Optimal. Leaf size=117 \[ -\log \left (\sqrt [3]{x^6-x^4-x^3+1}+x\right )-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^6-x^4-x^3+1}-x}\right )+\frac {1}{2} \log \left (x^2-\sqrt [3]{x^6-x^4-x^3+1} x+\left (x^6-x^4-x^3+1\right )^{2/3}\right ) \]
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Rubi [F] time = 0.89, 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 {-3-x^4+3 x^6}{\left (1-x^4+x^6\right ) \sqrt [3]{1-x^3-x^4+x^6}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {-3-x^4+3 x^6}{\left (1-x^4+x^6\right ) \sqrt [3]{1-x^3-x^4+x^6}} \, dx &=\int \left (\frac {3}{\sqrt [3]{1-x^3-x^4+x^6}}-\frac {2 \left (3-x^4\right )}{\left (1-x^4+x^6\right ) \sqrt [3]{1-x^3-x^4+x^6}}\right ) \, dx\\ &=-\left (2 \int \frac {3-x^4}{\left (1-x^4+x^6\right ) \sqrt [3]{1-x^3-x^4+x^6}} \, dx\right )+3 \int \frac {1}{\sqrt [3]{1-x^3-x^4+x^6}} \, dx\\ &=-\left (2 \int \left (\frac {3}{\left (1-x^4+x^6\right ) \sqrt [3]{1-x^3-x^4+x^6}}-\frac {x^4}{\left (1-x^4+x^6\right ) \sqrt [3]{1-x^3-x^4+x^6}}\right ) \, dx\right )+3 \int \frac {1}{\sqrt [3]{1-x^3-x^4+x^6}} \, dx\\ &=2 \int \frac {x^4}{\left (1-x^4+x^6\right ) \sqrt [3]{1-x^3-x^4+x^6}} \, dx+3 \int \frac {1}{\sqrt [3]{1-x^3-x^4+x^6}} \, dx-6 \int \frac {1}{\left (1-x^4+x^6\right ) \sqrt [3]{1-x^3-x^4+x^6}} \, dx\\ \end {align*}
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Mathematica [F] time = 0.25, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-3-x^4+3 x^6}{\left (1-x^4+x^6\right ) \sqrt [3]{1-x^3-x^4+x^6}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.42, size = 117, normalized size = 1.00 \begin {gather*} -\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{1-x^3-x^4+x^6}}\right )-\log \left (x+\sqrt [3]{1-x^3-x^4+x^6}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{1-x^3-x^4+x^6}+\left (1-x^3-x^4+x^6\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 2.88, size = 157, normalized size = 1.34 \begin {gather*} -\sqrt {3} \arctan \left (\frac {2 \, \sqrt {3} {\left (x^{6} - x^{4} - x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 2 \, \sqrt {3} {\left (x^{6} - x^{4} - x^{3} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (x^{6} - x^{4} + 1\right )}}{3 \, {\left (x^{6} - x^{4} - 2 \, x^{3} + 1\right )}}\right ) - \frac {1}{2} \, \log \left (\frac {x^{6} - x^{4} + 3 \, {\left (x^{6} - x^{4} - x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{6} - x^{4} - x^{3} + 1\right )}^{\frac {2}{3}} x + 1}{x^{6} - x^{4} + 1}\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 {3 \, x^{6} - x^{4} - 3}{{\left (x^{6} - x^{4} - x^{3} + 1\right )}^{\frac {1}{3}} {\left (x^{6} - x^{4} + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 3.49, size = 340, normalized size = 2.91
method | result | size |
trager | \(-\ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{6}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{4}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-x^{4}-x^{3}+1\right )^{\frac {2}{3}} x +\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-x^{4}-x^{3}+1\right )^{\frac {1}{3}} x^{2}-2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}-\left (x^{6}-x^{4}-x^{3}+1\right )^{\frac {2}{3}} x -2 \left (x^{6}-x^{4}-x^{3}+1\right )^{\frac {1}{3}} x^{2}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )}{x^{6}-x^{4}+1}\right )+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (\frac {x^{6}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-x^{4}-x^{3}+1\right )^{\frac {2}{3}} x +\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-x^{4}-x^{3}+1\right )^{\frac {1}{3}} x^{2}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}-x^{4}-\left (x^{6}-x^{4}-x^{3}+1\right )^{\frac {2}{3}} x -2 \left (x^{6}-x^{4}-x^{3}+1\right )^{\frac {1}{3}} x^{2}-x^{3}+1}{x^{6}-x^{4}+1}\right )\) | \(340\) |
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 {3 \, x^{6} - x^{4} - 3}{{\left (x^{6} - x^{4} - x^{3} + 1\right )}^{\frac {1}{3}} {\left (x^{6} - x^{4} + 1\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 {-3\,x^6+x^4+3}{\left (x^6-x^4+1\right )\,{\left (x^6-x^4-x^3+1\right )}^{1/3}} \,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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