Optimal. Leaf size=109 \[ \frac {\sqrt [3]{x^6-1} \left (x^6-3\right )}{6 x^2}-\frac {1}{9} \log \left (\sqrt [3]{x^6-1}-x^2\right )-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x^2}{2 \sqrt [3]{x^6-1}+x^2}\right )}{3 \sqrt {3}}+\frac {1}{18} \log \left (\left (x^6-1\right )^{2/3}+x^4+\sqrt [3]{x^6-1} x^2\right ) \]
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Rubi [A] time = 0.08, antiderivative size = 119, normalized size of antiderivative = 1.09, 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 = {453, 275, 279, 331, 292, 31, 634, 618, 204, 628} \begin {gather*} -\frac {1}{3} \sqrt [3]{x^6-1} x^4+\frac {\left (x^6-1\right )^{4/3}}{2 x^2}-\frac {1}{9} \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 )}{3 \sqrt {3}}+\frac {1}{18} \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 279
Rule 292
Rule 331
Rule 453
Rule 618
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^3} \, dx &=\frac {\left (-1+x^6\right )^{4/3}}{2 x^2}-2 \int x^3 \sqrt [3]{-1+x^6} \, dx\\ &=\frac {\left (-1+x^6\right )^{4/3}}{2 x^2}-\operatorname {Subst}\left (\int x \sqrt [3]{-1+x^3} \, dx,x,x^2\right )\\ &=-\frac {1}{3} x^4 \sqrt [3]{-1+x^6}+\frac {\left (-1+x^6\right )^{4/3}}{2 x^2}+\frac {1}{3} \operatorname {Subst}\left (\int \frac {x}{\left (-1+x^3\right )^{2/3}} \, dx,x,x^2\right )\\ &=-\frac {1}{3} x^4 \sqrt [3]{-1+x^6}+\frac {\left (-1+x^6\right )^{4/3}}{2 x^2}+\frac {1}{3} \operatorname {Subst}\left (\int \frac {x}{1-x^3} \, dx,x,\frac {x^2}{\sqrt [3]{-1+x^6}}\right )\\ &=-\frac {1}{3} x^4 \sqrt [3]{-1+x^6}+\frac {\left (-1+x^6\right )^{4/3}}{2 x^2}+\frac {1}{9} \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\frac {x^2}{\sqrt [3]{-1+x^6}}\right )-\frac {1}{9} \operatorname {Subst}\left (\int \frac {1-x}{1+x+x^2} \, dx,x,\frac {x^2}{\sqrt [3]{-1+x^6}}\right )\\ &=-\frac {1}{3} x^4 \sqrt [3]{-1+x^6}+\frac {\left (-1+x^6\right )^{4/3}}{2 x^2}-\frac {1}{9} \log \left (1-\frac {x^2}{\sqrt [3]{-1+x^6}}\right )+\frac {1}{18} \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}{6} \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\frac {x^2}{\sqrt [3]{-1+x^6}}\right )\\ &=-\frac {1}{3} x^4 \sqrt [3]{-1+x^6}+\frac {\left (-1+x^6\right )^{4/3}}{2 x^2}-\frac {1}{9} \log \left (1-\frac {x^2}{\sqrt [3]{-1+x^6}}\right )+\frac {1}{18} \log \left (1+\frac {x^4}{\left (-1+x^6\right )^{2/3}}+\frac {x^2}{\sqrt [3]{-1+x^6}}\right )+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 x^2}{\sqrt [3]{-1+x^6}}\right )\\ &=-\frac {1}{3} x^4 \sqrt [3]{-1+x^6}+\frac {\left (-1+x^6\right )^{4/3}}{2 x^2}-\frac {\tan ^{-1}\left (\frac {1+\frac {2 x^2}{\sqrt [3]{-1+x^6}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {1}{9} \log \left (1-\frac {x^2}{\sqrt [3]{-1+x^6}}\right )+\frac {1}{18} \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 = 0.03, size = 50, normalized size = 0.46 \begin {gather*} \frac {\sqrt [3]{x^6-1} \left (-\frac {x^6 \, _2F_1\left (-\frac {1}{3},\frac {2}{3};\frac {5}{3};x^6\right )}{\sqrt [3]{1-x^6}}+x^6-1\right )}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.48, size = 109, normalized size = 1.00 \begin {gather*} \frac {\sqrt [3]{x^6-1} \left (x^6-3\right )}{6 x^2}-\frac {1}{9} \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 )}{3 \sqrt {3}}+\frac {1}{18} \log \left (\left (x^6-1\right )^{2/3}+x^4+\sqrt [3]{x^6-1} x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 113, normalized size = 1.04 \begin {gather*} -\frac {2 \, \sqrt {3} x^{2} \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 ) + x^{2} \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 (x^{6} - 1\right )}^{\frac {1}{3}} {\left (x^{6} - 3\right )}}{18 \, x^{2}} \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^{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.30, size = 56, normalized size = 0.51 \begin {gather*} \frac {x^{12}-4 x^{6}+3}{6 x^{2} \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 )}{6 \mathrm {signum}\left (x^{6}-1\right )^{\frac {2}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 106, normalized size = 0.97 \begin {gather*} \frac {1}{9} \, \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 {1}{3}}}{6 \, x^{2} {\left (\frac {x^{6} - 1}{x^{6}} - 1\right )}} + \frac {1}{18} \, \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}{9} \, \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^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 3.25, size = 71, normalized size = 0.65 \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 )} + \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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