Optimal. Leaf size=109 \[ \frac {1}{12} \text {RootSum}\left [2 \text {$\#$1}^6-3 \text {$\#$1}^3-1\& ,\frac {-\text {$\#$1}^3 \log \left (\sqrt [3]{x^3-1}-\text {$\#$1} x\right )+\text {$\#$1}^3 \log (x)-\log \left (\sqrt [3]{x^3-1}-\text {$\#$1} x\right )+\log (x)}{4 \text {$\#$1}^4-3 \text {$\#$1}}\& \right ]+\frac {\left (x^3-1\right )^{2/3} \left (-x^3-4\right )}{40 x^5} \]
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Rubi [B] time = 0.55, antiderivative size = 225, normalized size of antiderivative = 2.06, number of steps used = 11, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6728, 264, 277, 239, 430, 429} \begin {gather*} -\frac {\left (17+\sqrt {17}\right ) x \left (x^3-1\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,-\frac {4 x^3}{1-\sqrt {17}}\right )}{34 \left (1-\sqrt {17}\right ) \left (1-x^3\right )^{2/3}}-\frac {\left (17-\sqrt {17}\right ) x \left (x^3-1\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,-\frac {4 x^3}{1+\sqrt {17}}\right )}{34 \left (1+\sqrt {17}\right ) \left (1-x^3\right )^{2/3}}-\frac {1}{8} \log \left (\sqrt [3]{x^3-1}-x\right )+\frac {\tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{4 \sqrt {3}}+\frac {\left (x^3-1\right )^{5/3}}{10 x^5}-\frac {\left (x^3-1\right )^{2/3}}{8 x^2} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 239
Rule 264
Rule 277
Rule 429
Rule 430
Rule 6728
Rubi steps
\begin {align*} \int \frac {\left (-1+x^3\right )^{2/3} \left (-1+x^6\right )}{x^6 \left (-2+x^3+2 x^6\right )} \, dx &=\int \left (\frac {\left (-1+x^3\right )^{2/3}}{2 x^6}+\frac {\left (-1+x^3\right )^{2/3}}{4 x^3}+\frac {\left (-1-2 x^3\right ) \left (-1+x^3\right )^{2/3}}{4 \left (-2+x^3+2 x^6\right )}\right ) \, dx\\ &=\frac {1}{4} \int \frac {\left (-1+x^3\right )^{2/3}}{x^3} \, dx+\frac {1}{4} \int \frac {\left (-1-2 x^3\right ) \left (-1+x^3\right )^{2/3}}{-2+x^3+2 x^6} \, dx+\frac {1}{2} \int \frac {\left (-1+x^3\right )^{2/3}}{x^6} \, dx\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{8 x^2}+\frac {\left (-1+x^3\right )^{5/3}}{10 x^5}+\frac {1}{4} \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx+\frac {1}{4} \int \left (\frac {\left (-2-\frac {2}{\sqrt {17}}\right ) \left (-1+x^3\right )^{2/3}}{1-\sqrt {17}+4 x^3}+\frac {\left (-2+\frac {2}{\sqrt {17}}\right ) \left (-1+x^3\right )^{2/3}}{1+\sqrt {17}+4 x^3}\right ) \, dx\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{8 x^2}+\frac {\left (-1+x^3\right )^{5/3}}{10 x^5}+\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{4 \sqrt {3}}-\frac {1}{8} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\frac {1}{34} \left (-17+\sqrt {17}\right ) \int \frac {\left (-1+x^3\right )^{2/3}}{1+\sqrt {17}+4 x^3} \, dx-\frac {1}{34} \left (17+\sqrt {17}\right ) \int \frac {\left (-1+x^3\right )^{2/3}}{1-\sqrt {17}+4 x^3} \, dx\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{8 x^2}+\frac {\left (-1+x^3\right )^{5/3}}{10 x^5}+\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{4 \sqrt {3}}-\frac {1}{8} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\frac {\left (\left (-17+\sqrt {17}\right ) \left (-1+x^3\right )^{2/3}\right ) \int \frac {\left (1-x^3\right )^{2/3}}{1+\sqrt {17}+4 x^3} \, dx}{34 \left (1-x^3\right )^{2/3}}-\frac {\left (\left (17+\sqrt {17}\right ) \left (-1+x^3\right )^{2/3}\right ) \int \frac {\left (1-x^3\right )^{2/3}}{1-\sqrt {17}+4 x^3} \, dx}{34 \left (1-x^3\right )^{2/3}}\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{8 x^2}+\frac {\left (-1+x^3\right )^{5/3}}{10 x^5}-\frac {\left (17+\sqrt {17}\right ) x \left (-1+x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,-\frac {4 x^3}{1-\sqrt {17}}\right )}{34 \left (1-\sqrt {17}\right ) \left (1-x^3\right )^{2/3}}-\frac {\left (17-\sqrt {17}\right ) x \left (-1+x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,-\frac {4 x^3}{1+\sqrt {17}}\right )}{34 \left (1+\sqrt {17}\right ) \left (1-x^3\right )^{2/3}}+\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{4 \sqrt {3}}-\frac {1}{8} \log \left (-x+\sqrt [3]{-1+x^3}\right )\\ \end {align*}
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Mathematica [B] time = 0.65, size = 372, normalized size = 3.41 \begin {gather*} \frac {-2 \sqrt [3]{43+13 \sqrt {17}} \log \left (\sqrt [3]{\sqrt {17}-3}-\frac {\sqrt [3]{2} x}{\sqrt [3]{x^3-1}}\right )-2 \sqrt [3]{13 \sqrt {17}-43} \log \left (\frac {\sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+\sqrt [3]{3+\sqrt {17}}\right )+2 \sqrt {3} \sqrt [3]{43+13 \sqrt {17}} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{\frac {2}{\sqrt {17}-3}} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )+2 \sqrt {3} \sqrt [3]{13 \sqrt {17}-43} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{\frac {2}{3+\sqrt {17}}} x}{\sqrt [3]{x^3-1}}}{\sqrt {3}}\right )+\sqrt [3]{43+13 \sqrt {17}} \log \left (\frac {\sqrt [3]{2 \left (\sqrt {17}-3\right )} x}{\sqrt [3]{x^3-1}}+\frac {2^{2/3} x^2}{\left (x^3-1\right )^{2/3}}+\left (\sqrt {17}-3\right )^{2/3}\right )+\sqrt [3]{13 \sqrt {17}-43} \log \left (-\frac {\sqrt [3]{2 \left (3+\sqrt {17}\right )} x}{\sqrt [3]{x^3-1}}+\frac {2^{2/3} x^2}{\left (x^3-1\right )^{2/3}}+\left (3+\sqrt {17}\right )^{2/3}\right )}{48 \sqrt {17}}-\frac {\left (x^3-1\right )^{2/3} \left (x^3+4\right )}{40 x^5} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.00, size = 109, normalized size = 1.00 \begin {gather*} \frac {\left (-4-x^3\right ) \left (-1+x^3\right )^{2/3}}{40 x^5}+\frac {1}{12} \text {RootSum}\left [-1-3 \text {$\#$1}^3+2 \text {$\#$1}^6\&,\frac {\log (x)-\log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^3-\log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-3 \text {$\#$1}+4 \text {$\#$1}^4}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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^{3} - 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{6} + x^{3} - 2\right )} x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 354.81, size = 7065, normalized size = 64.82
method | result | size |
risch | \(\text {Expression too large to display}\) | \(7065\) |
trager | \(\text {Expression too large to display}\) | \(8767\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{6} - 1\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{6} + x^{3} - 2\right )} x^{6}}\,{d x} \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^3-1\right )}^{2/3}\,\left (x^6-1\right )}{x^6\,\left (2\,x^6+x^3-2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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