Optimal. Leaf size=101 \[ \frac {1}{3} \log \left (\sqrt [3]{x^3-1}+x\right )+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3-1}-x}\right )}{\sqrt {3}}+\frac {\left (x^3-1\right )^{2/3} \left (7 x^3-2\right )}{10 x^5}-\frac {1}{6} \log \left (-\sqrt [3]{x^3-1} x+\left (x^3-1\right )^{2/3}+x^2\right ) \]
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Rubi [A] time = 0.13, antiderivative size = 111, normalized size of antiderivative = 1.10, number of steps used = 10, number of rules used = 10, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {580, 583, 12, 377, 200, 31, 634, 618, 204, 628} \begin {gather*} \frac {1}{3} \log \left (\frac {x}{\sqrt [3]{x^3-1}}+1\right )-\frac {\tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{x^3-1}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\left (x^3-1\right )^{2/3}}{5 x^5}+\frac {7 \left (x^3-1\right )^{2/3}}{10 x^2}-\frac {1}{6} \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 (-1+3 x^3\right )}{x^6 \left (-1+2 x^3\right )} \, dx &=-\frac {\left (-1+x^3\right )^{2/3}}{5 x^5}-\frac {1}{5} \int \frac {7-9 x^3}{x^3 \sqrt [3]{-1+x^3} \left (-1+2 x^3\right )} \, dx\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{5 x^5}+\frac {7 \left (-1+x^3\right )^{2/3}}{10 x^2}+\frac {1}{10} \int -\frac {10}{\sqrt [3]{-1+x^3} \left (-1+2 x^3\right )} \, dx\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{5 x^5}+\frac {7 \left (-1+x^3\right )^{2/3}}{10 x^2}-\int \frac {1}{\sqrt [3]{-1+x^3} \left (-1+2 x^3\right )} \, dx\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{5 x^5}+\frac {7 \left (-1+x^3\right )^{2/3}}{10 x^2}-\operatorname {Subst}\left (\int \frac {1}{-1-x^3} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{5 x^5}+\frac {7 \left (-1+x^3\right )^{2/3}}{10 x^2}-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{-1-x} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {-2+x}{1-x+x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{5 x^5}+\frac {7 \left (-1+x^3\right )^{2/3}}{10 x^2}+\frac {1}{3} \log \left (1+\frac {x}{\sqrt [3]{-1+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+x^3}}\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{5 x^5}+\frac {7 \left (-1+x^3\right )^{2/3}}{10 x^2}-\frac {1}{6} \log \left (1+\frac {x^2}{\left (-1+x^3\right )^{2/3}}-\frac {x}{\sqrt [3]{-1+x^3}}\right )+\frac {1}{3} \log \left (1+\frac {x}{\sqrt [3]{-1+x^3}}\right )-\operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+\frac {2 x}{\sqrt [3]{-1+x^3}}\right )\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{5 x^5}+\frac {7 \left (-1+x^3\right )^{2/3}}{10 x^2}+\frac {\tan ^{-1}\left (\frac {-1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{6} \log \left (1+\frac {x^2}{\left (-1+x^3\right )^{2/3}}-\frac {x}{\sqrt [3]{-1+x^3}}\right )+\frac {1}{3} \log \left (1+\frac {x}{\sqrt [3]{-1+x^3}}\right )\\ \end {align*}
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Mathematica [A] time = 0.12, size = 111, normalized size = 1.10 \begin {gather*} \frac {1}{3} \log \left (\frac {x}{\sqrt [3]{1-x^3}}+1\right )+\frac {\tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{1-x^3}}-1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{6} \log \left (-\frac {x}{\sqrt [3]{1-x^3}}+\frac {x^2}{\left (1-x^3\right )^{2/3}}+1\right )+\left (x^3-1\right )^{2/3} \left (\frac {7}{10 x^2}-\frac {1}{5 x^5}\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.19, size = 101, normalized size = 1.00 \begin {gather*} \frac {\left (-1+x^3\right )^{2/3} \left (-2+7 x^3\right )}{10 x^5}+\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 = 0.98, size = 124, normalized size = 1.23 \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 ) - 3 \, {\left (7 \, x^{3} - 2\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{30 \, 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 (3 \, x^{3} - 1\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 = 3.93, size = 400, normalized size = 3.96
method | result | size |
risch | \(\frac {7 x^{6}-9 x^{3}+2}{10 x^{5} \left (x^{3}-1\right )^{\frac {1}{3}}}+\RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (-\frac {-9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+3 x \left (x^{3}-1\right )^{\frac {2}{3}}-3 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}-3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+1}{2 x^{3}-1}\right )-\frac {\ln \left (-\frac {-9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}-6 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+2 x \left (x^{3}-1\right )^{\frac {2}{3}}-2 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}-x^{3}+3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+1}{2 x^{3}-1}\right )}{3}-\ln \left (-\frac {-9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}-6 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+2 x \left (x^{3}-1\right )^{\frac {2}{3}}-2 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}-x^{3}+3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+1}{2 x^{3}-1}\right ) \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )\) | \(400\) |
trager | \(\frac {\left (x^{3}-1\right )^{\frac {2}{3}} \left (7 x^{3}-2\right )}{10 x^{5}}+32 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) \ln \left (-\frac {-374952960 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2} x^{3}+10252800 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -10252800 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{2}-34273056 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{3}-421089 x \left (x^{3}-1\right )^{\frac {2}{3}}+421089 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}-252248 x^{3}+2999623680 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2}+51556128 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )+220717}{2 x^{3}-1}\right )-\frac {\ln \left (\frac {374952960 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2} x^{3}+10252800 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -10252800 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{2}-26461536 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{3}+527889 x \left (x^{3}-1\right )^{\frac {2}{3}}-527889 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}-64078 x^{3}-2999623680 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2}-10936032 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )-9154}{2 x^{3}-1}\right )}{3}-32 \ln \left (\frac {374952960 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2} x^{3}+10252800 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -10252800 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{2}-26461536 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right ) x^{3}+527889 x \left (x^{3}-1\right )^{\frac {2}{3}}-527889 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}-64078 x^{3}-2999623680 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )^{2}-10936032 \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )-9154}{2 x^{3}-1}\right ) \RootOf \left (9216 \textit {\_Z}^{2}+96 \textit {\_Z} +1\right )\) | \(488\) |
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^{3} - 1\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 (3\,x^3-1\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 (3 x^{3} - 1\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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