Optimal. Leaf size=89 \[ \frac {1}{4} \left (x^6-1\right )^{2/3}+\frac {1}{6} \log \left (\sqrt [3]{x^6-1}+1\right )-\frac {1}{12} \log \left (\left (x^6-1\right )^{2/3}-\sqrt [3]{x^6-1}+1\right )+\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{x^6-1}}{\sqrt {3}}\right )}{2 \sqrt {3}} \]
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Rubi [A] time = 0.04, antiderivative size = 65, normalized size of antiderivative = 0.73, number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {266, 50, 56, 618, 204, 31} \begin {gather*} \frac {1}{4} \left (x^6-1\right )^{2/3}+\frac {1}{4} \log \left (\sqrt [3]{x^6-1}+1\right )+\frac {\tan ^{-1}\left (\frac {1-2 \sqrt [3]{x^6-1}}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {\log (x)}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 50
Rule 56
Rule 204
Rule 266
Rule 618
Rubi steps
\begin {align*} \int \frac {\left (-1+x^6\right )^{2/3}}{x} \, dx &=\frac {1}{6} \operatorname {Subst}\left (\int \frac {(-1+x)^{2/3}}{x} \, dx,x,x^6\right )\\ &=\frac {1}{4} \left (-1+x^6\right )^{2/3}-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x} x} \, dx,x,x^6\right )\\ &=\frac {1}{4} \left (-1+x^6\right )^{2/3}-\frac {\log (x)}{2}+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sqrt [3]{-1+x^6}\right )-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [3]{-1+x^6}\right )\\ &=\frac {1}{4} \left (-1+x^6\right )^{2/3}-\frac {\log (x)}{2}+\frac {1}{4} \log \left (1+\sqrt [3]{-1+x^6}\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt [3]{-1+x^6}\right )\\ &=\frac {1}{4} \left (-1+x^6\right )^{2/3}+\frac {\tan ^{-1}\left (\frac {1-2 \sqrt [3]{-1+x^6}}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {\log (x)}{2}+\frac {1}{4} \log \left (1+\sqrt [3]{-1+x^6}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 30, normalized size = 0.34 \begin {gather*} -\frac {1}{4} \left (x^6-1\right )^{2/3} \left (\, _2F_1\left (\frac {2}{3},1;\frac {5}{3};1-x^6\right )-1\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.05, size = 89, normalized size = 1.00 \begin {gather*} \frac {1}{4} \left (-1+x^6\right )^{2/3}+\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-1+x^6}}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {1}{6} \log \left (1+\sqrt [3]{-1+x^6}\right )-\frac {1}{12} \log \left (1-\sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 67, normalized size = 0.75 \begin {gather*} -\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + \frac {1}{4} \, {\left (x^{6} - 1\right )}^{\frac {2}{3}} - \frac {1}{12} \, \log \left ({\left (x^{6} - 1\right )}^{\frac {2}{3}} - {\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{6} \, \log \left ({\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 66, normalized size = 0.74 \begin {gather*} -\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {1}{4} \, {\left (x^{6} - 1\right )}^{\frac {2}{3}} - \frac {1}{12} \, \log \left ({\left (x^{6} - 1\right )}^{\frac {2}{3}} - {\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{6} \, \log \left ({\left | {\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 8.29, size = 84, normalized size = 0.94
method | result | size |
meijerg | \(-\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \mathrm {signum}\left (x^{6}-1\right )^{\frac {2}{3}} \left (\frac {2 \pi \sqrt {3}\, x^{6} \hypergeom \left (\left [\frac {1}{3}, 1, 1\right ], \left [2, 2\right ], x^{6}\right )}{3 \Gamma \left (\frac {2}{3}\right )}-\frac {\left (\frac {3}{2}-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+6 \ln \relax (x )+i \pi \right ) \pi \sqrt {3}}{\Gamma \left (\frac {2}{3}\right )}\right )}{18 \pi \left (-\mathrm {signum}\left (x^{6}-1\right )\right )^{\frac {2}{3}}}\) | \(84\) |
trager | \(\frac {\left (x^{6}-1\right )^{\frac {2}{3}}}{4}+\frac {\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\frac {-1505024575790858565046 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{6}-94448748446849318458973 x^{6} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+23171389162410581752275 x^{6}+68267310132857019576606 \left (x^{6}-1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-142296551347461340528569 \left (x^{6}-1\right )^{\frac {2}{3}}-68267310132857019576606 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}}+96321572850614948162944 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}+142296551347461340528569 \left (x^{6}-1\right )^{\frac {1}{3}}+164588882983471967739550 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-45974978496846392365625}{x^{6}}\right )}{6}-\frac {\ln \left (-\frac {1505024575790858565046 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{6}-91438699295267601328881 x^{6} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-116115113033469041646202 x^{6}+68267310132857019576606 \left (x^{6}-1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+210563861480318360105175 \left (x^{6}-1\right )^{\frac {2}{3}}-68267310132857019576606 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}}-96321572850614948162944 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-210563861480318360105175 \left (x^{6}-1\right )^{\frac {1}{3}}-28054262717757928586338 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+114242288629703411942231}{x^{6}}\right ) \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{6}-\frac {\ln \left (-\frac {1505024575790858565046 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{6}-91438699295267601328881 x^{6} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-116115113033469041646202 x^{6}+68267310132857019576606 \left (x^{6}-1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+210563861480318360105175 \left (x^{6}-1\right )^{\frac {2}{3}}-68267310132857019576606 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}}-96321572850614948162944 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-210563861480318360105175 \left (x^{6}-1\right )^{\frac {1}{3}}-28054262717757928586338 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+114242288629703411942231}{x^{6}}\right )}{6}\) | \(357\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 65, normalized size = 0.73 \begin {gather*} -\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {1}{4} \, {\left (x^{6} - 1\right )}^{\frac {2}{3}} - \frac {1}{12} \, \log \left ({\left (x^{6} - 1\right )}^{\frac {2}{3}} - {\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{6} \, \log \left ({\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.84, size = 89, normalized size = 1.00 \begin {gather*} \frac {\ln \left (\frac {{\left (x^6-1\right )}^{1/3}}{4}+\frac {1}{4}\right )}{6}+\ln \left (9\,{\left (-\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )}^2+\frac {{\left (x^6-1\right )}^{1/3}}{4}\right )\,\left (-\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )-\ln \left (9\,{\left (\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )}^2+\frac {{\left (x^6-1\right )}^{1/3}}{4}\right )\,\left (\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )+\frac {{\left (x^6-1\right )}^{2/3}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.84, size = 37, normalized size = 0.42 \begin {gather*} - \frac {x^{4} \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 {e^{2 i \pi }}{x^{6}}} \right )}}{6 \Gamma \left (\frac {1}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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