Optimal. Leaf size=85 \[ \frac {1}{3} \log \left (\sqrt [3]{2 x^3-1}-x\right )+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{2 x^3-1}+x}\right )}{\sqrt {3}}-\frac {1}{6} \log \left (\sqrt [3]{2 x^3-1} x+\left (2 x^3-1\right )^{2/3}+x^2\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 86, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {494, 292, 31, 634, 618, 204, 628} \begin {gather*} \frac {1}{3} \log \left (1-\frac {x}{\sqrt [3]{2 x^3-1}}\right )+\frac {\tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{2 x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{6} \log \left (\frac {x}{\sqrt [3]{2 x^3-1}}+\frac {x^2}{\left (2 x^3-1\right )^{2/3}}+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 292
Rule 494
Rule 618
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {x}{\left (-1+x^3\right ) \left (-1+2 x^3\right )^{2/3}} \, dx &=\operatorname {Subst}\left (\int \frac {x}{-1+x^3} \, dx,x,\frac {x}{\sqrt [3]{-1+2 x^3}}\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{-1+x} \, dx,x,\frac {x}{\sqrt [3]{-1+2 x^3}}\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {-1+x}{1+x+x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+2 x^3}}\right )\\ &=\frac {1}{3} \log \left (1-\frac {x}{\sqrt [3]{-1+2 x^3}}\right )-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+2 x^3}}\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+2 x^3}}\right )\\ &=\frac {1}{3} \log \left (1-\frac {x}{\sqrt [3]{-1+2 x^3}}\right )-\frac {1}{6} \log \left (1+\frac {x^2}{\left (-1+2 x^3\right )^{2/3}}+\frac {x}{\sqrt [3]{-1+2 x^3}}\right )-\operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 x}{\sqrt [3]{-1+2 x^3}}\right )\\ &=\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+2 x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (1-\frac {x}{\sqrt [3]{-1+2 x^3}}\right )-\frac {1}{6} \log \left (1+\frac {x^2}{\left (-1+2 x^3\right )^{2/3}}+\frac {x}{\sqrt [3]{-1+2 x^3}}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 40, normalized size = 0.47 \begin {gather*} -\frac {x^2 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};-\frac {x^3}{1-2 x^3}\right )}{2 \left (2 x^3-1\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.19, size = 85, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+2 x^3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (-x+\sqrt [3]{-1+2 x^3}\right )-\frac {1}{6} \log \left (x^2+x \sqrt [3]{-1+2 x^3}+\left (-1+2 x^3\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 104, normalized size = 1.22 \begin {gather*} \frac {1}{3} \, \sqrt {3} \arctan \left (-\frac {4 \, \sqrt {3} {\left (2 \, x^{3} - 1\right )}^{\frac {1}{3}} x^{2} - 2 \, \sqrt {3} {\left (2 \, x^{3} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (2 \, x^{3} - 1\right )}}{10 \, x^{3} - 1}\right ) + \frac {1}{6} \, \log \left (\frac {x^{3} + 3 \, {\left (2 \, x^{3} - 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (2 \, x^{3} - 1\right )}^{\frac {2}{3}} x - 1}{x^{3} - 1}\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 {x}{{\left (2 \, x^{3} - 1\right )}^{\frac {2}{3}} {\left (x^{3} - 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.88, size = 447, normalized size = 5.26
method | result | size |
trager | \(\frac {\ln \left (-\frac {-27 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+15 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (2 x^{3}-1\right )^{\frac {2}{3}} x -3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (2 x^{3}-1\right )^{\frac {1}{3}} x^{2}-30 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+4 \left (2 x^{3}-1\right )^{\frac {2}{3}} x -5 \left (2 x^{3}-1\right )^{\frac {1}{3}} x^{2}-3 x^{3}+9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+1}{\left (-1+x \right ) \left (x^{2}+x +1\right )}\right )}{3}-\frac {\ln \left (\frac {-9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+15 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (2 x^{3}-1\right )^{\frac {2}{3}} x -12 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (2 x^{3}-1\right )^{\frac {1}{3}} x^{2}-3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+\left (2 x^{3}-1\right )^{\frac {2}{3}} x -5 \left (2 x^{3}-1\right )^{\frac {1}{3}} x^{2}+6 x^{3}-3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-3}{\left (-1+x \right ) \left (x^{2}+x +1\right )}\right )}{3}-\ln \left (\frac {-9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+15 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (2 x^{3}-1\right )^{\frac {2}{3}} x -12 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (2 x^{3}-1\right )^{\frac {1}{3}} x^{2}-3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+\left (2 x^{3}-1\right )^{\frac {2}{3}} x -5 \left (2 x^{3}-1\right )^{\frac {1}{3}} x^{2}+6 x^{3}-3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-3}{\left (-1+x \right ) \left (x^{2}+x +1\right )}\right ) \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )\) | \(447\) |
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 {x}{{\left (2 \, x^{3} - 1\right )}^{\frac {2}{3}} {\left (x^{3} - 1\right )}}\,{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 {x}{\left (x^3-1\right )\,{\left (2\,x^3-1\right )}^{2/3}} \,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 {x}{\left (x - 1\right ) \left (2 x^{3} - 1\right )^{\frac {2}{3}} \left (x^{2} + x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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