Optimal. Leaf size=97 \[ \log \left (\sqrt [3]{x^3+1}+x\right )+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3+1}-x}\right )+\frac {\left (x^3+1\right )^{2/3} \left (11 x^3-4\right )}{10 x^5}-\frac {1}{2} \log \left (-\sqrt [3]{x^3+1} x+\left (x^3+1\right )^{2/3}+x^2\right ) \]
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Rubi [A] time = 0.14, antiderivative size = 107, normalized size of antiderivative = 1.10, number of steps used = 10, number of rules used = 10, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {580, 583, 12, 377, 200, 31, 634, 618, 204, 628} \begin {gather*} \log \left (\frac {x}{\sqrt [3]{x^3+1}}+1\right )-\sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{x^3+1}}}{\sqrt {3}}\right )-\frac {2 \left (x^3+1\right )^{2/3}}{5 x^5}+\frac {11 \left (x^3+1\right )^{2/3}}{10 x^2}-\frac {1}{2} \log \left (-\frac {x}{\sqrt [3]{x^3+1}}+\frac {x^2}{\left (x^3+1\right )^{2/3}}+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 31
Rule 200
Rule 204
Rule 377
Rule 580
Rule 583
Rule 618
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (1+2 x^3\right )} \, dx &=-\frac {2 \left (1+x^3\right )^{2/3}}{5 x^5}+\frac {1}{5} \int \frac {-11-7 x^3}{x^3 \sqrt [3]{1+x^3} \left (1+2 x^3\right )} \, dx\\ &=-\frac {2 \left (1+x^3\right )^{2/3}}{5 x^5}+\frac {11 \left (1+x^3\right )^{2/3}}{10 x^2}-\frac {1}{10} \int -\frac {30}{\sqrt [3]{1+x^3} \left (1+2 x^3\right )} \, dx\\ &=-\frac {2 \left (1+x^3\right )^{2/3}}{5 x^5}+\frac {11 \left (1+x^3\right )^{2/3}}{10 x^2}+3 \int \frac {1}{\sqrt [3]{1+x^3} \left (1+2 x^3\right )} \, dx\\ &=-\frac {2 \left (1+x^3\right )^{2/3}}{5 x^5}+\frac {11 \left (1+x^3\right )^{2/3}}{10 x^2}+3 \operatorname {Subst}\left (\int \frac {1}{1+x^3} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )\\ &=-\frac {2 \left (1+x^3\right )^{2/3}}{5 x^5}+\frac {11 \left (1+x^3\right )^{2/3}}{10 x^2}+\operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )+\operatorname {Subst}\left (\int \frac {2-x}{1-x+x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )\\ &=-\frac {2 \left (1+x^3\right )^{2/3}}{5 x^5}+\frac {11 \left (1+x^3\right )^{2/3}}{10 x^2}+\log \left (1+\frac {x}{\sqrt [3]{1+x^3}}\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )+\frac {3}{2} \operatorname {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )\\ &=-\frac {2 \left (1+x^3\right )^{2/3}}{5 x^5}+\frac {11 \left (1+x^3\right )^{2/3}}{10 x^2}-\frac {1}{2} \log \left (1+\frac {x^2}{\left (1+x^3\right )^{2/3}}-\frac {x}{\sqrt [3]{1+x^3}}\right )+\log \left (1+\frac {x}{\sqrt [3]{1+x^3}}\right )-3 \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+\frac {2 x}{\sqrt [3]{1+x^3}}\right )\\ &=-\frac {2 \left (1+x^3\right )^{2/3}}{5 x^5}+\frac {11 \left (1+x^3\right )^{2/3}}{10 x^2}-\sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )-\frac {1}{2} \log \left (1+\frac {x^2}{\left (1+x^3\right )^{2/3}}-\frac {x}{\sqrt [3]{1+x^3}}\right )+\log \left (1+\frac {x}{\sqrt [3]{1+x^3}}\right )\\ \end {align*}
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Mathematica [A] time = 0.11, size = 106, normalized size = 1.09 \begin {gather*} 3 \left (\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 )\right )+\left (x^3+1\right )^{2/3} \left (\frac {11}{10 x^2}-\frac {2}{5 x^5}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.19, size = 97, normalized size = 1.00 \begin {gather*} \frac {\left (1+x^3\right )^{2/3} \left (-4+11 x^3\right )}{10 x^5}+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{1+x^3}}\right )+\log \left (x+\sqrt [3]{1+x^3}\right )-\frac {1}{2} \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 = 0.88, size = 124, normalized size = 1.28 \begin {gather*} -\frac {10 \, \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 ) - 5 \, 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 ) - {\left (11 \, x^{3} - 4\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{10 \, 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^{3} + 2\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 [C] time = 4.27, size = 258, normalized size = 2.66
method | result | size |
risch | \(\frac {11 x^{6}+7 x^{3}-4}{10 x^{5} \left (x^{3}+1\right )^{\frac {1}{3}}}+\ln \left (-\frac {3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x +6 \left (x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-x \left (x^{3}+1\right )^{\frac {2}{3}}+x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}-1}{2 x^{3}+1}\right )+3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (\frac {9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x -3 \left (x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+2 x \left (x^{3}+1\right )^{\frac {2}{3}}+x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}+3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )}{2 x^{3}+1}\right )\) | \(258\) |
trager | \(\frac {\left (x^{3}+1\right )^{\frac {2}{3}} \left (11 x^{3}-4\right )}{10 x^{5}}+\ln \left (\frac {-18 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+18 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x +63 \left (x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+21 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-15 x \left (x^{3}+1\right )^{\frac {2}{3}}+6 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}-5 x^{3}+18 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}-3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-10}{2 x^{3}+1}\right )+3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (-\frac {45 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+18 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x -45 \left (x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+24 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+21 x \left (x^{3}+1\right )^{\frac {2}{3}}+6 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}-4 x^{3}-45 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+21 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-2}{2 x^{3}+1}\right )\) | \(327\) |
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^{3} + 2\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 = -0.01 \begin {gather*} \int \frac {{\left (x^3+1\right )}^{2/3}\,\left (x^3+2\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 (x^{3} + 2\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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