Optimal. Leaf size=265 \[ \frac {\log \left (\sqrt [3]{2} \sqrt [3]{x^3-1}-x\right )}{6\ 2^{2/3}}-\frac {\log \left (6^{2/3} \sqrt [3]{x^3-1}-3 x\right )}{8 \sqrt [3]{6}}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{2} \sqrt [3]{x^3-1}+x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\sqrt [6]{3} \tan ^{-1}\left (\frac {3^{5/6} x}{2\ 2^{2/3} \sqrt [3]{x^3-1}+\sqrt [3]{3} x}\right )}{8 \sqrt [3]{2}}+\frac {\left (x^3-1\right )^{2/3} \left (13 x^3-8\right )}{40 x^5}-\frac {\log \left (\sqrt [3]{2} \sqrt [3]{x^3-1} x+2^{2/3} \left (x^3-1\right )^{2/3}+x^2\right )}{12\ 2^{2/3}}+\frac {\log \left (6^{2/3} \sqrt [3]{x^3-1} x+2 \sqrt [3]{6} \left (x^3-1\right )^{2/3}+3 x^2\right )}{16 \sqrt [3]{6}} \]
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Rubi [C] time = 0.85, antiderivative size = 173, normalized size of antiderivative = 0.65, number of steps used = 9, number of rules used = 6, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {6725, 264, 277, 239, 430, 429} \begin {gather*} \frac {x \left (x^3-1\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,\frac {x^3}{4}\right )}{16 \left (1-x^3\right )^{2/3}}-\frac {x \left (x^3-1\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,\frac {x^3}{2}\right )}{4 \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}}{5 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 6725
Rubi steps
\begin {align*} \int \frac {\left (-1+x^3\right )^{2/3} \left (8-8 x^3+x^6\right )}{x^6 \left (-4+x^3\right ) \left (-2+x^3\right )} \, dx &=\int \left (\frac {\left (-1+x^3\right )^{2/3}}{x^6}-\frac {\left (-1+x^3\right )^{2/3}}{4 x^3}-\frac {\left (-1+x^3\right )^{2/3}}{4 \left (-4+x^3\right )}+\frac {\left (-1+x^3\right )^{2/3}}{2 \left (-2+x^3\right )}\right ) \, dx\\ &=-\left (\frac {1}{4} \int \frac {\left (-1+x^3\right )^{2/3}}{x^3} \, dx\right )-\frac {1}{4} \int \frac {\left (-1+x^3\right )^{2/3}}{-4+x^3} \, dx+\frac {1}{2} \int \frac {\left (-1+x^3\right )^{2/3}}{-2+x^3} \, dx+\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}}{5 x^5}-\frac {1}{4} \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx-\frac {\left (-1+x^3\right )^{2/3} \int \frac {\left (1-x^3\right )^{2/3}}{-4+x^3} \, dx}{4 \left (1-x^3\right )^{2/3}}+\frac {\left (-1+x^3\right )^{2/3} \int \frac {\left (1-x^3\right )^{2/3}}{-2+x^3} \, dx}{2 \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}}{5 x^5}+\frac {x \left (-1+x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,\frac {x^3}{4}\right )}{16 \left (1-x^3\right )^{2/3}}-\frac {x \left (-1+x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,\frac {x^3}{2}\right )}{4 \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 [A] time = 0.41, size = 259, normalized size = 0.98 \begin {gather*} \frac {1}{96} \left (8 \sqrt [3]{2} \log \left (\sqrt [3]{2}-\frac {x}{\sqrt [3]{x^3-1}}\right )-2\ 6^{2/3} \log \left (2^{2/3}-\frac {\sqrt [3]{3} x}{\sqrt [3]{x^3-1}}\right )-8 \sqrt [3]{2} \sqrt {3} \tan ^{-1}\left (\frac {\frac {2^{2/3} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )+6\ 2^{2/3} \sqrt [6]{3} \tan ^{-1}\left (\frac {\frac {\sqrt [3]{6} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )-4 \sqrt [3]{2} \log \left (\frac {\sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+\frac {x^2}{\left (x^3-1\right )^{2/3}}+2^{2/3}\right )+6^{2/3} \log \left (\frac {2^{2/3} \sqrt [3]{3} x}{\sqrt [3]{x^3-1}}+\frac {3^{2/3} x^2}{\left (x^3-1\right )^{2/3}}+2 \sqrt [3]{2}\right )\right )+\left (x^3-1\right )^{2/3} \left (\frac {13}{40 x^2}-\frac {1}{5 x^5}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.77, size = 265, normalized size = 1.00 \begin {gather*} \frac {\log \left (\sqrt [3]{2} \sqrt [3]{x^3-1}-x\right )}{6\ 2^{2/3}}-\frac {\log \left (6^{2/3} \sqrt [3]{x^3-1}-3 x\right )}{8 \sqrt [3]{6}}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{2} \sqrt [3]{x^3-1}+x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\sqrt [6]{3} \tan ^{-1}\left (\frac {3^{5/6} x}{2\ 2^{2/3} \sqrt [3]{x^3-1}+\sqrt [3]{3} x}\right )}{8 \sqrt [3]{2}}+\frac {\left (x^3-1\right )^{2/3} \left (13 x^3-8\right )}{40 x^5}-\frac {\log \left (\sqrt [3]{2} \sqrt [3]{x^3-1} x+2^{2/3} \left (x^3-1\right )^{2/3}+x^2\right )}{12\ 2^{2/3}}+\frac {\log \left (6^{2/3} \sqrt [3]{x^3-1} x+2 \sqrt [3]{6} \left (x^3-1\right )^{2/3}+3 x^2\right )}{16 \sqrt [3]{6}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 11.22, size = 552, normalized size = 2.08 \begin {gather*} \frac {30 \cdot 6^{\frac {1}{6}} \sqrt {2} \left (-1\right )^{\frac {1}{3}} x^{5} \arctan \left (\frac {6^{\frac {1}{6}} {\left (24 \cdot 6^{\frac {2}{3}} \sqrt {2} \left (-1\right )^{\frac {2}{3}} {\left (5 \, x^{7} - 22 \, x^{4} + 8 \, x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} - 36 \, \sqrt {2} \left (-1\right )^{\frac {1}{3}} {\left (109 \, x^{8} - 116 \, x^{5} + 16 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 6^{\frac {1}{3}} \sqrt {2} {\left (1189 \, x^{9} - 2064 \, x^{6} + 912 \, x^{3} - 64\right )}\right )}}{6 \, {\left (971 \, x^{9} - 960 \, x^{6} - 48 \, x^{3} + 64\right )}}\right ) + 10 \cdot 6^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{5} \log \left (\frac {18 \cdot 6^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} - 6^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{3} - 4\right )} - 36 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x}{x^{3} - 4}\right ) - 5 \cdot 6^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {12 \cdot 6^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (5 \, x^{4} - 2 \, x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} - 6^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (109 \, x^{6} - 116 \, x^{3} + 16\right )} - 18 \, {\left (11 \, x^{5} - 8 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x^{6} - 8 \, x^{3} + 16}\right ) + 40 \cdot 4^{\frac {1}{6}} \sqrt {3} x^{5} \arctan \left (\frac {4^{\frac {1}{6}} {\left (12 \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (2 \, x^{7} - 5 \, x^{4} + 2 \, x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} \sqrt {3} {\left (91 \, x^{9} - 168 \, x^{6} + 84 \, x^{3} - 8\right )} + 12 \, \sqrt {3} {\left (19 \, x^{8} - 22 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (53 \, x^{9} - 48 \, x^{6} - 12 \, x^{3} + 8\right )}}\right ) + 20 \cdot 4^{\frac {2}{3}} x^{5} \log \left (\frac {6 \cdot 4^{\frac {1}{3}} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 4^{\frac {2}{3}} {\left (x^{3} - 2\right )} - 12 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x}{x^{3} - 2}\right ) - 10 \cdot 4^{\frac {2}{3}} x^{5} \log \left (\frac {6 \cdot 4^{\frac {2}{3}} {\left (2 \, x^{4} - x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (19 \, x^{6} - 22 \, x^{3} + 4\right )} + 6 \, {\left (5 \, x^{5} - 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x^{6} - 4 \, x^{3} + 4}\right ) + 36 \, {\left (13 \, x^{3} - 8\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{1440 \, 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} - 8 \, x^{3} + 8\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{3} - 2\right )} {\left (x^{3} - 4\right )} x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.46, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x^{3}-1\right )^{\frac {2}{3}} \left (x^{6}-8 x^{3}+8\right )}{x^{6} \left (x^{3}-4\right ) \left (x^{3}-2\right )}\, dx \end {gather*}
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} - 8 \, x^{3} + 8\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{3} - 2\right )} {\left (x^{3} - 4\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 = -0.00 \begin {gather*} \int \frac {{\left (x^3-1\right )}^{2/3}\,\left (x^6-8\,x^3+8\right )}{x^6\,\left (x^3-2\right )\,\left (x^3-4\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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