Optimal. Leaf size=140 \[ \frac {\log \left (\sqrt [3]{2} \sqrt [3]{x^3-1}-x\right )}{4\ 2^{2/3}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{2} \sqrt [3]{x^3-1}+x}\right )}{4\ 2^{2/3}}+\frac {\left (x^3-1\right )^{2/3} \left (11 x^3+4\right )}{40 x^5}-\frac {\log \left (\sqrt [3]{2} \sqrt [3]{x^3-1} x+2^{2/3} \left (x^3-1\right )^{2/3}+x^2\right )}{8\ 2^{2/3}} \]
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Rubi [A] time = 0.14, antiderivative size = 145, normalized size of antiderivative = 1.04, number of steps used = 10, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {580, 583, 12, 377, 200, 31, 634, 617, 204, 628} \begin {gather*} \frac {\log \left (\sqrt [3]{2}-\frac {x}{\sqrt [3]{x^3-1}}\right )}{4\ 2^{2/3}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\frac {2^{2/3} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{4\ 2^{2/3}}+\frac {\left (x^3-1\right )^{2/3}}{10 x^5}+\frac {11 \left (x^3-1\right )^{2/3}}{40 x^2}-\frac {\log \left (\frac {\sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+\frac {x^2}{\left (x^3-1\right )^{2/3}}+2^{2/3}\right )}{8\ 2^{2/3}} \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 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {\left (-1+x^3\right )^{2/3} \left (1+x^3\right )}{x^6 \left (-2+x^3\right )} \, dx &=\frac {\left (-1+x^3\right )^{2/3}}{10 x^5}-\frac {1}{10} \int \frac {11-13 x^3}{x^3 \left (-2+x^3\right ) \sqrt [3]{-1+x^3}} \, dx\\ &=\frac {\left (-1+x^3\right )^{2/3}}{10 x^5}+\frac {11 \left (-1+x^3\right )^{2/3}}{40 x^2}+\frac {1}{40} \int \frac {30}{\left (-2+x^3\right ) \sqrt [3]{-1+x^3}} \, dx\\ &=\frac {\left (-1+x^3\right )^{2/3}}{10 x^5}+\frac {11 \left (-1+x^3\right )^{2/3}}{40 x^2}+\frac {3}{4} \int \frac {1}{\left (-2+x^3\right ) \sqrt [3]{-1+x^3}} \, dx\\ &=\frac {\left (-1+x^3\right )^{2/3}}{10 x^5}+\frac {11 \left (-1+x^3\right )^{2/3}}{40 x^2}+\frac {3}{4} \operatorname {Subst}\left (\int \frac {1}{-2+x^3} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )\\ &=\frac {\left (-1+x^3\right )^{2/3}}{10 x^5}+\frac {11 \left (-1+x^3\right )^{2/3}}{40 x^2}+\frac {\operatorname {Subst}\left (\int \frac {1}{-\sqrt [3]{2}+x} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )}{4\ 2^{2/3}}+\frac {\operatorname {Subst}\left (\int \frac {-2 \sqrt [3]{2}-x}{2^{2/3}+\sqrt [3]{2} x+x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )}{4\ 2^{2/3}}\\ &=\frac {\left (-1+x^3\right )^{2/3}}{10 x^5}+\frac {11 \left (-1+x^3\right )^{2/3}}{40 x^2}+\frac {\log \left (\sqrt [3]{2}-\frac {x}{\sqrt [3]{-1+x^3}}\right )}{4\ 2^{2/3}}-\frac {\operatorname {Subst}\left (\int \frac {\sqrt [3]{2}+2 x}{2^{2/3}+\sqrt [3]{2} x+x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )}{8\ 2^{2/3}}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{2^{2/3}+\sqrt [3]{2} x+x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )}{8 \sqrt [3]{2}}\\ &=\frac {\left (-1+x^3\right )^{2/3}}{10 x^5}+\frac {11 \left (-1+x^3\right )^{2/3}}{40 x^2}+\frac {\log \left (\sqrt [3]{2}-\frac {x}{\sqrt [3]{-1+x^3}}\right )}{4\ 2^{2/3}}-\frac {\log \left (2^{2/3}+\frac {x^2}{\left (-1+x^3\right )^{2/3}}+\frac {\sqrt [3]{2} x}{\sqrt [3]{-1+x^3}}\right )}{8\ 2^{2/3}}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2^{2/3} x}{\sqrt [3]{-1+x^3}}\right )}{4\ 2^{2/3}}\\ &=\frac {\left (-1+x^3\right )^{2/3}}{10 x^5}+\frac {11 \left (-1+x^3\right )^{2/3}}{40 x^2}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2^{2/3} x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{4\ 2^{2/3}}+\frac {\log \left (\sqrt [3]{2}-\frac {x}{\sqrt [3]{-1+x^3}}\right )}{4\ 2^{2/3}}-\frac {\log \left (2^{2/3}+\frac {x^2}{\left (-1+x^3\right )^{2/3}}+\frac {\sqrt [3]{2} x}{\sqrt [3]{-1+x^3}}\right )}{8\ 2^{2/3}}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 135, normalized size = 0.96 \begin {gather*} \left (\frac {1}{10 x^5}+\frac {11}{40 x^2}\right ) \left (x^3-1\right )^{2/3}-\frac {-2 \log \left (2-\frac {2^{2/3} x}{\sqrt [3]{1-x^3}}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2^{2/3} x}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )+\log \left (\frac {2^{2/3} x}{\sqrt [3]{1-x^3}}+\frac {\sqrt [3]{2} x^2}{\left (1-x^3\right )^{2/3}}+2\right )}{8\ 2^{2/3}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.31, size = 140, normalized size = 1.00 \begin {gather*} \frac {\left (-1+x^3\right )^{2/3} \left (4+11 x^3\right )}{40 x^5}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{-1+x^3}}\right )}{4\ 2^{2/3}}+\frac {\log \left (-x+\sqrt [3]{2} \sqrt [3]{-1+x^3}\right )}{4\ 2^{2/3}}-\frac {\log \left (x^2+\sqrt [3]{2} x \sqrt [3]{-1+x^3}+2^{2/3} \left (-1+x^3\right )^{2/3}\right )}{8\ 2^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 3.15, size = 266, normalized size = 1.90 \begin {gather*} \frac {20 \cdot 4^{\frac {1}{6}} \sqrt {3} x^{5} \arctan \left (\frac {4^{\frac {1}{6}} \sqrt {3} {\left (12 \cdot 4^{\frac {2}{3}} {\left (2 \, x^{7} - 5 \, x^{4} + 2 \, x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (91 \, x^{9} - 168 \, x^{6} + 84 \, x^{3} - 8\right )} + 12 \, {\left (19 \, x^{8} - 22 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (53 \, x^{9} - 48 \, x^{6} - 12 \, x^{3} + 8\right )}}\right ) + 10 \cdot 4^{\frac {2}{3}} x^{5} \log \left (\frac {6 \cdot 4^{\frac {1}{3}} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 4^{\frac {2}{3}} {\left (x^{3} - 2\right )} - 12 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x}{x^{3} - 2}\right ) - 5 \cdot 4^{\frac {2}{3}} x^{5} \log \left (\frac {6 \cdot 4^{\frac {2}{3}} {\left (2 \, x^{4} - x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (19 \, x^{6} - 22 \, x^{3} + 4\right )} + 6 \, {\left (5 \, x^{5} - 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x^{6} - 4 \, x^{3} + 4}\right ) + 12 \, {\left (11 \, x^{3} + 4\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{480 \, 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} + 1\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{3} - 2\right )} x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 18.11, size = 889, normalized size = 6.35
method | result | size |
risch | \(\frac {11 x^{6}-7 x^{3}-4}{40 x^{5} \left (x^{3}-1\right )^{\frac {1}{3}}}+\frac {\RootOf \left (\textit {\_Z}^{3}-2\right ) \ln \left (\frac {3 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{3} x^{3}+27 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{3}+15 \left (x^{3}-1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x +2 \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}-3 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{2}-2 \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{3}-18 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) x^{3}+x \left (x^{3}-1\right )^{\frac {2}{3}}+2 \RootOf \left (\textit {\_Z}^{3}-2\right )+18 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )}{x^{3}-2}\right )}{8}-\frac {\ln \left (\frac {6 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{3} x^{3}-18 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{3}-30 \left (x^{3}-1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x -\RootOf \left (\textit {\_Z}^{3}-2\right )^{2} \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+24 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{2}+6 \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{3}-18 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) x^{3}-8 x \left (x^{3}-1\right )^{\frac {2}{3}}-4 \RootOf \left (\textit {\_Z}^{3}-2\right )+12 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )}{x^{3}-2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )}{8}-\frac {3 \ln \left (\frac {6 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{3} x^{3}-18 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{3}-30 \left (x^{3}-1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x -\RootOf \left (\textit {\_Z}^{3}-2\right )^{2} \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+24 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{2}+6 \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{3}-18 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) x^{3}-8 x \left (x^{3}-1\right )^{\frac {2}{3}}-4 \RootOf \left (\textit {\_Z}^{3}-2\right )+12 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )}{x^{3}-2}\right ) \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )}{4}\) | \(889\) |
trager | \(\text {Expression too large to display}\) | \(1116\) |
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} + 1\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{3} - 2\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+1\right )}{x^6\,\left (x^3-2\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 + 1\right ) \left (x^{2} - x + 1\right )}{x^{6} \left (x^{3} - 2\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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