Optimal. Leaf size=104 \[ \frac {1}{3} \log \left (\sqrt [3]{x^3+1}+x\right )+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3+1}-x}\right )}{\sqrt {3}}-\frac {1}{6} \log \left (-\sqrt [3]{x^3+1} x+\left (x^3+1\right )^{2/3}+x^2\right )+\frac {\left (x^3+1\right )^{2/3} \left (x^6-14 x^3+5\right )}{40 x^8} \]
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Rubi [C] time = 0.40, antiderivative size = 119, normalized size of antiderivative = 1.14, number of steps used = 8, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {6725, 271, 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 )+\frac {1}{2} \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 )}{\sqrt {3}}+\frac {\left (x^3+1\right )^{5/3}}{8 x^8}-\frac {19 \left (x^3+1\right )^{5/3}}{40 x^5}+\frac {\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 271
Rule 277
Rule 429
Rule 6725
Rubi steps
\begin {align*} \int \frac {\left (1+x^3\right )^{2/3} \left (-1+3 x^6\right )}{x^9 \left (1+2 x^3\right )} \, dx &=\int \left (-\frac {\left (1+x^3\right )^{2/3}}{x^9}+\frac {2 \left (1+x^3\right )^{2/3}}{x^6}-\frac {\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^6} \, 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^9} \, dx-\int \frac {\left (1+x^3\right )^{2/3}}{x^3} \, dx\\ &=\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\left (1+x^3\right )^{5/3}}{8 x^8}-\frac {2 \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 {3}{8} \int \frac {\left (1+x^3\right )^{2/3}}{x^6} \, dx-\int \frac {1}{\sqrt [3]{1+x^3}} \, dx\\ &=\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\left (1+x^3\right )^{5/3}}{8 x^8}-\frac {19 \left (1+x^3\right )^{5/3}}{40 x^5}+2 x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,-2 x^3\right )-\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} \log \left (-x+\sqrt [3]{1+x^3}\right )\\ \end {align*}
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Mathematica [A] time = 0.12, size = 104, normalized size = 1.00 \begin {gather*} \frac {1}{3} \log \left (\frac {x}{\sqrt [3]{x^3+1}}+1\right )+\frac {\tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}-1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{6} \log \left (-\frac {x}{\sqrt [3]{x^3+1}}+\frac {x^2}{\left (x^3+1\right )^{2/3}}+1\right )+\frac {\left (x^3+1\right )^{2/3} \left (x^6-14 x^3+5\right )}{40 x^8} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.20, size = 104, normalized size = 1.00 \begin {gather*} \frac {\left (1+x^3\right )^{2/3} \left (5-14 x^3+x^6\right )}{40 x^8}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{1+x^3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (x+\sqrt [3]{1+x^3}\right )-\frac {1}{6} \log \left (x^2-x \sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.01, size = 127, normalized size = 1.22 \begin {gather*} -\frac {40 \, \sqrt {3} x^{8} \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 ) - 20 \, x^{8} \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 ) - 3 \, {\left (x^{6} - 14 \, x^{3} + 5\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{120 \, x^{8}} \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 (3 \, x^{6} - 1\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{3} + 1\right )} x^{9}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 3.96, size = 425, normalized size = 4.09
method | result | size |
risch | \(\frac {x^{9}-13 x^{6}-9 x^{3}+5}{40 x^{8} \left (x^{3}+1\right )^{\frac {1}{3}}}-\frac {\ln \left (-\frac {9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}-3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x +3 \left (x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+6 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-2 x \left (x^{3}+1\right )^{\frac {2}{3}}+2 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}+x^{3}+3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+1}{2 x^{3}+1}\right )}{3}-\ln \left (-\frac {9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}-3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x +3 \left (x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+6 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-2 x \left (x^{3}+1\right )^{\frac {2}{3}}+2 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}+x^{3}+3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+1}{2 x^{3}+1}\right ) \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+\RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (-\frac {9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x -3 \left (x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}-x \left (x^{3}+1\right )^{\frac {2}{3}}+x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}-3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )}{2 x^{3}+1}\right )\) | \(425\) |
trager | \(\frac {\left (x^{3}+1\right )^{\frac {2}{3}} \left (x^{6}-14 x^{3}+5\right )}{40 x^{8}}+\RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (\frac {63 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+18 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x -18 \left (x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}-15 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-15 x \left (x^{3}+1\right )^{\frac {2}{3}}+15 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}-2 x^{3}-63 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}-48 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-4}{2 x^{3}+1}\right )-\frac {\ln \left (\frac {63 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}-18 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x +18 \left (x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+57 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-21 x \left (x^{3}+1\right )^{\frac {2}{3}}+21 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}+10 x^{3}-63 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+6 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+5}{2 x^{3}+1}\right )}{3}-\ln \left (\frac {63 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}-18 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x +18 \left (x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+57 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-21 x \left (x^{3}+1\right )^{\frac {2}{3}}+21 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}+10 x^{3}-63 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+6 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+5}{2 x^{3}+1}\right ) \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )\) | \(489\) |
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 (3 \, x^{6} - 1\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{3} + 1\right )} x^{9}}\,{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 (3\,x^6-1\right )}{x^9\,\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 (3 x^{6} - 1\right )}{x^{9} \left (2 x^{3} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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