Optimal. Leaf size=104 \[ \frac {1}{6} x^4 \sqrt [3]{-1+x^6}+\frac {\text {ArcTan}\left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{-1+x^6}}\right )}{6 \sqrt {3}}+\frac {1}{18} \log \left (-x^2+\sqrt [3]{-1+x^6}\right )-\frac {1}{36} \log \left (x^4+x^2 \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 69, normalized size of antiderivative = 0.66, number of steps
used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {281, 285, 337}
\begin {gather*} \frac {\text {ArcTan}\left (\frac {\frac {2 x^2}{\sqrt [3]{x^6-1}}+1}{\sqrt {3}}\right )}{6 \sqrt {3}}+\frac {1}{6} \sqrt [3]{x^6-1} x^4+\frac {1}{12} \log \left (x^2-\sqrt [3]{x^6-1}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 281
Rule 285
Rule 337
Rubi steps
\begin {align*} \int x^3 \sqrt [3]{-1+x^6} \, dx &=\frac {1}{2} \text {Subst}\left (\int x \sqrt [3]{-1+x^3} \, dx,x,x^2\right )\\ &=\frac {1}{6} x^4 \sqrt [3]{-1+x^6}-\frac {1}{6} \text {Subst}\left (\int \frac {x}{\left (-1+x^3\right )^{2/3}} \, dx,x,x^2\right )\\ &=\frac {1}{6} x^4 \sqrt [3]{-1+x^6}-\frac {1}{6} \text {Subst}\left (\int \frac {x}{1-x^3} \, dx,x,\frac {x^2}{\sqrt [3]{-1+x^6}}\right )\\ &=\frac {1}{6} x^4 \sqrt [3]{-1+x^6}-\frac {1}{18} \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,\frac {x^2}{\sqrt [3]{-1+x^6}}\right )+\frac {1}{18} \text {Subst}\left (\int \frac {1-x}{1+x+x^2} \, dx,x,\frac {x^2}{\sqrt [3]{-1+x^6}}\right )\\ &=\frac {1}{6} x^4 \sqrt [3]{-1+x^6}+\frac {1}{18} \log \left (1-\frac {x^2}{\sqrt [3]{-1+x^6}}\right )-\frac {1}{36} \text {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\frac {x^2}{\sqrt [3]{-1+x^6}}\right )+\frac {1}{12} \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\frac {x^2}{\sqrt [3]{-1+x^6}}\right )\\ &=\frac {1}{6} x^4 \sqrt [3]{-1+x^6}+\frac {1}{18} \log \left (1-\frac {x^2}{\sqrt [3]{-1+x^6}}\right )-\frac {1}{36} \log \left (1+\frac {x^4}{\left (-1+x^6\right )^{2/3}}+\frac {x^2}{\sqrt [3]{-1+x^6}}\right )-\frac {1}{6} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 x^2}{\sqrt [3]{-1+x^6}}\right )\\ &=\frac {1}{6} x^4 \sqrt [3]{-1+x^6}+\frac {\tan ^{-1}\left (\frac {1+\frac {2 x^2}{\sqrt [3]{-1+x^6}}}{\sqrt {3}}\right )}{6 \sqrt {3}}+\frac {1}{18} \log \left (1-\frac {x^2}{\sqrt [3]{-1+x^6}}\right )-\frac {1}{36} \log \left (1+\frac {x^4}{\left (-1+x^6\right )^{2/3}}+\frac {x^2}{\sqrt [3]{-1+x^6}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.43, size = 100, normalized size = 0.96 \begin {gather*} \frac {1}{36} \left (6 x^4 \sqrt [3]{-1+x^6}+2 \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{-1+x^6}}\right )+2 \log \left (-x^2+\sqrt [3]{-1+x^6}\right )-\log \left (x^4+x^2 \sqrt [3]{-1+x^6}+\left (-1+x^6\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 = 2.99, size = 33, normalized size = 0.32
method | result | size |
meijerg | \(\frac {\mathrm {signum}\left (x^{6}-1\right )^{\frac {1}{3}} x^{4} \hypergeom \left (\left [-\frac {1}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x^{6}\right )}{4 \left (-\mathrm {signum}\left (x^{6}-1\right )\right )^{\frac {1}{3}}}\) | \(33\) |
risch | \(\frac {x^{4} \left (x^{6}-1\right )^{\frac {1}{3}}}{6}-\frac {\left (-\mathrm {signum}\left (x^{6}-1\right )\right )^{\frac {2}{3}} x^{4} \hypergeom \left (\left [\frac {2}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x^{6}\right )}{12 \mathrm {signum}\left (x^{6}-1\right )^{\frac {2}{3}}}\) | \(46\) |
trager | \(\frac {x^{4} \left (x^{6}-1\right )^{\frac {1}{3}}}{6}+\frac {\ln \left (-2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{6}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{6}+x^{6}-3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x^{2}-3 x^{4} \left (x^{6}-1\right )^{\frac {1}{3}}-2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1\right )}{18}+\frac {\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{6}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{6}-3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{4}-2 x^{6}+3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x^{2}+3 x^{2} \left (x^{6}-1\right )^{\frac {2}{3}}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+1\right )}{18}\) | \(187\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 94, normalized size = 0.90 \begin {gather*} -\frac {1}{18} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} + 1\right )}\right ) - \frac {{\left (x^{6} - 1\right )}^{\frac {1}{3}}}{6 \, x^{2} {\left (\frac {x^{6} - 1}{x^{6}} - 1\right )}} - \frac {1}{36} \, \log \left (\frac {{\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} + \frac {{\left (x^{6} - 1\right )}^{\frac {2}{3}}}{x^{4}} + 1\right ) + \frac {1}{18} \, \log \left (\frac {{\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 94, normalized size = 0.90 \begin {gather*} \frac {1}{6} \, {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{4} - \frac {1}{18} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2} + 2 \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{3 \, x^{2}}\right ) + \frac {1}{18} \, \log \left (-\frac {x^{2} - {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}}\right ) - \frac {1}{36} \, \log \left (\frac {x^{4} + {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{2} + {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{x^{4}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.54, size = 36, normalized size = 0.35 \begin {gather*} - \frac {x^{4} e^{- \frac {2 i \pi }{3}} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {x^{6}} \right )}}{6 \Gamma \left (\frac {5}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^3\,{\left (x^6-1\right )}^{1/3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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