Optimal. Leaf size=209 \[ -\frac {1}{3} \log \left (\sqrt [3]{x^3+1}-x\right )-\frac {\log \left (3^{2/3} \sqrt [3]{x^3+1}-3 x\right )}{\sqrt [3]{3}}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3+1}+x}\right )}{\sqrt {3}}+\sqrt [6]{3} \tan ^{-1}\left (\frac {3^{5/6} x}{2 \sqrt [3]{x^3+1}+\sqrt [3]{3} x}\right )+\frac {\left (x^3+1\right )^{2/3} \left (-6 x^3-1\right )}{5 x^5}+\frac {1}{6} \log \left (\sqrt [3]{x^3+1} x+\left (x^3+1\right )^{2/3}+x^2\right )+\frac {\log \left (3^{2/3} \sqrt [3]{x^3+1} x+\sqrt [3]{3} \left (x^3+1\right )^{2/3}+3 x^2\right )}{2 \sqrt [3]{3}} \]
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Rubi [C] time = 0.39, antiderivative size = 99, normalized size of antiderivative = 0.47, number of steps used = 6, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {6725, 264, 277, 239, 429} \begin {gather*} 2 x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,2 x^3\right )-\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 )}{\sqrt {3}}-\frac {\left (x^3+1\right )^{5/3}}{5 x^5}-\frac {\left (x^3+1\right )^{2/3}}{x^2} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 239
Rule 264
Rule 277
Rule 429
Rule 6725
Rubi steps
\begin {align*} \int \frac {\left (1+x^3\right )^{2/3} \left (-1+2 x^6\right )}{x^6 \left (-1+2 x^3\right )} \, dx &=\int \left (\frac {\left (1+x^3\right )^{2/3}}{x^6}+\frac {2 \left (1+x^3\right )^{2/3}}{x^3}-\frac {2 \left (1+x^3\right )^{2/3}}{-1+2 x^3}\right ) \, dx\\ &=2 \int \frac {\left (1+x^3\right )^{2/3}}{x^3} \, dx-2 \int \frac {\left (1+x^3\right )^{2/3}}{-1+2 x^3} \, dx+\int \frac {\left (1+x^3\right )^{2/3}}{x^6} \, dx\\ &=-\frac {\left (1+x^3\right )^{2/3}}{x^2}-\frac {\left (1+x^3\right )^{5/3}}{5 x^5}+2 x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,2 x^3\right )+2 \int \frac {1}{\sqrt [3]{1+x^3}} \, dx\\ &=-\frac {\left (1+x^3\right )^{2/3}}{x^2}-\frac {\left (1+x^3\right )^{5/3}}{5 x^5}+2 x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,2 x^3\right )+\frac {2 \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\log \left (-x+\sqrt [3]{1+x^3}\right )\\ \end {align*}
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Mathematica [C] time = 0.41, size = 164, normalized size = 0.78 \begin {gather*} \frac {1}{45} \left (-20\ 3^{2/3} \log \left (1-\frac {\sqrt [3]{3} x}{\sqrt [3]{x^3+1}}\right )+60 \sqrt [6]{3} \tan ^{-1}\left (\frac {2 x}{\sqrt [6]{3} \sqrt [3]{x^3+1}}+\frac {1}{\sqrt {3}}\right )-\frac {9 \left (x^3+1\right )^{2/3}}{x^5}-\frac {54 \left (x^3+1\right )^{2/3}}{x^2}+10\ 3^{2/3} \log \left (\frac {\sqrt [3]{3} x}{\sqrt [3]{x^3+1}}+\frac {3^{2/3} x^2}{\left (x^3+1\right )^{2/3}}+1\right )\right )-\frac {1}{2} x^4 F_1\left (\frac {4}{3};\frac {1}{3},1;\frac {7}{3};-x^3,2 x^3\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.41, size = 209, normalized size = 1.00 \begin {gather*} \frac {\left (-1-6 x^3\right ) \left (1+x^3\right )^{2/3}}{5 x^5}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^3}}\right )}{\sqrt {3}}+\sqrt [6]{3} \tan ^{-1}\left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{1+x^3}}\right )-\frac {1}{3} \log \left (-x+\sqrt [3]{1+x^3}\right )-\frac {\log \left (-3 x+3^{2/3} \sqrt [3]{1+x^3}\right )}{\sqrt [3]{3}}+\frac {1}{6} \log \left (x^2+x \sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right )+\frac {\log \left (3 x^2+3^{2/3} x \sqrt [3]{1+x^3}+\sqrt [3]{3} \left (1+x^3\right )^{2/3}\right )}{2 \sqrt [3]{3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 30.55, size = 382, normalized size = 1.83 \begin {gather*} \frac {10 \cdot 3^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{5} \log \left (\frac {9 \cdot 3^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 3^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (2 \, x^{3} - 1\right )} - 9 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} x}{2 \, x^{3} - 1}\right ) - 5 \cdot 3^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {3 \cdot 3^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (7 \, x^{4} + x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} - 3^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (31 \, x^{6} + 23 \, x^{3} + 1\right )} - 9 \, {\left (5 \, x^{5} + 2 \, x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{4 \, x^{6} - 4 \, x^{3} + 1}\right ) - 30 \cdot 3^{\frac {1}{6}} \left (-1\right )^{\frac {1}{3}} x^{5} \arctan \left (\frac {3^{\frac {1}{6}} {\left (6 \cdot 3^{\frac {2}{3}} \left (-1\right )^{\frac {2}{3}} {\left (14 \, x^{7} - 5 \, x^{4} - x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} + 18 \, \left (-1\right )^{\frac {1}{3}} {\left (31 \, x^{8} + 23 \, x^{5} + x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}} - 3^{\frac {1}{3}} {\left (127 \, x^{9} + 201 \, x^{6} + 48 \, x^{3} + 1\right )}\right )}}{3 \, {\left (251 \, x^{9} + 231 \, x^{6} + 6 \, x^{3} - 1\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 ) - 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 ) - 18 \, {\left (6 \, x^{3} + 1\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 (2 \, x^{6} - 1\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{3} - 1\right )} x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{3}+1\right )^{\frac {2}{3}} \left (2 x^{6}-1\right )}{x^{6} \left (2 x^{3}-1\right )}\, dx\]
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 (2 \, x^{6} - 1\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{3} - 1\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 (2\,x^6-1\right )}{x^6\,\left (2\,x^3-1\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 (2 x^{6} - 1\right )}{x^{6} \left (2 x^{3} - 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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