Optimal. Leaf size=108 \[ \frac {2}{9} \log \left (\sqrt [3]{x^3-1}-x\right )-\frac {2 \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3-1}+x}\right )}{3 \sqrt {3}}-\frac {1}{9} \log \left (\sqrt [3]{x^3-1} x+\left (x^3-1\right )^{2/3}+x^2\right )+\frac {\left (x^3-1\right )^{2/3} \left (5 x^6+6 x^3-6\right )}{15 x^5} \]
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Rubi [A] time = 0.03, antiderivative size = 97, normalized size of antiderivative = 0.90, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1487, 451, 277, 239} \begin {gather*} \frac {1}{3} \log \left (\sqrt [3]{x^3-1}-x\right )-\frac {2 \tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {2 \left (x^3-1\right )^{5/3}}{5 x^5}+\frac {\left (x^3-1\right )^{5/3}}{3 x^2}+\frac {\left (x^3-1\right )^{2/3}}{3 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 239
Rule 277
Rule 451
Rule 1487
Rubi steps
\begin {align*} \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^6\right )}{x^6} \, dx &=\frac {\left (-1+x^3\right )^{5/3}}{3 x^2}+\frac {1}{3} \int \frac {\left (6-2 x^3\right ) \left (-1+x^3\right )^{2/3}}{x^6} \, dx\\ &=\frac {2 \left (-1+x^3\right )^{5/3}}{5 x^5}+\frac {\left (-1+x^3\right )^{5/3}}{3 x^2}-\frac {2}{3} \int \frac {\left (-1+x^3\right )^{2/3}}{x^3} \, dx\\ &=\frac {\left (-1+x^3\right )^{2/3}}{3 x^2}+\frac {2 \left (-1+x^3\right )^{5/3}}{5 x^5}+\frac {\left (-1+x^3\right )^{5/3}}{3 x^2}-\frac {2}{3} \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx\\ &=\frac {\left (-1+x^3\right )^{2/3}}{3 x^2}+\frac {2 \left (-1+x^3\right )^{5/3}}{5 x^5}+\frac {\left (-1+x^3\right )^{5/3}}{3 x^2}-\frac {2 \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {1}{3} \log \left (-x+\sqrt [3]{-1+x^3}\right )\\ \end {align*}
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Mathematica [C] time = 0.04, size = 52, normalized size = 0.48 \begin {gather*} \frac {x \left (x^3-1\right )^{2/3} \, _2F_1\left (-\frac {2}{3},\frac {1}{3};\frac {4}{3};x^3\right )}{\left (1-x^3\right )^{2/3}}+\frac {2 \left (x^3-1\right )^{5/3}}{5 x^5} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.21, size = 108, normalized size = 1.00 \begin {gather*} \frac {2}{9} \log \left (\sqrt [3]{x^3-1}-x\right )-\frac {2 \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3-1}+x}\right )}{3 \sqrt {3}}-\frac {1}{9} \log \left (\sqrt [3]{x^3-1} x+\left (x^3-1\right )^{2/3}+x^2\right )+\frac {\left (x^3-1\right )^{2/3} \left (5 x^6+6 x^3-6\right )}{15 x^5} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 117, normalized size = 1.08 \begin {gather*} -\frac {10 \, \sqrt {3} x^{5} \arctan \left (-\frac {25382 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} - 13720 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (5831 \, x^{3} - 7200\right )}}{58653 \, x^{3} - 8000}\right ) - 5 \, x^{5} \log \left (-3 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x + 1\right ) - 3 \, {\left (5 \, x^{6} + 6 \, x^{3} - 6\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{45 \, x^{5}} \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} + 2\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.31, size = 59, normalized size = 0.55 \begin {gather*} \frac {5 x^{9}+x^{6}-12 x^{3}+6}{15 x^{5} \left (x^{3}-1\right )^{\frac {1}{3}}}-\frac {2 \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {1}{3}} x \hypergeom \left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{3}\right )}{3 \mathrm {signum}\left (x^{3}-1\right )^{\frac {1}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 106, normalized size = 0.98 \begin {gather*} \frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) - \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}}}{3 \, x^{2} {\left (\frac {x^{3} - 1}{x^{3}} - 1\right )}} + \frac {2 \, {\left (x^{3} - 1\right )}^{\frac {5}{3}}}{5 \, x^{5}} - \frac {1}{9} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) + \frac {2}{9} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.13, size = 54, normalized size = 0.50 \begin {gather*} \frac {x\,{\left (x^3-1\right )}^{2/3}\,{{}}_2{\mathrm {F}}_1\left (-\frac {2}{3},\frac {1}{3};\ \frac {4}{3};\ x^3\right )}{{\left (1-x^3\right )}^{2/3}}-\frac {2\,{\left (x^3-1\right )}^{2/3}-2\,x^3\,{\left (x^3-1\right )}^{2/3}}{5\,x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.52, size = 160, normalized size = 1.48 \begin {gather*} - \frac {x e^{- \frac {i \pi }{3}} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {x^{3}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} + 2 \left (\begin {cases} \frac {\left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{- \frac {i \pi }{3}} \Gamma \left (- \frac {5}{3}\right )}{3 \Gamma \left (- \frac {2}{3}\right )} - \frac {\left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{- \frac {i \pi }{3}} \Gamma \left (- \frac {5}{3}\right )}{3 x^{3} \Gamma \left (- \frac {2}{3}\right )} & \text {for}\: \frac {1}{\left |{x^{3}}\right |} > 1 \\- \frac {\left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {5}{3}\right )}{3 \Gamma \left (- \frac {2}{3}\right )} + \frac {\left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {5}{3}\right )}{3 x^{3} \Gamma \left (- \frac {2}{3}\right )} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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