Optimal. Leaf size=111 \[ \frac {\sqrt [3]{x^6-1} \left (-3 x^6-1\right )}{8 x^8}-\frac {1}{6} \log \left (\sqrt [3]{x^6-1}-x^2\right )-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x^2}{2 \sqrt [3]{x^6-1}+x^2}\right )}{2 \sqrt {3}}+\frac {1}{12} \log \left (\left (x^6-1\right )^{2/3}+x^4+\sqrt [3]{x^6-1} x^2\right ) \]
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Rubi [A] time = 0.07, antiderivative size = 119, normalized size of antiderivative = 1.07, number of steps used = 10, number of rules used = 10, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {451, 275, 277, 331, 292, 31, 634, 618, 204, 628} \begin {gather*} \frac {\left (x^6-1\right )^{4/3}}{8 x^8}-\frac {\sqrt [3]{x^6-1}}{2 x^2}-\frac {1}{6} \log \left (1-\frac {x^2}{\sqrt [3]{x^6-1}}\right )-\frac {\tan ^{-1}\left (\frac {\frac {2 x^2}{\sqrt [3]{x^6-1}}+1}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {1}{12} \log \left (\frac {x^4}{\left (x^6-1\right )^{2/3}}+\frac {x^2}{\sqrt [3]{x^6-1}}+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 275
Rule 277
Rule 292
Rule 331
Rule 451
Rule 618
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^9} \, dx &=\frac {\left (-1+x^6\right )^{4/3}}{8 x^8}+\int \frac {\sqrt [3]{-1+x^6}}{x^3} \, dx\\ &=\frac {\left (-1+x^6\right )^{4/3}}{8 x^8}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1+x^3}}{x^2} \, dx,x,x^2\right )\\ &=-\frac {\sqrt [3]{-1+x^6}}{2 x^2}+\frac {\left (-1+x^6\right )^{4/3}}{8 x^8}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {x}{\left (-1+x^3\right )^{2/3}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt [3]{-1+x^6}}{2 x^2}+\frac {\left (-1+x^6\right )^{4/3}}{8 x^8}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {x}{1-x^3} \, dx,x,\frac {x^2}{\sqrt [3]{-1+x^6}}\right )\\ &=-\frac {\sqrt [3]{-1+x^6}}{2 x^2}+\frac {\left (-1+x^6\right )^{4/3}}{8 x^8}+\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\frac {x^2}{\sqrt [3]{-1+x^6}}\right )-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1-x}{1+x+x^2} \, dx,x,\frac {x^2}{\sqrt [3]{-1+x^6}}\right )\\ &=-\frac {\sqrt [3]{-1+x^6}}{2 x^2}+\frac {\left (-1+x^6\right )^{4/3}}{8 x^8}-\frac {1}{6} \log \left (1-\frac {x^2}{\sqrt [3]{-1+x^6}}\right )+\frac {1}{12} \operatorname {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\frac {x^2}{\sqrt [3]{-1+x^6}}\right )-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\frac {x^2}{\sqrt [3]{-1+x^6}}\right )\\ &=-\frac {\sqrt [3]{-1+x^6}}{2 x^2}+\frac {\left (-1+x^6\right )^{4/3}}{8 x^8}-\frac {1}{6} \log \left (1-\frac {x^2}{\sqrt [3]{-1+x^6}}\right )+\frac {1}{12} \log \left (1+\frac {x^4}{\left (-1+x^6\right )^{2/3}}+\frac {x^2}{\sqrt [3]{-1+x^6}}\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 x^2}{\sqrt [3]{-1+x^6}}\right )\\ &=-\frac {\sqrt [3]{-1+x^6}}{2 x^2}+\frac {\left (-1+x^6\right )^{4/3}}{8 x^8}-\frac {\tan ^{-1}\left (\frac {1+\frac {2 x^2}{\sqrt [3]{-1+x^6}}}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{6} \log \left (1-\frac {x^2}{\sqrt [3]{-1+x^6}}\right )+\frac {1}{12} \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 [C] time = 0.02, size = 50, normalized size = 0.45 \begin {gather*} \frac {\sqrt [3]{x^6-1} \left (-\frac {4 x^6 \, _2F_1\left (-\frac {1}{3},-\frac {1}{3};\frac {2}{3};x^6\right )}{\sqrt [3]{1-x^6}}+x^6-1\right )}{8 x^8} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.50, size = 111, normalized size = 1.00 \begin {gather*} \frac {\left (-1-3 x^6\right ) \sqrt [3]{-1+x^6}}{8 x^8}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{-1+x^6}}\right )}{2 \sqrt {3}}-\frac {1}{6} \log \left (-x^2+\sqrt [3]{-1+x^6}\right )+\frac {1}{12} \log \left (x^4+x^2 \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 = 0.71, size = 116, normalized size = 1.05 \begin {gather*} -\frac {4 \, \sqrt {3} x^{8} \arctan \left (-\frac {25382 \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{4} - 13720 \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {2}{3}} x^{2} + \sqrt {3} {\left (5831 \, x^{6} - 7200\right )}}{58653 \, x^{6} - 8000}\right ) + 2 \, x^{8} \log \left (-3 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{4} + 3 \, {\left (x^{6} - 1\right )}^{\frac {2}{3}} x^{2} + 1\right ) + 3 \, {\left (3 \, x^{6} + 1\right )} {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{24 \, x^{8}} \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*} \int \frac {{\left (x^{6} + 1\right )} {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{9}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 7.69, size = 58, normalized size = 0.52
method | result | size |
risch | \(-\frac {3 x^{12}-2 x^{6}-1}{8 x^{8} \left (x^{6}-1\right )^{\frac {2}{3}}}+\frac {\left (-\mathrm {signum}\left (x^{6}-1\right )\right )^{\frac {2}{3}} x^{4} \hypergeom \left (\left [\frac {2}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x^{6}\right )}{4 \mathrm {signum}\left (x^{6}-1\right )^{\frac {2}{3}}}\) | \(58\) |
meijerg | \(-\frac {\mathrm {signum}\left (x^{6}-1\right )^{\frac {1}{3}} \hypergeom \left (\left [-\frac {1}{3}, -\frac {1}{3}\right ], \left [\frac {2}{3}\right ], x^{6}\right )}{2 \left (-\mathrm {signum}\left (x^{6}-1\right )\right )^{\frac {1}{3}} x^{2}}-\frac {\mathrm {signum}\left (x^{6}-1\right )^{\frac {1}{3}} \left (-x^{6}+1\right )^{\frac {4}{3}}}{8 \left (-\mathrm {signum}\left (x^{6}-1\right )\right )^{\frac {1}{3}} x^{8}}\) | \(66\) |
trager | \(-\frac {\left (3 x^{6}+1\right ) \left (x^{6}-1\right )^{\frac {1}{3}}}{8 x^{8}}-\frac {\ln \left (-95839020187648 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )^{2} x^{6}+71103429967360 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) x^{6}-147783054114048 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{4}+299529781823 x^{6}+77053995319296 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x^{2}+276285385917 x^{4} \left (x^{6}-1\right )^{\frac {1}{3}}-577277555133 x^{2} \left (x^{6}-1\right )^{\frac {2}{3}}+6133697292009472 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )^{2}-21851442872064 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )-199163781631\right )}{6}+\frac {128 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) \ln \left (198613804384256 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )^{2} x^{6}+147007218940672 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) x^{6}-70729058794752 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{4}-297961563070 x^{6}-77053995319296 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x^{2}+577277555133 x^{4} \left (x^{6}-1\right )^{\frac {1}{3}}-276285385917 x^{2} \left (x^{6}-1\right )^{\frac {2}{3}}-12711283480592384 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )^{2}-23183919873536 \RootOf \left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )+98797781439\right )}{3}\) | \(316\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 93, normalized size = 0.84 \begin {gather*} \frac {1}{6} \, \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}}}{2 \, x^{2}} + \frac {{\left (x^{6} - 1\right )}^{\frac {4}{3}}}{8 \, x^{8}} + \frac {1}{12} \, \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}{6} \, \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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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (x^6-1\right )}^{1/3}\,\left (x^6+1\right )}{x^9} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 3.37, size = 167, normalized size = 1.50 \begin {gather*} \begin {cases} \frac {\sqrt [3]{-1 + \frac {1}{x^{6}}} e^{- \frac {2 i \pi }{3}} \Gamma \left (- \frac {4}{3}\right )}{6 \Gamma \left (- \frac {1}{3}\right )} - \frac {\sqrt [3]{-1 + \frac {1}{x^{6}}} e^{- \frac {2 i \pi }{3}} \Gamma \left (- \frac {4}{3}\right )}{6 x^{6} \Gamma \left (- \frac {1}{3}\right )} & \text {for}\: \frac {1}{\left |{x^{6}}\right |} > 1 \\- \frac {\sqrt [3]{1 - \frac {1}{x^{6}}} \Gamma \left (- \frac {4}{3}\right )}{6 \Gamma \left (- \frac {1}{3}\right )} + \frac {\sqrt [3]{1 - \frac {1}{x^{6}}} \Gamma \left (- \frac {4}{3}\right )}{6 x^{6} \Gamma \left (- \frac {1}{3}\right )} & \text {otherwise} \end {cases} + \frac {e^{\frac {i \pi }{3}} \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, - \frac {1}{3} \\ \frac {2}{3} \end {matrix}\middle | {x^{6}} \right )}}{6 x^{2} \Gamma \left (\frac {2}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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