Optimal. Leaf size=214 \[ -\frac {1}{3} \log \left (\sqrt [3]{x^3-1}-x\right )+\frac {2}{3} \sqrt [3]{2} \log \left (\sqrt [3]{2} \sqrt [3]{x^3-1}-x\right )+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3-1}+x}\right )}{\sqrt {3}}-\frac {2 \sqrt [3]{2} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{2} \sqrt [3]{x^3-1}+x}\right )}{\sqrt {3}}+\frac {\left (x^3-1\right )^{2/3} \left (7 x^3+8\right )}{10 x^5}+\frac {1}{6} \log \left (\sqrt [3]{x^3-1} x+\left (x^3-1\right )^{2/3}+x^2\right )-\frac {1}{3} \sqrt [3]{2} \log \left (\sqrt [3]{2} \sqrt [3]{x^3-1} x+2^{2/3} \left (x^3-1\right )^{2/3}+x^2\right ) \]
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Rubi [C] time = 0.39, antiderivative size = 123, normalized size of antiderivative = 0.57, number of steps used = 7, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6725, 264, 277, 239, 430, 429} \begin {gather*} -\frac {2 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 )}{\left (1-x^3\right )^{2/3}}+\frac {3}{2} \log \left (\sqrt [3]{x^3-1}-x\right )-\sqrt {3} \tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )-\frac {4 \left (x^3-1\right )^{5/3}}{5 x^5}+\frac {3 \left (x^3-1\right )^{2/3}}{2 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+2 x^3+x^6\right )}{x^6 \left (-2+x^3\right )} \, dx &=\int \left (-\frac {4 \left (-1+x^3\right )^{2/3}}{x^6}-\frac {3 \left (-1+x^3\right )^{2/3}}{x^3}+\frac {4 \left (-1+x^3\right )^{2/3}}{-2+x^3}\right ) \, dx\\ &=-\left (3 \int \frac {\left (-1+x^3\right )^{2/3}}{x^3} \, dx\right )-4 \int \frac {\left (-1+x^3\right )^{2/3}}{x^6} \, dx+4 \int \frac {\left (-1+x^3\right )^{2/3}}{-2+x^3} \, dx\\ &=\frac {3 \left (-1+x^3\right )^{2/3}}{2 x^2}-\frac {4 \left (-1+x^3\right )^{5/3}}{5 x^5}-3 \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx+\frac {\left (4 \left (-1+x^3\right )^{2/3}\right ) \int \frac {\left (1-x^3\right )^{2/3}}{-2+x^3} \, dx}{\left (1-x^3\right )^{2/3}}\\ &=\frac {3 \left (-1+x^3\right )^{2/3}}{2 x^2}-\frac {4 \left (-1+x^3\right )^{5/3}}{5 x^5}-\frac {2 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 )}{\left (1-x^3\right )^{2/3}}-\sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )+\frac {3}{2} \log \left (-x+\sqrt [3]{-1+x^3}\right )\\ \end {align*}
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Mathematica [C] time = 0.27, size = 181, normalized size = 0.85 \begin {gather*} -\frac {\sqrt [3]{1-x^3} x^4 F_1\left (\frac {4}{3};\frac {1}{3},1;\frac {7}{3};x^3,\frac {x^3}{2}\right )}{8 \sqrt [3]{x^3-1}}+\frac {\left (x^3-1\right )^{2/3} \left (7 x^3+8\right )}{10 x^5}-\frac {-2 \log \left (2-\frac {2^{2/3} x}{\sqrt [3]{1-x^3}}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2^{2/3} x}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )+\log \left (\frac {2^{2/3} x}{\sqrt [3]{1-x^3}}+\frac {\sqrt [3]{2} x^2}{\left (1-x^3\right )^{2/3}}+2\right )}{3\ 2^{2/3}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.50, size = 214, normalized size = 1.00 \begin {gather*} -\frac {1}{3} \log \left (\sqrt [3]{x^3-1}-x\right )+\frac {2}{3} \sqrt [3]{2} \log \left (\sqrt [3]{2} \sqrt [3]{x^3-1}-x\right )+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3-1}+x}\right )}{\sqrt {3}}-\frac {2 \sqrt [3]{2} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{2} \sqrt [3]{x^3-1}+x}\right )}{\sqrt {3}}+\frac {\left (x^3-1\right )^{2/3} \left (7 x^3+8\right )}{10 x^5}+\frac {1}{6} \log \left (\sqrt [3]{x^3-1} x+\left (x^3-1\right )^{2/3}+x^2\right )-\frac {1}{3} \sqrt [3]{2} \log \left (\sqrt [3]{2} \sqrt [3]{x^3-1} x+2^{2/3} \left (x^3-1\right )^{2/3}+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 26.72, size = 360, normalized size = 1.68 \begin {gather*} \frac {20 \, \sqrt {3} 2^{\frac {1}{3}} x^{5} \arctan \left (\frac {12 \, \sqrt {3} 2^{\frac {2}{3}} {\left (2 \, x^{7} - 5 \, x^{4} + 2 \, x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 6 \, \sqrt {3} 2^{\frac {1}{3}} {\left (19 \, x^{8} - 22 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}} + \sqrt {3} {\left (91 \, x^{9} - 168 \, x^{6} + 84 \, x^{3} - 8\right )}}{3 \, {\left (53 \, x^{9} - 48 \, x^{6} - 12 \, x^{3} + 8\right )}}\right ) + 30 \, \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 ) + 20 \cdot 2^{\frac {1}{3}} x^{5} \log \left (-\frac {3 \cdot 2^{\frac {2}{3}} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} - 6 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x + 2^{\frac {1}{3}} {\left (x^{3} - 2\right )}}{x^{3} - 2}\right ) - 10 \cdot 2^{\frac {1}{3}} x^{5} \log \left (\frac {12 \cdot 2^{\frac {1}{3}} {\left (2 \, x^{4} - x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 2^{\frac {2}{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 ) - 15 \, 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 ) + 9 \, {\left (7 \, x^{3} + 8\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{90 \, 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 \, x^{3} + 8\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (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 [F] time = 0.98, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x^{3}-1\right )^{\frac {2}{3}} \left (x^{6}+2 x^{3}+8\right )}{x^{6} \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} + 2 \, x^{3} + 8\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (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 = -0.00 \begin {gather*} \int \frac {{\left (x^3-1\right )}^{2/3}\,\left (x^6+2\,x^3+8\right )}{x^6\,\left (x^3-2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (x^{6} + 2 x^{3} + 8\right )}{x^{6} \left (x^{3} - 2\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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