Optimal. Leaf size=115 \[ \log \left (\sqrt [3]{x^2-x+1}+x-1\right )-\frac {1}{2} \log \left (x^2+\left (x^2-x+1\right )^{2/3}+(1-x) \sqrt [3]{x^2-x+1}-2 x+1\right )-\sqrt {3} \tan ^{-1}\left (\frac {\frac {\sqrt [3]{x^2-x+1}}{\sqrt {3}}-\frac {2 x}{\sqrt {3}}+\frac {2}{\sqrt {3}}}{\sqrt [3]{x^2-x+1}}\right ) \]
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Rubi [F] time = 0.47, 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 {2+x^2}{x \left (2-2 x+x^2\right ) \sqrt [3]{1-x+x^2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {2+x^2}{x \left (2-2 x+x^2\right ) \sqrt [3]{1-x+x^2}} \, dx &=\int \left (\frac {1}{x \sqrt [3]{1-x+x^2}}+\frac {2}{\left (2-2 x+x^2\right ) \sqrt [3]{1-x+x^2}}\right ) \, dx\\ &=2 \int \frac {1}{\left (2-2 x+x^2\right ) \sqrt [3]{1-x+x^2}} \, dx+\int \frac {1}{x \sqrt [3]{1-x+x^2}} \, dx\\ &=2 \int \frac {1}{\left (2-2 x+x^2\right ) \sqrt [3]{1-x+x^2}} \, dx-\frac {\left (\sqrt [3]{\frac {-1-i \sqrt {3}+2 x}{x}} \sqrt [3]{\frac {-1+i \sqrt {3}+2 x}{x}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{x} \sqrt [3]{1-\frac {1}{2} \left (1-i \sqrt {3}\right ) x} \sqrt [3]{1-\frac {1}{2} \left (1+i \sqrt {3}\right ) x}} \, dx,x,\frac {1}{x}\right )}{2^{2/3} \left (\frac {1}{x}\right )^{2/3} \sqrt [3]{1-x+x^2}}\\ &=-\frac {3 \sqrt [3]{-\frac {1-i \sqrt {3}-2 x}{x}} \sqrt [3]{-\frac {1+i \sqrt {3}-2 x}{x}} F_1\left (\frac {2}{3};\frac {1}{3},\frac {1}{3};\frac {5}{3};\frac {1-i \sqrt {3}}{2 x},\frac {1+i \sqrt {3}}{2 x}\right )}{2\ 2^{2/3} \sqrt [3]{1-x+x^2}}+2 \int \frac {1}{\left (2-2 x+x^2\right ) \sqrt [3]{1-x+x^2}} \, dx\\ \end {align*}
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Mathematica [F] time = 0.23, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2+x^2}{x \left (2-2 x+x^2\right ) \sqrt [3]{1-x+x^2}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.13, size = 115, normalized size = 1.00 \begin {gather*} \log \left (\sqrt [3]{x^2-x+1}+x-1\right )-\frac {1}{2} \log \left (x^2+\left (x^2-x+1\right )^{2/3}+(1-x) \sqrt [3]{x^2-x+1}-2 x+1\right )-\sqrt {3} \tan ^{-1}\left (\frac {\frac {\sqrt [3]{x^2-x+1}}{\sqrt {3}}-\frac {2 x}{\sqrt {3}}+\frac {2}{\sqrt {3}}}{\sqrt [3]{x^2-x+1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.07, size = 147, normalized size = 1.28 \begin {gather*} -\sqrt {3} \arctan \left (\frac {4 \, \sqrt {3} {\left (x^{2} - x + 1\right )}^{\frac {2}{3}} {\left (x - 1\right )} + 2 \, \sqrt {3} {\left (x^{2} - x + 1\right )}^{\frac {1}{3}} {\left (x^{2} - 2 \, x + 1\right )} + \sqrt {3} {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}}{x^{3} - 11 \, x^{2} + 11 \, x - 9}\right ) + \frac {1}{2} \, \log \left (\frac {x^{3} - 2 \, x^{2} + 3 \, {\left (x^{2} - x + 1\right )}^{\frac {2}{3}} {\left (x - 1\right )} + 3 \, {\left (x^{2} - x + 1\right )}^{\frac {1}{3}} {\left (x^{2} - 2 \, x + 1\right )} + 2 \, x}{x^{3} - 2 \, x^{2} + 2 \, x}\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}{{\left (x^{2} - x + 1\right )}^{\frac {1}{3}} {\left (x^{2} - 2 \, x + 2\right )} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.85, size = 615, normalized size = 5.35 \begin {gather*} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}-x +1\right )^{\frac {2}{3}} x -\left (x^{2}-x +1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}-x +1\right )^{\frac {2}{3}}+2 x \left (x^{2}-x +1\right )^{\frac {2}{3}}+2 \left (x^{2}-x +1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x -2 \left (x^{2}-x +1\right )^{\frac {1}{3}} x^{2}-3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+x^{3}-2 \left (x^{2}-x +1\right )^{\frac {2}{3}}-\left (x^{2}-x +1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+4 \left (x^{2}-x +1\right )^{\frac {1}{3}} x +3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x -4 x^{2}-2 \left (x^{2}-x +1\right )^{\frac {1}{3}}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+4 x -2}{\left (x^{2}-2 x +2\right ) x}\right )-\ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}-2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+3 x \left (x^{2}-x +1\right )^{\frac {2}{3}}-3 \left (x^{2}-x +1\right )^{\frac {1}{3}} x^{2}-3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x +5 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-3 \left (x^{2}-x +1\right )^{\frac {2}{3}}+6 \left (x^{2}-x +1\right )^{\frac {1}{3}} x +\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-5 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x -2 x^{2}-3 \left (x^{2}-x +1\right )^{\frac {1}{3}}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+2 x -2}{\left (x^{2}-2 x +2\right ) x}\right ) \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-\ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}-2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+3 x \left (x^{2}-x +1\right )^{\frac {2}{3}}-3 \left (x^{2}-x +1\right )^{\frac {1}{3}} x^{2}-3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x +5 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-3 \left (x^{2}-x +1\right )^{\frac {2}{3}}+6 \left (x^{2}-x +1\right )^{\frac {1}{3}} x +\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-5 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x -2 x^{2}-3 \left (x^{2}-x +1\right )^{\frac {1}{3}}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+2 x -2}{\left (x^{2}-2 x +2\right ) x}\right ) \end {gather*}
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}{{\left (x^{2} - x + 1\right )}^{\frac {1}{3}} {\left (x^{2} - 2 \, x + 2\right )} x}\,{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\,{\left (x^2-x+1\right )}^{1/3}\,\left (x^2-2\,x+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 {x^{2} + 2}{x \left (x^{2} - 2 x + 2\right ) \sqrt [3]{x^{2} - x + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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