Optimal. Leaf size=131 \[ -\frac {2 \log \left (3^{2/3} \sqrt [3]{x^3+1}-3 x\right )}{\sqrt [3]{3}}+2 \sqrt [6]{3} \tan ^{-1}\left (\frac {3^{5/6} x}{2 \sqrt [3]{x^3+1}+\sqrt [3]{3} x}\right )+\frac {\left (x^3+1\right )^{2/3} \left (-17 x^3-2\right )}{10 x^5}+\frac {\log \left (3^{2/3} \sqrt [3]{x^3+1} x+\sqrt [3]{3} \left (x^3+1\right )^{2/3}+3 x^2\right )}{\sqrt [3]{3}} \]
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Rubi [A] time = 0.53, antiderivative size = 134, normalized size of antiderivative = 1.02, number of steps used = 13, number of rules used = 11, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.297, Rules used = {1586, 6725, 271, 264, 377, 200, 31, 634, 617, 204, 628} \begin {gather*} -\frac {2 \log \left (1-\frac {\sqrt [3]{3} x}{\sqrt [3]{x^3+1}}\right )}{\sqrt [3]{3}}+2 \sqrt [6]{3} \tan ^{-1}\left (\frac {2 x}{\sqrt [6]{3} \sqrt [3]{x^3+1}}+\frac {1}{\sqrt {3}}\right )-\frac {\left (x^3+1\right )^{2/3}}{5 x^5}-\frac {17 \left (x^3+1\right )^{2/3}}{10 x^2}+\frac {\log \left (\frac {\sqrt [3]{3} x}{\sqrt [3]{x^3+1}}+\frac {3^{2/3} x^2}{\left (x^3+1\right )^{2/3}}+1\right )}{\sqrt [3]{3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 200
Rule 204
Rule 264
Rule 271
Rule 377
Rule 617
Rule 628
Rule 634
Rule 1586
Rule 6725
Rubi steps
\begin {align*} \int \frac {\left (1+x^3\right )^{2/3} \left (-1-2 x^3+2 x^6\right )}{x^6 \left (-1+x^3+2 x^6\right )} \, dx &=\int \frac {-1-2 x^3+2 x^6}{x^6 \sqrt [3]{1+x^3} \left (-1+2 x^3\right )} \, dx\\ &=\int \left (\frac {1}{x^6 \sqrt [3]{1+x^3}}+\frac {4}{x^3 \sqrt [3]{1+x^3}}-\frac {6}{\sqrt [3]{1+x^3} \left (-1+2 x^3\right )}\right ) \, dx\\ &=4 \int \frac {1}{x^3 \sqrt [3]{1+x^3}} \, dx-6 \int \frac {1}{\sqrt [3]{1+x^3} \left (-1+2 x^3\right )} \, dx+\int \frac {1}{x^6 \sqrt [3]{1+x^3}} \, dx\\ &=-\frac {\left (1+x^3\right )^{2/3}}{5 x^5}-\frac {2 \left (1+x^3\right )^{2/3}}{x^2}-\frac {3}{5} \int \frac {1}{x^3 \sqrt [3]{1+x^3}} \, dx-6 \operatorname {Subst}\left (\int \frac {1}{-1+3 x^3} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )\\ &=-\frac {\left (1+x^3\right )^{2/3}}{5 x^5}-\frac {17 \left (1+x^3\right )^{2/3}}{10 x^2}-2 \operatorname {Subst}\left (\int \frac {1}{-1+\sqrt [3]{3} x} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )-2 \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]{1+x^3}}\right )\\ &=-\frac {\left (1+x^3\right )^{2/3}}{5 x^5}-\frac {17 \left (1+x^3\right )^{2/3}}{10 x^2}-\frac {2 \log \left (1-\frac {\sqrt [3]{3} x}{\sqrt [3]{1+x^3}}\right )}{\sqrt [3]{3}}+3 \operatorname {Subst}\left (\int \frac {1}{1+\sqrt [3]{3} x+3^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{1+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]{1+x^3}}\right )}{\sqrt [3]{3}}\\ &=-\frac {\left (1+x^3\right )^{2/3}}{5 x^5}-\frac {17 \left (1+x^3\right )^{2/3}}{10 x^2}-\frac {2 \log \left (1-\frac {\sqrt [3]{3} x}{\sqrt [3]{1+x^3}}\right )}{\sqrt [3]{3}}+\frac {\log \left (1+\frac {3^{2/3} x^2}{\left (1+x^3\right )^{2/3}}+\frac {\sqrt [3]{3} x}{\sqrt [3]{1+x^3}}\right )}{\sqrt [3]{3}}-\left (2\ 3^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{3} x}{\sqrt [3]{1+x^3}}\right )\\ &=-\frac {\left (1+x^3\right )^{2/3}}{5 x^5}-\frac {17 \left (1+x^3\right )^{2/3}}{10 x^2}+2 \sqrt [6]{3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{3} x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )-\frac {2 \log \left (1-\frac {\sqrt [3]{3} x}{\sqrt [3]{1+x^3}}\right )}{\sqrt [3]{3}}+\frac {\log \left (1+\frac {3^{2/3} x^2}{\left (1+x^3\right )^{2/3}}+\frac {\sqrt [3]{3} x}{\sqrt [3]{1+x^3}}\right )}{\sqrt [3]{3}}\\ \end {align*}
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Mathematica [F] time = 0.33, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1+x^3\right )^{2/3} \left (-1-2 x^3+2 x^6\right )}{x^6 \left (-1+x^3+2 x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.32, size = 131, normalized size = 1.00 \begin {gather*} \frac {\left (-2-17 x^3\right ) \left (1+x^3\right )^{2/3}}{10 x^5}+2 \sqrt [6]{3} \tan ^{-1}\left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{1+x^3}}\right )-\frac {2 \log \left (-3 x+3^{2/3} \sqrt [3]{1+x^3}\right )}{\sqrt [3]{3}}+\frac {\log \left (3 x^2+3^{2/3} x \sqrt [3]{1+x^3}+\sqrt [3]{3} \left (1+x^3\right )^{2/3}\right )}{\sqrt [3]{3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 2.76, size = 292, normalized size = 2.23 \begin {gather*} \frac {20 \cdot 3^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{5} \log \left (\frac {9 \cdot 3^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 3^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (2 \, x^{3} - 1\right )} - 9 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} x}{2 \, x^{3} - 1}\right ) - 10 \cdot 3^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {3 \cdot 3^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (7 \, x^{4} + x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} - 3^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (31 \, x^{6} + 23 \, x^{3} + 1\right )} - 9 \, {\left (5 \, x^{5} + 2 \, x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{4 \, x^{6} - 4 \, x^{3} + 1}\right ) - 60 \cdot 3^{\frac {1}{6}} \left (-1\right )^{\frac {1}{3}} x^{5} \arctan \left (\frac {3^{\frac {1}{6}} {\left (6 \cdot 3^{\frac {2}{3}} \left (-1\right )^{\frac {2}{3}} {\left (14 \, x^{7} - 5 \, x^{4} - x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} + 18 \, \left (-1\right )^{\frac {1}{3}} {\left (31 \, x^{8} + 23 \, x^{5} + x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}} - 3^{\frac {1}{3}} {\left (127 \, x^{9} + 201 \, x^{6} + 48 \, x^{3} + 1\right )}\right )}}{3 \, {\left (251 \, x^{9} + 231 \, x^{6} + 6 \, x^{3} - 1\right )}}\right ) - 9 \, {\left (17 \, x^{3} + 2\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{90 \, 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 (2 \, x^{6} - 2 \, x^{3} - 1\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{6} + 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 = 17.01, size = 893, normalized size = 6.82
method | result | size |
risch | \(-\frac {17 x^{6}+19 x^{3}+2}{10 x^{5} \left (x^{3}+1\right )^{\frac {1}{3}}}-\frac {2 \ln \left (\frac {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 )^{3} x^{3}-81 \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}+15 \left (x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{3}+9\right )^{2} \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 +2 \RootOf \left (\textit {\_Z}^{3}+9\right )^{2} \left (x^{3}+1\right )^{\frac {1}{3}} x^{2}+63 \left (x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+9\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 ) x^{2}-4 \RootOf \left (\textit {\_Z}^{3}+9\right ) x^{3}+108 \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}-21 x \left (x^{3}+1\right )^{\frac {2}{3}}-\RootOf \left (\textit {\_Z}^{3}+9\right )+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 x^{3}-1}\right ) \RootOf \left (\textit {\_Z}^{3}+9\right )}{3}-6 \ln \left (\frac {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 )^{3} x^{3}-81 \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}+15 \left (x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{3}+9\right )^{2} \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 +2 \RootOf \left (\textit {\_Z}^{3}+9\right )^{2} \left (x^{3}+1\right )^{\frac {1}{3}} x^{2}+63 \left (x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+9\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 ) x^{2}-4 \RootOf \left (\textit {\_Z}^{3}+9\right ) x^{3}+108 \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}-21 x \left (x^{3}+1\right )^{\frac {2}{3}}-\RootOf \left (\textit {\_Z}^{3}+9\right )+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 x^{3}-1}\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 )+\frac {2 \RootOf \left (\textit {\_Z}^{3}+9\right ) \ln \left (\frac {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 )^{3} x^{3}+108 \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}-15 \left (x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{3}+9\right )^{2} \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 +7 \RootOf \left (\textit {\_Z}^{3}+9\right )^{2} \left (x^{3}+1\right )^{\frac {1}{3}} x^{2}+18 \left (x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+9\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 ) x^{2}+\RootOf \left (\textit {\_Z}^{3}+9\right ) x^{3}+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}-6 x \left (x^{3}+1\right )^{\frac {2}{3}}+\RootOf \left (\textit {\_Z}^{3}+9\right )+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 )}{2 x^{3}-1}\right )}{3}\) | \(893\) |
trager | \(\text {Expression too large to display}\) | \(1104\) |
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 (2 \, x^{6} - 2 \, x^{3} - 1\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{6} + 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 (-2\,x^6+2\,x^3+1\right )}{x^6\,\left (2\,x^6+x^3-1\right )} \,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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