Optimal. Leaf size=151 \[ -\frac {5}{3} 2^{2/3} \log \left (2^{2/3} \sqrt [3]{x^3+1}-2 x\right )+\frac {5\ 2^{2/3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^3+1}+x}\right )}{\sqrt {3}}+\frac {5 \log \left (2^{2/3} \sqrt [3]{x^3+1} x+\sqrt [3]{2} \left (x^3+1\right )^{2/3}+2 x^2\right )}{3 \sqrt [3]{2}}+\frac {\left (x^3+1\right )^{2/3} \left (-17 x^6+10 x^3+2\right )}{5 x^5 \left (x^3-1\right )} \]
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Rubi [F] time = 1.86, 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 (2+x^6\right )}{x^6 \left (-1+x^3\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^6\right )}{x^6 \left (-1+x^3\right )^2} \, dx &=\int \left (\frac {\left (1+x^3\right )^{2/3}}{3 (-1+x)^2}-\frac {2 \left (1+x^3\right )^{2/3}}{-1+x}+\frac {2 \left (1+x^3\right )^{2/3}}{x^6}+\frac {4 \left (1+x^3\right )^{2/3}}{x^3}+\frac {(1+x) \left (1+x^3\right )^{2/3}}{\left (1+x+x^2\right )^2}+\frac {(11+6 x) \left (1+x^3\right )^{2/3}}{3 \left (1+x+x^2\right )}\right ) \, dx\\ &=\frac {1}{3} \int \frac {\left (1+x^3\right )^{2/3}}{(-1+x)^2} \, dx+\frac {1}{3} \int \frac {(11+6 x) \left (1+x^3\right )^{2/3}}{1+x+x^2} \, dx-2 \int \frac {\left (1+x^3\right )^{2/3}}{-1+x} \, dx+2 \int \frac {\left (1+x^3\right )^{2/3}}{x^6} \, dx+4 \int \frac {\left (1+x^3\right )^{2/3}}{x^3} \, dx+\int \frac {(1+x) \left (1+x^3\right )^{2/3}}{\left (1+x+x^2\right )^2} \, dx\\ &=-\frac {2 \left (1+x^3\right )^{2/3}}{x^2}-\frac {2 \left (1+x^3\right )^{5/3}}{5 x^5}+\frac {1}{3} \int \frac {\left (1+x^3\right )^{2/3}}{(-1+x)^2} \, dx+\frac {1}{3} \int \left (\frac {\left (6-\frac {16 i}{\sqrt {3}}\right ) \left (1+x^3\right )^{2/3}}{1-i \sqrt {3}+2 x}+\frac {\left (6+\frac {16 i}{\sqrt {3}}\right ) \left (1+x^3\right )^{2/3}}{1+i \sqrt {3}+2 x}\right ) \, dx-2 \int \frac {\left (1+x^3\right )^{2/3}}{-1+x} \, dx+4 \int \frac {1}{\sqrt [3]{1+x^3}} \, dx+\int \left (\frac {\left (1+x^3\right )^{2/3}}{\left (1+x+x^2\right )^2}+\frac {x \left (1+x^3\right )^{2/3}}{\left (1+x+x^2\right )^2}\right ) \, dx\\ &=-\frac {2 \left (1+x^3\right )^{2/3}}{x^2}-\frac {2 \left (1+x^3\right )^{5/3}}{5 x^5}+\frac {4 \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-2 \log \left (-x+\sqrt [3]{1+x^3}\right )+\frac {1}{3} \int \frac {\left (1+x^3\right )^{2/3}}{(-1+x)^2} \, dx-2 \int \frac {\left (1+x^3\right )^{2/3}}{-1+x} \, dx+\frac {1}{9} \left (2 \left (9-8 i \sqrt {3}\right )\right ) \int \frac {\left (1+x^3\right )^{2/3}}{1-i \sqrt {3}+2 x} \, dx+\frac {1}{9} \left (2 \left (9+8 i \sqrt {3}\right )\right ) \int \frac {\left (1+x^3\right )^{2/3}}{1+i \sqrt {3}+2 x} \, dx+\int \frac {\left (1+x^3\right )^{2/3}}{\left (1+x+x^2\right )^2} \, dx+\int \frac {x \left (1+x^3\right )^{2/3}}{\left (1+x+x^2\right )^2} \, dx\\ &=-\frac {2 \left (1+x^3\right )^{2/3}}{x^2}-\frac {2 \left (1+x^3\right )^{5/3}}{5 x^5}+\frac {4 \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-2 \log \left (-x+\sqrt [3]{1+x^3}\right )+\frac {1}{3} \int \frac {\left (1+x^3\right )^{2/3}}{(-1+x)^2} \, dx-2 \int \frac {\left (1+x^3\right )^{2/3}}{-1+x} \, dx+\frac {1}{9} \left (2 \left (9-8 i \sqrt {3}\right )\right ) \int \frac {\left (1+x^3\right )^{2/3}}{1-i \sqrt {3}+2 x} \, dx+\frac {1}{9} \left (2 \left (9+8 i \sqrt {3}\right )\right ) \int \frac {\left (1+x^3\right )^{2/3}}{1+i \sqrt {3}+2 x} \, dx+\int \left (-\frac {2 \left (-1+i \sqrt {3}\right ) \left (1+x^3\right )^{2/3}}{3 \left (-1+i \sqrt {3}-2 x\right )^2}-\frac {2 i \left (1+x^3\right )^{2/3}}{3 \sqrt {3} \left (-1+i \sqrt {3}-2 x\right )}-\frac {2 \left (-1-i \sqrt {3}\right ) \left (1+x^3\right )^{2/3}}{3 \left (1+i \sqrt {3}+2 x\right )^2}-\frac {2 i \left (1+x^3\right )^{2/3}}{3 \sqrt {3} \left (1+i \sqrt {3}+2 x\right )}\right ) \, dx+\int \left (-\frac {4 \left (1+x^3\right )^{2/3}}{3 \left (-1+i \sqrt {3}-2 x\right )^2}+\frac {4 i \left (1+x^3\right )^{2/3}}{3 \sqrt {3} \left (-1+i \sqrt {3}-2 x\right )}-\frac {4 \left (1+x^3\right )^{2/3}}{3 \left (1+i \sqrt {3}+2 x\right )^2}+\frac {4 i \left (1+x^3\right )^{2/3}}{3 \sqrt {3} \left (1+i \sqrt {3}+2 x\right )}\right ) \, dx\\ &=-\frac {2 \left (1+x^3\right )^{2/3}}{x^2}-\frac {2 \left (1+x^3\right )^{5/3}}{5 x^5}+\frac {4 \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-2 \log \left (-x+\sqrt [3]{1+x^3}\right )+\frac {1}{3} \int \frac {\left (1+x^3\right )^{2/3}}{(-1+x)^2} \, dx-\frac {4}{3} \int \frac {\left (1+x^3\right )^{2/3}}{\left (-1+i \sqrt {3}-2 x\right )^2} \, dx-\frac {4}{3} \int \frac {\left (1+x^3\right )^{2/3}}{\left (1+i \sqrt {3}+2 x\right )^2} \, dx-2 \int \frac {\left (1+x^3\right )^{2/3}}{-1+x} \, dx-\frac {(2 i) \int \frac {\left (1+x^3\right )^{2/3}}{-1+i \sqrt {3}-2 x} \, dx}{3 \sqrt {3}}-\frac {(2 i) \int \frac {\left (1+x^3\right )^{2/3}}{1+i \sqrt {3}+2 x} \, dx}{3 \sqrt {3}}+\frac {(4 i) \int \frac {\left (1+x^3\right )^{2/3}}{-1+i \sqrt {3}-2 x} \, dx}{3 \sqrt {3}}+\frac {(4 i) \int \frac {\left (1+x^3\right )^{2/3}}{1+i \sqrt {3}+2 x} \, dx}{3 \sqrt {3}}+\frac {1}{3} \left (2 \left (1-i \sqrt {3}\right )\right ) \int \frac {\left (1+x^3\right )^{2/3}}{\left (-1+i \sqrt {3}-2 x\right )^2} \, dx+\frac {1}{3} \left (2 \left (1+i \sqrt {3}\right )\right ) \int \frac {\left (1+x^3\right )^{2/3}}{\left (1+i \sqrt {3}+2 x\right )^2} \, dx+\frac {1}{9} \left (2 \left (9-8 i \sqrt {3}\right )\right ) \int \frac {\left (1+x^3\right )^{2/3}}{1-i \sqrt {3}+2 x} \, dx+\frac {1}{9} \left (2 \left (9+8 i \sqrt {3}\right )\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 = 0.50, size = 138, normalized size = 0.91 \begin {gather*} \frac {5 \left (-2 \log \left (1-\frac {\sqrt [3]{2} x}{\sqrt [3]{x^3+1}}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{2} x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )+\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 )}{3 \sqrt [3]{2}}+\left (x^3+1\right )^{2/3} \left (-\frac {2}{5 x^5}-\frac {x}{x^3-1}-\frac {12}{5 x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.46, size = 151, normalized size = 1.00 \begin {gather*} -\frac {5}{3} 2^{2/3} \log \left (2^{2/3} \sqrt [3]{x^3+1}-2 x\right )+\frac {5\ 2^{2/3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^3+1}+x}\right )}{\sqrt {3}}+\frac {5 \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 )}{3 \sqrt [3]{2}}+\frac {\left (x^3+1\right )^{2/3} \left (-17 x^6+10 x^3+2\right )}{5 x^5 \left (x^3-1\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 2.83, size = 297, normalized size = 1.97 \begin {gather*} -\frac {50 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} {\left (x^{8} - x^{5}\right )} \arctan \left (\frac {3 \, \sqrt {3} \left (-4\right )^{\frac {2}{3}} {\left (5 \, x^{7} - 4 \, x^{4} - x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} + 6 \, \sqrt {3} \left (-4\right )^{\frac {1}{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 ) - 50 \, \left (-4\right )^{\frac {1}{3}} {\left (x^{8} - x^{5}\right )} \log \left (\frac {3 \, \left (-4\right )^{\frac {2}{3}} {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 6 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} x + \left (-4\right )^{\frac {1}{3}} {\left (x^{3} - 1\right )}}{x^{3} - 1}\right ) + 25 \, \left (-4\right )^{\frac {1}{3}} {\left (x^{8} - x^{5}\right )} \log \left (-\frac {6 \, \left (-4\right )^{\frac {1}{3}} {\left (5 \, x^{4} + x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} - \left (-4\right )^{\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 ) + 18 \, {\left (17 \, x^{6} - 10 \, x^{3} - 2\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{90 \, {\left (x^{8} - x^{5}\right )}} \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\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (x^{3} - 1\right )}^{2} x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.38, size = 939, normalized size = 6.22
result too large to display
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\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (x^{3} - 1\right )}^{2} 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+2\right )}{x^6\,{\left (x^3-1\right )}^2} \,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\right )}{x^{6} \left (x - 1\right )^{2} \left (x^{2} + x + 1\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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