Optimal. Leaf size=78 \[ -\frac {\log \left (x^3-1\right )}{6 \sqrt [3]{3}}+\frac {\log \left (\sqrt [3]{3} x-\sqrt [3]{x^3+2}\right )}{2 \sqrt [3]{3}}-\frac {\tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{3} x}{\sqrt [3]{x^3+2}}+1}{\sqrt {3}}\right )}{3^{5/6}} \]
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Rubi [A] time = 0.07, antiderivative size = 107, normalized size of antiderivative = 1.37, number of steps used = 7, number of rules used = 7, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {377, 200, 31, 634, 617, 204, 628} \[ \frac {\log \left (1-\frac {\sqrt [3]{3} x}{\sqrt [3]{x^3+2}}\right )}{3 \sqrt [3]{3}}-\frac {\log \left (\frac {3^{2/3} x^2}{\left (x^3+2\right )^{2/3}}+\frac {\sqrt [3]{3} x}{\sqrt [3]{x^3+2}}+1\right )}{6 \sqrt [3]{3}}-\frac {\tan ^{-1}\left (\frac {2 x}{\sqrt [6]{3} \sqrt [3]{x^3+2}}+\frac {1}{\sqrt {3}}\right )}{3^{5/6}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 200
Rule 204
Rule 377
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {1}{\left (-1+x^3\right ) \sqrt [3]{2+x^3}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{-1+3 x^3} \, dx,x,\frac {x}{\sqrt [3]{2+x^3}}\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{-1+\sqrt [3]{3} x} \, dx,x,\frac {x}{\sqrt [3]{2+x^3}}\right )+\frac {1}{3} \operatorname {Subst}\left (\int \frac {-2-\sqrt [3]{3} x}{1+\sqrt [3]{3} x+3^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{2+x^3}}\right )\\ &=\frac {\log \left (1-\frac {\sqrt [3]{3} x}{\sqrt [3]{2+x^3}}\right )}{3 \sqrt [3]{3}}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt [3]{3} x+3^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{2+x^3}}\right )-\frac {\operatorname {Subst}\left (\int \frac {\sqrt [3]{3}+2\ 3^{2/3} x}{1+\sqrt [3]{3} x+3^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{2+x^3}}\right )}{6 \sqrt [3]{3}}\\ &=\frac {\log \left (1-\frac {\sqrt [3]{3} x}{\sqrt [3]{2+x^3}}\right )}{3 \sqrt [3]{3}}-\frac {\log \left (1+\frac {3^{2/3} x^2}{\left (2+x^3\right )^{2/3}}+\frac {\sqrt [3]{3} x}{\sqrt [3]{2+x^3}}\right )}{6 \sqrt [3]{3}}+\frac {\operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{3} x}{\sqrt [3]{2+x^3}}\right )}{\sqrt [3]{3}}\\ &=-\frac {\tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{3} x}{\sqrt [3]{2+x^3}}}{\sqrt {3}}\right )}{3^{5/6}}+\frac {\log \left (1-\frac {\sqrt [3]{3} x}{\sqrt [3]{2+x^3}}\right )}{3 \sqrt [3]{3}}-\frac {\log \left (1+\frac {3^{2/3} x^2}{\left (2+x^3\right )^{2/3}}+\frac {\sqrt [3]{3} x}{\sqrt [3]{2+x^3}}\right )}{6 \sqrt [3]{3}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 104, normalized size = 1.33 \[ \frac {\sqrt {3} \left (2 \log \left (1-\frac {\sqrt [3]{3} x}{\sqrt [3]{x^3+2}}\right )-\log \left (\frac {\sqrt [3]{3} x}{\sqrt [3]{x^3+2}}+\frac {3^{2/3} x^2}{\left (x^3+2\right )^{2/3}}+1\right )\right )-6 \tan ^{-1}\left (\frac {2 x}{\sqrt [6]{3} \sqrt [3]{x^3+2}}+\frac {1}{\sqrt {3}}\right )}{6\ 3^{5/6}} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.24, size = 113, normalized size = 1.45 \[ \frac {\log \left (3^{2/3} \sqrt [3]{x^3+2}-3 x\right )}{3 \sqrt [3]{3}}-\frac {\tan ^{-1}\left (\frac {3^{5/6} x}{2 \sqrt [3]{x^3+2}+\sqrt [3]{3} x}\right )}{3^{5/6}}-\frac {\log \left (3^{2/3} \sqrt [3]{x^3+2} x+\sqrt [3]{3} \left (x^3+2\right )^{2/3}+3 x^2\right )}{6 \sqrt [3]{3}} \]
Antiderivative was successfully verified.
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fricas [B] time = 3.67, size = 232, normalized size = 2.97 \[ \frac {1}{27} \cdot 3^{\frac {2}{3}} \log \left (\frac {9 \cdot 3^{\frac {1}{3}} {\left (x^{3} + 2\right )}^{\frac {1}{3}} x^{2} - 2 \cdot 3^{\frac {2}{3}} {\left (x^{3} - 1\right )} - 9 \, {\left (x^{3} + 2\right )}^{\frac {2}{3}} x}{x^{3} - 1}\right ) - \frac {1}{54} \cdot 3^{\frac {2}{3}} \log \left (\frac {3 \cdot 3^{\frac {2}{3}} {\left (7 \, x^{4} + 2 \, x\right )} {\left (x^{3} + 2\right )}^{\frac {2}{3}} + 3^{\frac {1}{3}} {\left (31 \, x^{6} + 46 \, x^{3} + 4\right )} + 9 \, {\left (5 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{3} + 2\right )}^{\frac {1}{3}}}{x^{6} - 2 \, x^{3} + 1}\right ) - \frac {1}{9} \cdot 3^{\frac {1}{6}} \arctan \left (\frac {3^{\frac {1}{6}} {\left (12 \cdot 3^{\frac {2}{3}} {\left (7 \, x^{7} - 5 \, x^{4} - 2 \, x\right )} {\left (x^{3} + 2\right )}^{\frac {2}{3}} - 3^{\frac {1}{3}} {\left (127 \, x^{9} + 402 \, x^{6} + 192 \, x^{3} + 8\right )} - 18 \, {\left (31 \, x^{8} + 46 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{3} + 2\right )}^{\frac {1}{3}}\right )}}{3 \, {\left (251 \, x^{9} + 462 \, x^{6} + 24 \, x^{3} - 8\right )}}\right ) \]
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 \[ \int \frac {1}{{\left (x^{3} + 2\right )}^{\frac {1}{3}} {\left (x^{3} - 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.74, size = 902, normalized size = 11.56
method | result | size |
trager | \(\RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+81 \textit {\_Z}^{2}\right ) \ln \left (-\frac {27 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+81 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-9\right )^{2} x^{3}+12 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+81 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-9\right )^{3} x^{3}+15 \left (x^{3}+2\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+81 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-9\right )^{2} x +45 \left (x^{3}+2\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+81 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-9\right ) x^{2}+7 \left (x^{3}+2\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-9\right )^{2} x^{2}+36 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+81 \textit {\_Z}^{2}\right ) x^{3}+16 \RootOf \left (\textit {\_Z}^{3}-9\right ) x^{3}+21 x \left (x^{3}+2\right )^{\frac {2}{3}}+18 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+81 \textit {\_Z}^{2}\right )+8 \RootOf \left (\textit {\_Z}^{3}-9\right )}{\left (-1+x \right ) \left (x^{2}+x +1\right )}\right )-\frac {\ln \left (\frac {-27 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+81 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-9\right )^{2} x^{3}+9 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+81 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-9\right )^{3} x^{3}+15 \left (x^{3}+2\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+81 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-9\right )^{2} x +45 \left (x^{3}+2\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+81 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-9\right ) x^{2}-2 \left (x^{3}+2\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-9\right )^{2} x^{2}+9 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+81 \textit {\_Z}^{2}\right ) x^{3}-3 \RootOf \left (\textit {\_Z}^{3}-9\right ) x^{3}-6 x \left (x^{3}+2\right )^{\frac {2}{3}}+18 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+81 \textit {\_Z}^{2}\right )-6 \RootOf \left (\textit {\_Z}^{3}-9\right )}{\left (-1+x \right ) \left (x^{2}+x +1\right )}\right ) \RootOf \left (\textit {\_Z}^{3}-9\right )}{9}-\ln \left (\frac {-27 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+81 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-9\right )^{2} x^{3}+9 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+81 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-9\right )^{3} x^{3}+15 \left (x^{3}+2\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+81 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-9\right )^{2} x +45 \left (x^{3}+2\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+81 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-9\right ) x^{2}-2 \left (x^{3}+2\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-9\right )^{2} x^{2}+9 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+81 \textit {\_Z}^{2}\right ) x^{3}-3 \RootOf \left (\textit {\_Z}^{3}-9\right ) x^{3}-6 x \left (x^{3}+2\right )^{\frac {2}{3}}+18 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+81 \textit {\_Z}^{2}\right )-6 \RootOf \left (\textit {\_Z}^{3}-9\right )}{\left (-1+x \right ) \left (x^{2}+x +1\right )}\right ) \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+81 \textit {\_Z}^{2}\right )\) | \(902\) |
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 \[ \int \frac {1}{{\left (x^{3} + 2\right )}^{\frac {1}{3}} {\left (x^{3} - 1\right )}}\,{d x} \]
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 \[ \int \frac {1}{\left (x^3-1\right )\,{\left (x^3+2\right )}^{1/3}} \,d x \]
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 \[ \int \frac {1}{\left (x - 1\right ) \sqrt [3]{x^{3} + 2} \left (x^{2} + x + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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