Optimal. Leaf size=118 \[ \frac {7}{9} \log \left (\sqrt [3]{x^3-1}+x\right )+\frac {7 \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3-1}-x}\right )}{3 \sqrt {3}}-\frac {7}{18} \log \left (-\sqrt [3]{x^3-1} x+\left (x^3-1\right )^{2/3}+x^2\right )+\frac {\left (x^3-1\right )^{2/3} \left (62 x^6-33 x^3+6\right )}{30 x^5 \left (2 x^3-1\right )} \]
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Rubi [C] time = 0.48, antiderivative size = 224, normalized size of antiderivative = 1.90, number of steps used = 15, number of rules used = 14, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {6742, 264, 277, 239, 378, 377, 200, 31, 634, 618, 204, 628, 430, 429} \begin {gather*} -\frac {2 x \left (x^3-1\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,2 x^3\right )}{\left (1-x^3\right )^{2/3}}-\frac {2 x \left (x^3-1\right )^{2/3}}{3 \left (1-2 x^3\right )}+\frac {4}{9} \log \left (\frac {x}{\sqrt [3]{x^3-1}}+1\right )+\frac {1}{2} \log \left (\sqrt [3]{x^3-1}-x\right )-\frac {4 \tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{x^3-1}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\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}}{5 x^5}+\frac {\left (x^3-1\right )^{2/3}}{2 x^2}-\frac {2}{9} \log \left (-\frac {x}{\sqrt [3]{x^3-1}}+\frac {x^2}{\left (x^3-1\right )^{2/3}}+1\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 31
Rule 200
Rule 204
Rule 239
Rule 264
Rule 277
Rule 377
Rule 378
Rule 429
Rule 430
Rule 618
Rule 628
Rule 634
Rule 6742
Rubi steps
\begin {align*} \int \frac {\left (-1+x^3\right )^{2/3} \left (1-5 x^3+4 x^6\right )}{x^6 \left (-1+2 x^3\right )^2} \, dx &=\int \left (\frac {\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}}{\left (-1+2 x^3\right )^2}+\frac {2 \left (-1+x^3\right )^{2/3}}{-1+2 x^3}\right ) \, dx\\ &=-\left (2 \int \frac {\left (-1+x^3\right )^{2/3}}{\left (-1+2 x^3\right )^2} \, dx\right )+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^6} \, 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 {2 x \left (-1+x^3\right )^{2/3}}{3 \left (1-2 x^3\right )}+\frac {\left (-1+x^3\right )^{5/3}}{5 x^5}-\frac {4}{3} \int \frac {1}{\sqrt [3]{-1+x^3} \left (-1+2 x^3\right )} \, dx+\frac {\left (2 \left (-1+x^3\right )^{2/3}\right ) \int \frac {\left (1-x^3\right )^{2/3}}{-1+2 x^3} \, dx}{\left (1-x^3\right )^{2/3}}-\int \frac {1}{\sqrt [3]{-1+x^3}} \, dx\\ &=\frac {\left (-1+x^3\right )^{2/3}}{2 x^2}-\frac {2 x \left (-1+x^3\right )^{2/3}}{3 \left (1-2 x^3\right )}+\frac {\left (-1+x^3\right )^{5/3}}{5 x^5}-\frac {2 x \left (-1+x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,2 x^3\right )}{\left (1-x^3\right )^{2/3}}-\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 )-\frac {4}{3} \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}}{2 x^2}-\frac {2 x \left (-1+x^3\right )^{2/3}}{3 \left (1-2 x^3\right )}+\frac {\left (-1+x^3\right )^{5/3}}{5 x^5}-\frac {2 x \left (-1+x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,2 x^3\right )}{\left (1-x^3\right )^{2/3}}-\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 )-\frac {4}{9} \operatorname {Subst}\left (\int \frac {1}{-1-x} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )-\frac {4}{9} \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}}{2 x^2}-\frac {2 x \left (-1+x^3\right )^{2/3}}{3 \left (1-2 x^3\right )}+\frac {\left (-1+x^3\right )^{5/3}}{5 x^5}-\frac {2 x \left (-1+x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,2 x^3\right )}{\left (1-x^3\right )^{2/3}}-\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {4}{9} \log \left (1+\frac {x}{\sqrt [3]{-1+x^3}}\right )+\frac {1}{2} \log \left (-x+\sqrt [3]{-1+x^3}\right )-\frac {2}{9} \operatorname {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )+\frac {2}{3} \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}}{2 x^2}-\frac {2 x \left (-1+x^3\right )^{2/3}}{3 \left (1-2 x^3\right )}+\frac {\left (-1+x^3\right )^{5/3}}{5 x^5}-\frac {2 x \left (-1+x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,2 x^3\right )}{\left (1-x^3\right )^{2/3}}-\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {2}{9} \log \left (1+\frac {x^2}{\left (-1+x^3\right )^{2/3}}-\frac {x}{\sqrt [3]{-1+x^3}}\right )+\frac {4}{9} \log \left (1+\frac {x}{\sqrt [3]{-1+x^3}}\right )+\frac {1}{2} \log \left (-x+\sqrt [3]{-1+x^3}\right )-\frac {4}{3} \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}}{2 x^2}-\frac {2 x \left (-1+x^3\right )^{2/3}}{3 \left (1-2 x^3\right )}+\frac {\left (-1+x^3\right )^{5/3}}{5 x^5}-\frac {2 x \left (-1+x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,2 x^3\right )}{\left (1-x^3\right )^{2/3}}-\frac {4 \tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {2}{9} \log \left (1+\frac {x^2}{\left (-1+x^3\right )^{2/3}}-\frac {x}{\sqrt [3]{-1+x^3}}\right )+\frac {4}{9} \log \left (1+\frac {x}{\sqrt [3]{-1+x^3}}\right )+\frac {1}{2} \log \left (-x+\sqrt [3]{-1+x^3}\right )\\ \end {align*}
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Mathematica [A] time = 0.22, size = 125, normalized size = 1.06 \begin {gather*} \frac {7}{18} \left (2 \log \left (\frac {x}{\sqrt [3]{1-x^3}}+1\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{1-x^3}}-1}{\sqrt {3}}\right )-\log \left (-\frac {x}{\sqrt [3]{1-x^3}}+\frac {x^2}{\left (1-x^3\right )^{2/3}}+1\right )\right )+\frac {\left (x^3-1\right )^{2/3} \left (62 x^6-33 x^3+6\right )}{30 x^5 \left (2 x^3-1\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.27, size = 118, normalized size = 1.00 \begin {gather*} \frac {7}{9} \log \left (\sqrt [3]{x^3-1}+x\right )+\frac {7 \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3-1}-x}\right )}{3 \sqrt {3}}-\frac {7}{18} \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 (62 x^6-33 x^3+6\right )}{30 x^5 \left (2 x^3-1\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.07, size = 155, normalized size = 1.31 \begin {gather*} -\frac {70 \, \sqrt {3} {\left (2 \, x^{8} - x^{5}\right )} \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 ) - 35 \, {\left (2 \, x^{8} - x^{5}\right )} \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 (62 \, x^{6} - 33 \, x^{3} + 6\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{90 \, {\left (2 \, 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 (4 \, x^{6} - 5 \, x^{3} + 1\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (2 \, 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 = 1.37, size = 407, normalized size = 3.45 \begin {gather*} \frac {62 x^{9}-95 x^{6}+39 x^{3}-6}{30 \left (2 x^{3}-1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{5}}+\frac {7 \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 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} 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}}+x^{3}-1}{2 x^{3}-1}\right )}{9}-\frac {7 \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}+3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+x \left (x^{3}-1\right )^{\frac {2}{3}}+2 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 )}{9}-\frac {7 \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}+3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+x \left (x^{3}-1\right )^{\frac {2}{3}}+2 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 ) \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )}{3} \end {gather*}
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 (4 \, x^{6} - 5 \, x^{3} + 1\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (2 \, 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 (4\,x^6-5\,x^3+1\right )}{x^6\,{\left (2\,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(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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