Optimal. Leaf size=132 \[ \frac {\log \left (2 \sqrt [3]{x^2+x+2}+2^{2/3} x\right )}{2^{2/3}}-\frac {\log \left (\sqrt [3]{2} x^2-2^{2/3} \sqrt [3]{x^2+x+2} x+2 \left (x^2+x+2\right )^{2/3}\right )}{2\ 2^{2/3}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\frac {\sqrt [3]{x^2+x+2}}{\sqrt {3}}-\frac {2^{2/3} x}{\sqrt {3}}}{\sqrt [3]{x^2+x+2}}\right )}{2^{2/3}} \]
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Rubi [F] time = 0.38, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {6+2 x+x^2}{(2+x) \left (2+x^2\right ) \sqrt [3]{2+x+x^2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {6+2 x+x^2}{(2+x) \left (2+x^2\right ) \sqrt [3]{2+x+x^2}} \, dx &=\int \left (\frac {1}{(2+x) \sqrt [3]{2+x+x^2}}+\frac {2}{\left (2+x^2\right ) \sqrt [3]{2+x+x^2}}\right ) \, dx\\ &=2 \int \frac {1}{\left (2+x^2\right ) \sqrt [3]{2+x+x^2}} \, dx+\int \frac {1}{(2+x) \sqrt [3]{2+x+x^2}} \, dx\\ &=2 \int \frac {1}{\left (2+x^2\right ) \sqrt [3]{2+x+x^2}} \, dx-\frac {\left (\sqrt [3]{\frac {1-i \sqrt {7}+2 x}{2+x}} \sqrt [3]{\frac {1+i \sqrt {7}+2 x}{2+x}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{x} \sqrt [3]{1-\frac {1}{2} \left (3-i \sqrt {7}\right ) x} \sqrt [3]{1-\frac {1}{2} \left (3+i \sqrt {7}\right ) x}} \, dx,x,\frac {1}{2+x}\right )}{2^{2/3} \left (\frac {1}{2+x}\right )^{2/3} \sqrt [3]{2+x+x^2}}\\ &=-\frac {3 \sqrt [3]{\frac {1-i \sqrt {7}+2 x}{2+x}} \sqrt [3]{\frac {1+i \sqrt {7}+2 x}{2+x}} F_1\left (\frac {2}{3};\frac {1}{3},\frac {1}{3};\frac {5}{3};\frac {3-i \sqrt {7}}{2 (2+x)},\frac {3+i \sqrt {7}}{2 (2+x)}\right )}{2\ 2^{2/3} \sqrt [3]{2+x+x^2}}+2 \int \frac {1}{\left (2+x^2\right ) \sqrt [3]{2+x+x^2}} \, dx\\ \end {align*}
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Mathematica [F] time = 0.50, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {6+2 x+x^2}{(2+x) \left (2+x^2\right ) \sqrt [3]{2+x+x^2}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.18, size = 132, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {3} \tan ^{-1}\left (\frac {-\frac {2^{2/3} x}{\sqrt {3}}+\frac {\sqrt [3]{2+x+x^2}}{\sqrt {3}}}{\sqrt [3]{2+x+x^2}}\right )}{2^{2/3}}+\frac {\log \left (2^{2/3} x+2 \sqrt [3]{2+x+x^2}\right )}{2^{2/3}}-\frac {\log \left (\sqrt [3]{2} x^2-2^{2/3} x \sqrt [3]{2+x+x^2}+2 \left (2+x+x^2\right )^{2/3}\right )}{2\ 2^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 10.83, size = 395, normalized size = 2.99 \begin {gather*} -\frac {1}{6} \cdot 4^{\frac {1}{6}} \sqrt {3} \arctan \left (\frac {4^{\frac {1}{6}} \sqrt {3} {\left (12 \cdot 4^{\frac {2}{3}} {\left (x^{7} + x^{6} - x^{5} - 2 \, x^{4} - 10 \, x^{3} - 8 \, x^{2} - 8 \, x\right )} {\left (x^{2} + x + 2\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (x^{9} + 24 \, x^{8} - 36 \, x^{7} - 64 \, x^{6} - 276 \, x^{5} - 168 \, x^{4} - 136 \, x^{3} + 144 \, x^{2} + 96 \, x + 64\right )} + 12 \, {\left (x^{8} - 14 \, x^{7} - 10 \, x^{6} - 20 \, x^{5} + 20 \, x^{4} + 16 \, x^{3} + 16 \, x^{2}\right )} {\left (x^{2} + x + 2\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (x^{9} - 48 \, x^{8} - 36 \, x^{7} - 64 \, x^{6} + 84 \, x^{5} + 120 \, x^{4} + 152 \, x^{3} + 144 \, x^{2} + 96 \, x + 64\right )}}\right ) + \frac {1}{12} \cdot 4^{\frac {2}{3}} \log \left (-\frac {6 \cdot 4^{\frac {1}{3}} {\left (x^{2} + x + 2\right )}^{\frac {1}{3}} x^{2} + 4^{\frac {2}{3}} {\left (x^{3} + 2 \, x^{2} + 2 \, x + 4\right )} + 12 \, {\left (x^{2} + x + 2\right )}^{\frac {2}{3}} x}{x^{3} + 2 \, x^{2} + 2 \, x + 4}\right ) - \frac {1}{24} \cdot 4^{\frac {2}{3}} \log \left (\frac {6 \cdot 4^{\frac {2}{3}} {\left (x^{4} - x^{3} - x^{2} - 2 \, x\right )} {\left (x^{2} + x + 2\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (x^{6} - 14 \, x^{5} - 10 \, x^{4} - 20 \, x^{3} + 20 \, x^{2} + 16 \, x + 16\right )} - 6 \, {\left (x^{5} - 4 \, x^{4} - 4 \, x^{3} - 8 \, x^{2}\right )} {\left (x^{2} + x + 2\right )}^{\frac {1}{3}}}{x^{6} + 4 \, x^{5} + 8 \, x^{4} + 16 \, x^{3} + 20 \, x^{2} + 16 \, x + 16}\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 {x^{2} + 2 \, x + 6}{{\left (x^{2} + x + 2\right )}^{\frac {1}{3}} {\left (x^{2} + 2\right )} {\left (x + 2\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 5.52, size = 883, normalized size = 6.69
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trager | \(-\frac {\ln \left (\frac {2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{3}+2 \left (x^{2}+x +2\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) x +\left (x^{2}+x +2\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{2}+4 \left (x^{2}+x +2\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-2\right ) \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) x^{2}+2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) x^{3}-4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) x^{2}+4 \left (x^{2}+x +2\right )^{\frac {2}{3}} x -4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) x -8 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right )}{\left (2+x \right ) \left (x^{2}+2\right )}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )}{2}-\ln \left (\frac {2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{3}+2 \left (x^{2}+x +2\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) x +\left (x^{2}+x +2\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{2}+4 \left (x^{2}+x +2\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-2\right ) \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) x^{2}+2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) x^{3}-4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) x^{2}+4 \left (x^{2}+x +2\right )^{\frac {2}{3}} x -4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) x -8 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right )}{\left (2+x \right ) \left (x^{2}+2\right )}\right ) \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right )+\frac {\RootOf \left (\textit {\_Z}^{3}-2\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{3}-2\right )^{3} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) x^{3}-2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{3}+3 \left (x^{2}+x +2\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{2}+6 \left (x^{2}+x +2\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-2\right ) \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) x^{2}+2 \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{2}-4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) x^{2}+6 \left (x^{2}+x +2\right )^{\frac {2}{3}} x +2 \RootOf \left (\textit {\_Z}^{3}-2\right ) x -4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) x +4 \RootOf \left (\textit {\_Z}^{3}-2\right )-8 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right )}{\left (2+x \right ) \left (x^{2}+2\right )}\right )}{2}\) | \(883\) |
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 {x^{2} + 2 \, x + 6}{{\left (x^{2} + x + 2\right )}^{\frac {1}{3}} {\left (x^{2} + 2\right )} {\left (x + 2\right )}}\,{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 {x^2+2\,x+6}{\left (x^2+2\right )\,\left (x+2\right )\,{\left (x^2+x+2\right )}^{1/3}} \,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 {x^{2} + 2 x + 6}{\left (x + 2\right ) \left (x^{2} + 2\right ) \sqrt [3]{x^{2} + x + 2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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