Optimal. Leaf size=164 \[ \frac {\log \left (\sqrt [3]{2} 3^{2/3} \sqrt [3]{x^3+1}-3 x\right )}{3\ 2^{2/3} \sqrt [3]{3}}-\frac {\tan ^{-1}\left (\frac {3^{5/6} x}{2 \sqrt [3]{2} \sqrt [3]{x^3+1}+\sqrt [3]{3} x}\right )}{2^{2/3} 3^{5/6}}+\frac {\left (x^3+1\right )^{5/3}}{5 x^5}-\frac {\log \left (\sqrt [3]{2} 3^{2/3} \sqrt [3]{x^3+1} x+2^{2/3} \sqrt [3]{3} \left (x^3+1\right )^{2/3}+3 x^2\right )}{6\ 2^{2/3} \sqrt [3]{3}} \]
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Rubi [A] time = 0.24, antiderivative size = 166, normalized size of antiderivative = 1.01, number of steps used = 11, number of rules used = 10, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1586, 583, 12, 377, 200, 31, 634, 617, 204, 628} \begin {gather*} \frac {\log \left (\sqrt [3]{2}-\frac {\sqrt [3]{3} x}{\sqrt [3]{x^3+1}}\right )}{3\ 2^{2/3} \sqrt [3]{3}}-\frac {\tan ^{-1}\left (\frac {2^{2/3} x}{\sqrt [6]{3} \sqrt [3]{x^3+1}}+\frac {1}{\sqrt {3}}\right )}{2^{2/3} 3^{5/6}}+\frac {\left (x^3+1\right )^{2/3}}{5 x^5}+\frac {\left (x^3+1\right )^{2/3}}{5 x^2}-\frac {\log \left (\frac {\sqrt [3]{6} x}{\sqrt [3]{x^3+1}}+\frac {3^{2/3} x^2}{\left (x^3+1\right )^{2/3}}+2^{2/3}\right )}{6\ 2^{2/3} \sqrt [3]{3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 31
Rule 200
Rule 204
Rule 377
Rule 583
Rule 617
Rule 628
Rule 634
Rule 1586
Rubi steps
\begin {align*} \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (-2-x^3+x^6\right )} \, dx &=\int \frac {2+x^3}{x^6 \left (-2+x^3\right ) \sqrt [3]{1+x^3}} \, dx\\ &=\frac {\left (1+x^3\right )^{2/3}}{5 x^5}+\frac {1}{10} \int \frac {8+6 x^3}{x^3 \left (-2+x^3\right ) \sqrt [3]{1+x^3}} \, dx\\ &=\frac {\left (1+x^3\right )^{2/3}}{5 x^5}+\frac {\left (1+x^3\right )^{2/3}}{5 x^2}+\frac {1}{40} \int \frac {40}{\left (-2+x^3\right ) \sqrt [3]{1+x^3}} \, dx\\ &=\frac {\left (1+x^3\right )^{2/3}}{5 x^5}+\frac {\left (1+x^3\right )^{2/3}}{5 x^2}+\int \frac {1}{\left (-2+x^3\right ) \sqrt [3]{1+x^3}} \, dx\\ &=\frac {\left (1+x^3\right )^{2/3}}{5 x^5}+\frac {\left (1+x^3\right )^{2/3}}{5 x^2}+\operatorname {Subst}\left (\int \frac {1}{-2+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 {\left (1+x^3\right )^{2/3}}{5 x^2}+\frac {\operatorname {Subst}\left (\int \frac {1}{-\sqrt [3]{2}+\sqrt [3]{3} x} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )}{3\ 2^{2/3}}+\frac {\operatorname {Subst}\left (\int \frac {-2 \sqrt [3]{2}-\sqrt [3]{3} x}{2^{2/3}+\sqrt [3]{6} x+3^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )}{3\ 2^{2/3}}\\ &=\frac {\left (1+x^3\right )^{2/3}}{5 x^5}+\frac {\left (1+x^3\right )^{2/3}}{5 x^2}+\frac {\log \left (\sqrt [3]{2}-\frac {\sqrt [3]{3} x}{\sqrt [3]{1+x^3}}\right )}{3\ 2^{2/3} \sqrt [3]{3}}-\frac {\operatorname {Subst}\left (\int \frac {1}{2^{2/3}+\sqrt [3]{6} x+3^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )}{2 \sqrt [3]{2}}-\frac {\operatorname {Subst}\left (\int \frac {\sqrt [3]{6}+2\ 3^{2/3} x}{2^{2/3}+\sqrt [3]{6} x+3^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )}{6\ 2^{2/3} \sqrt [3]{3}}\\ &=\frac {\left (1+x^3\right )^{2/3}}{5 x^5}+\frac {\left (1+x^3\right )^{2/3}}{5 x^2}+\frac {\log \left (\sqrt [3]{2}-\frac {\sqrt [3]{3} x}{\sqrt [3]{1+x^3}}\right )}{3\ 2^{2/3} \sqrt [3]{3}}-\frac {\log \left (2^{2/3}+\frac {3^{2/3} x^2}{\left (1+x^3\right )^{2/3}}+\frac {\sqrt [3]{6} x}{\sqrt [3]{1+x^3}}\right )}{6\ 2^{2/3} \sqrt [3]{3}}+\frac {\operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2^{2/3} \sqrt [3]{3} x}{\sqrt [3]{1+x^3}}\right )}{2^{2/3} \sqrt [3]{3}}\\ &=\frac {\left (1+x^3\right )^{2/3}}{5 x^5}+\frac {\left (1+x^3\right )^{2/3}}{5 x^2}-\frac {\tan ^{-1}\left (\frac {1+\frac {2^{2/3} \sqrt [3]{3} x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{2^{2/3} 3^{5/6}}+\frac {\log \left (\sqrt [3]{2}-\frac {\sqrt [3]{3} x}{\sqrt [3]{1+x^3}}\right )}{3\ 2^{2/3} \sqrt [3]{3}}-\frac {\log \left (2^{2/3}+\frac {3^{2/3} x^2}{\left (1+x^3\right )^{2/3}}+\frac {\sqrt [3]{6} x}{\sqrt [3]{1+x^3}}\right )}{6\ 2^{2/3} \sqrt [3]{3}}\\ \end {align*}
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Mathematica [F] time = 0.31, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (-2-x^3+x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.42, size = 164, normalized size = 1.00 \begin {gather*} \frac {\left (1+x^3\right )^{5/3}}{5 x^5}-\frac {\tan ^{-1}\left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{2} \sqrt [3]{1+x^3}}\right )}{2^{2/3} 3^{5/6}}+\frac {\log \left (-3 x+\sqrt [3]{2} 3^{2/3} \sqrt [3]{1+x^3}\right )}{3\ 2^{2/3} \sqrt [3]{3}}-\frac {\log \left (3 x^2+\sqrt [3]{2} 3^{2/3} x \sqrt [3]{1+x^3}+2^{2/3} \sqrt [3]{3} \left (1+x^3\right )^{2/3}\right )}{6\ 2^{2/3} \sqrt [3]{3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 3.00, size = 253, normalized size = 1.54 \begin {gather*} \frac {10 \cdot 12^{\frac {2}{3}} x^{5} \log \left (\frac {18 \cdot 12^{\frac {1}{3}} {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 12^{\frac {2}{3}} {\left (x^{3} - 2\right )} - 36 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} x}{x^{3} - 2}\right ) - 5 \cdot 12^{\frac {2}{3}} x^{5} \log \left (\frac {6 \cdot 12^{\frac {2}{3}} {\left (4 \, x^{4} + x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} + 12^{\frac {1}{3}} {\left (55 \, x^{6} + 50 \, x^{3} + 4\right )} + 18 \, {\left (7 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x^{6} - 4 \, x^{3} + 4}\right ) - 60 \cdot 12^{\frac {1}{6}} x^{5} \arctan \left (\frac {12^{\frac {1}{6}} {\left (12 \cdot 12^{\frac {2}{3}} {\left (4 \, x^{7} - 7 \, x^{4} - 2 \, x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} - 12^{\frac {1}{3}} {\left (377 \, x^{9} + 600 \, x^{6} + 204 \, x^{3} + 8\right )} - 36 \, {\left (55 \, x^{8} + 50 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (487 \, x^{9} + 480 \, x^{6} + 12 \, x^{3} - 8\right )}}\right ) + 216 \, {\left (x^{3} + 1\right )}^{\frac {5}{3}}}{1080 \, 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} + 2\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - 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 = 18.09, size = 886, normalized size = 5.40
method | result | size |
risch | \(\frac {x^{6}+2 x^{3}+1}{5 x^{5} \left (x^{3}+1\right )^{\frac {1}{3}}}-\frac {\ln \left (\frac {6 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-18\right )+324 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-18\right )^{3} x^{3}-162 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-18\right )+324 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-18\right )^{2} x^{3}-42 \left (x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{3}-18\right )^{2} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-18\right )+324 \textit {\_Z}^{2}\right ) x +\RootOf \left (\textit {\_Z}^{3}-18\right )^{2} \left (x^{3}+1\right )^{\frac {1}{3}} x^{2}+144 \left (x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-18\right ) \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-18\right )+324 \textit {\_Z}^{2}\right ) x^{2}+10 \RootOf \left (\textit {\_Z}^{3}-18\right ) x^{3}-270 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-18\right )+324 \textit {\_Z}^{2}\right ) x^{3}-48 x \left (x^{3}+1\right )^{\frac {2}{3}}+4 \RootOf \left (\textit {\_Z}^{3}-18\right )-108 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-18\right )+324 \textit {\_Z}^{2}\right )}{x^{3}-2}\right ) \RootOf \left (\textit {\_Z}^{3}-18\right )}{18}-\ln \left (\frac {6 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-18\right )+324 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-18\right )^{3} x^{3}-162 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-18\right )+324 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-18\right )^{2} x^{3}-42 \left (x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{3}-18\right )^{2} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-18\right )+324 \textit {\_Z}^{2}\right ) x +\RootOf \left (\textit {\_Z}^{3}-18\right )^{2} \left (x^{3}+1\right )^{\frac {1}{3}} x^{2}+144 \left (x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-18\right ) \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-18\right )+324 \textit {\_Z}^{2}\right ) x^{2}+10 \RootOf \left (\textit {\_Z}^{3}-18\right ) x^{3}-270 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-18\right )+324 \textit {\_Z}^{2}\right ) x^{3}-48 x \left (x^{3}+1\right )^{\frac {2}{3}}+4 \RootOf \left (\textit {\_Z}^{3}-18\right )-108 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-18\right )+324 \textit {\_Z}^{2}\right )}{x^{3}-2}\right ) \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-18\right )+324 \textit {\_Z}^{2}\right )+\frac {\RootOf \left (\textit {\_Z}^{3}-18\right ) \ln \left (\frac {3 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-18\right )+324 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-18\right )^{3} x^{3}+135 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-18\right )+324 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-18\right )^{2} x^{3}+21 \left (x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{3}-18\right )^{2} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-18\right )+324 \textit {\_Z}^{2}\right ) x +4 \RootOf \left (\textit {\_Z}^{3}-18\right )^{2} \left (x^{3}+1\right )^{\frac {1}{3}} x^{2}+9 \left (x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-18\right ) \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-18\right )+324 \textit {\_Z}^{2}\right ) x^{2}-2 \RootOf \left (\textit {\_Z}^{3}-18\right ) x^{3}-90 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-18\right )+324 \textit {\_Z}^{2}\right ) x^{3}-3 x \left (x^{3}+1\right )^{\frac {2}{3}}-2 \RootOf \left (\textit {\_Z}^{3}-18\right )-90 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-18\right )^{2}+18 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-18\right )+324 \textit {\_Z}^{2}\right )}{x^{3}-2}\right )}{18}\) | \(886\) |
trager | \(\text {Expression too large to display}\) | \(1108\) |
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} + 2\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - 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+2\right )}{x^6\,\left (-x^6+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^{3} + 2\right )}{x^{6} \left (x + 1\right ) \left (x^{3} - 2\right ) \left (x^{2} - x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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