x^2}{\sqrt [3]{1-x} (1+x)} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (-(1-x)^{2/3}+\frac {1}{\sqrt [3]{1-x} (1+x)}\right ) \, dx,x,x^3\right )\\ &=\frac {1}{5} \left (1-x^3\right )^{5/3}+\frac {1}{3} \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} (1+x)} \, dx,x,x^3\right )\\ &=\frac {1}{5} \left (1-x^3\right )^{5/3}-\frac {\log \left (1+x^3\right )}{6 \sqrt [3]{2}}+\frac {1}{2} \text {Subst}\left (\int \frac {1}{2^{2/3}+\sqrt [3]{2} x+x^2} \, dx,x,\sqrt [3]{1-x^3}\right )-\frac {\text {Subst}\left (\int \frac {1}{\sqrt [3]{2}-x} \, dx,x,\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}\\ &=\frac {1}{5} \left (1-x^3\right )^{5/3}-\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}}-\frac {\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2^{2/3} \sqrt [3]{1-x^3}\right )}{\sqrt [3]{2}}\\ &=\frac {1}{5} \left (1-x^3\right )^{5/3}+\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 = 0.09, size = 127, normalized size = 1.31 \begin {gather*} \frac {1}{60} \left (12 \left (1-x^3\right )^{5/3}+10\ 2^{2/3} \sqrt {3} \tan ^{-1}\left (\frac {1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}\right )+10\ 2^{2/3} \log \left (-2+2^{2/3} \sqrt [3]{1-x^3}\right )-5\ 2^{2/3} \log \left (2+2^{2/3} \sqrt [3]{1-x^3}+\sqrt [3]{2} \left (1-x^3\right )^{2/3}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 4.78, size = 493, normalized size = 5.08
method | result | size |
trager | \(\left (-\frac {x^{3}}{5}+\frac {1}{5}\right ) \left (-x^{3}+1\right )^{\frac {2}{3}}+\frac {\RootOf \left (\textit {\_Z}^{3}-4\right ) \ln \left (-\frac {45 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x^{3}+6 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{3} x^{3}-15 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) x^{3}-2 \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{3}-63 \left (-x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-4\right ) \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right )-21 \left (-x^{3}+1\right )^{\frac {2}{3}}+105 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right )+14 \RootOf \left (\textit {\_Z}^{3}-4\right )}{\left (x +1\right ) \left (x^{2}-x +1\right )}\right )}{6}+\RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) \ln \left (\frac {72 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x^{3}+15 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{3} x^{3}+72 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) x^{3}+15 \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{3}+126 \left (-x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-4\right ) \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right )+42 \left (-x^{3}+1\right )^{\frac {2}{3}}-168 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right )-35 \RootOf \left (\textit {\_Z}^{3}-4\right )}{\left (x +1\right ) \left (x^{2}-x +1\right )}\right )\) | \(493\) |
risch | \(\text {Expression too large to display}\) | \(778\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.61, size = 97, normalized size = 1.00 \begin {gather*} \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}{5} \, {\left (-x^{3} + 1\right )}^{\frac {5}{3}} - \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.22, size = 106, normalized size = 1.09 \begin {gather*} -\frac {1}{5} \, {\left (x^{3} - 1\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} + \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 ) \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^{8}}{\sqrt [3]{- \left (x - 1\right ) \left (x^{2} + x + 1\right )} \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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Giac [A]
time = 0.93, size = 98, normalized size = 1.01 \begin {gather*} \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}{5} \, {\left (-x^{3} + 1\right )}^{\frac {5}{3}} - \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.65, size = 111, normalized size = 1.14 \begin {gather*} \frac {2^{2/3}\,\ln \left ({\left (1-x^3\right )}^{1/3}-2^{1/3}\right )}{6}+\frac {{\left (1-x^3\right )}^{5/3}}{5}+\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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