Optimal. Leaf size=221 \[ -\frac {10}{9} \log \left (\sqrt [3]{x^3+1}+x\right )+\frac {1}{9} 2^{2/3} \log \left (2^{2/3} \sqrt [3]{x^3+1}-2 x\right )-\frac {10 \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3+1}-x}\right )}{3 \sqrt {3}}-\frac {2^{2/3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^3+1}+x}\right )}{3 \sqrt {3}}+\frac {\left (x^3+1\right )^{2/3} \left (2-13 x^3\right )}{10 x^5}+\frac {5}{9} \log \left (-\sqrt [3]{x^3+1} x+\left (x^3+1\right )^{2/3}+x^2\right )-\frac {\log \left (2^{2/3} \sqrt [3]{x^3+1} x+\sqrt [3]{2} \left (x^3+1\right )^{2/3}+2 x^2\right )}{9 \sqrt [3]{2}} \]
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Rubi [F] time = 0.78, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (1+x^3\right )^{2/3} \left (1-2 x^3+2 x^6\right )}{x^6 \left (-1-x^3+2 x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {align*} \int \frac {\left (1+x^3\right )^{2/3} \left (1-2 x^3+2 x^6\right )}{x^6 \left (-1-x^3+2 x^6\right )} \, dx &=\int \left (\frac {\left (1+x^3\right )^{2/3}}{9 (-1+x)}-\frac {\left (1+x^3\right )^{2/3}}{x^6}+\frac {3 \left (1+x^3\right )^{2/3}}{x^3}+\frac {(-2-x) \left (1+x^3\right )^{2/3}}{9 \left (1+x+x^2\right )}-\frac {20 \left (1+x^3\right )^{2/3}}{3 \left (1+2 x^3\right )}\right ) \, dx\\ &=\frac {1}{9} \int \frac {\left (1+x^3\right )^{2/3}}{-1+x} \, dx+\frac {1}{9} \int \frac {(-2-x) \left (1+x^3\right )^{2/3}}{1+x+x^2} \, dx+3 \int \frac {\left (1+x^3\right )^{2/3}}{x^3} \, dx-\frac {20}{3} \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 {3 \left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\left (1+x^3\right )^{5/3}}{5 x^5}-\frac {20}{3} x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,-2 x^3\right )+\frac {1}{9} \int \frac {\left (1+x^3\right )^{2/3}}{-1+x} \, dx+\frac {1}{9} \int \left (\frac {\left (-1+i \sqrt {3}\right ) \left (1+x^3\right )^{2/3}}{1-i \sqrt {3}+2 x}+\frac {\left (-1-i \sqrt {3}\right ) \left (1+x^3\right )^{2/3}}{1+i \sqrt {3}+2 x}\right ) \, dx+3 \int \frac {1}{\sqrt [3]{1+x^3}} \, dx\\ &=-\frac {3 \left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\left (1+x^3\right )^{5/3}}{5 x^5}-\frac {20}{3} x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,-2 x^3\right )+\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 )+\frac {1}{9} \int \frac {\left (1+x^3\right )^{2/3}}{-1+x} \, dx+\frac {1}{9} \left (-1-i \sqrt {3}\right ) \int \frac {\left (1+x^3\right )^{2/3}}{1+i \sqrt {3}+2 x} \, dx+\frac {1}{9} \left (-1+i \sqrt {3}\right ) \int \frac {\left (1+x^3\right )^{2/3}}{1-i \sqrt {3}+2 x} \, dx\\ \end {align*}
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Mathematica [A] time = 0.52, size = 214, normalized size = 0.97 \begin {gather*} \frac {1}{18} \left (-20 \log \left (\frac {x}{\sqrt [3]{x^3+1}}+1\right )+2\ 2^{2/3} \log \left (1-\frac {\sqrt [3]{2} x}{\sqrt [3]{x^3+1}}\right )+20 \sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{x^3+1}}}{\sqrt {3}}\right )-2\ 2^{2/3} \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{2} x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )+10 \log \left (-\frac {x}{\sqrt [3]{x^3+1}}+\frac {x^2}{\left (x^3+1\right )^{2/3}}+1\right )-2^{2/3} \log \left (\frac {\sqrt [3]{2} x}{\sqrt [3]{x^3+1}}+\frac {2^{2/3} x^2}{\left (x^3+1\right )^{2/3}}+1\right )\right )+\left (x^3+1\right )^{2/3} \left (\frac {1}{5 x^5}-\frac {13}{10 x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.54, size = 221, normalized size = 1.00 \begin {gather*} -\frac {10}{9} \log \left (\sqrt [3]{x^3+1}+x\right )+\frac {1}{9} 2^{2/3} \log \left (2^{2/3} \sqrt [3]{x^3+1}-2 x\right )-\frac {10 \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3+1}-x}\right )}{3 \sqrt {3}}-\frac {2^{2/3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^3+1}+x}\right )}{3 \sqrt {3}}+\frac {\left (x^3+1\right )^{2/3} \left (2-13 x^3\right )}{10 x^5}+\frac {5}{9} \log \left (-\sqrt [3]{x^3+1} x+\left (x^3+1\right )^{2/3}+x^2\right )-\frac {\log \left (2^{2/3} \sqrt [3]{x^3+1} x+\sqrt [3]{2} \left (x^3+1\right )^{2/3}+2 x^2\right )}{9 \sqrt [3]{2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 4.98, size = 367, normalized size = 1.66 \begin {gather*} -\frac {10 \cdot 4^{\frac {1}{3}} \sqrt {3} x^{5} \arctan \left (\frac {3 \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (5 \, x^{7} - 4 \, x^{4} - x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} - 6 \cdot 4^{\frac {1}{3}} \sqrt {3} {\left (19 \, x^{8} + 16 \, x^{5} + x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}} - \sqrt {3} {\left (71 \, x^{9} + 111 \, x^{6} + 33 \, x^{3} + 1\right )}}{3 \, {\left (109 \, x^{9} + 105 \, x^{6} + 3 \, x^{3} - 1\right )}}\right ) - 300 \, \sqrt {3} x^{5} \arctan \left (\frac {4 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 2 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (x^{3} + 1\right )}}{7 \, x^{3} - 1}\right ) - 10 \cdot 4^{\frac {1}{3}} x^{5} \log \left (\frac {3 \cdot 4^{\frac {2}{3}} {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 6 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} x - 4^{\frac {1}{3}} {\left (x^{3} - 1\right )}}{x^{3} - 1}\right ) + 5 \cdot 4^{\frac {1}{3}} x^{5} \log \left (\frac {6 \cdot 4^{\frac {1}{3}} {\left (5 \, x^{4} + x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} + 4^{\frac {2}{3}} {\left (19 \, x^{6} + 16 \, x^{3} + 1\right )} + 24 \, {\left (2 \, x^{5} + x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x^{6} - 2 \, x^{3} + 1}\right ) + 150 \, x^{5} \log \left (\frac {2 \, x^{3} + 3 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} x + 1}{2 \, x^{3} + 1}\right ) + 27 \, {\left (13 \, x^{3} - 2\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{270 \, 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} - 2 \, x^{3} + 1\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{6} - 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 [C] time = 5.62, size = 1246, normalized size = 5.64 \begin {gather*} \text {Expression too large to display} \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 (2 \, x^{6} - 2 \, x^{3} + 1\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{6} - 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^6-2\,x^3+1\right )}{x^6\,\left (-2\,x^6+x^3+1\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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