Optimal. Leaf size=67 \[ -\frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt {-2 x^6+x^2+1}}\right )-\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x \sqrt {-2 x^6+x^2+1}}{2 x^6-x^2-1}\right ) \]
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Rubi [F] time = 1.39, 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 {\sqrt {1+x^2-2 x^6} \left (1+4 x^6\right )}{\left (-1-4 x^2+2 x^6\right ) \left (-1-2 x^2+2 x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt {1+x^2-2 x^6} \left (1+4 x^6\right )}{\left (-1-4 x^2+2 x^6\right ) \left (-1-2 x^2+2 x^6\right )} \, dx &=\int \left (\frac {\left (-2+3 x^4\right ) \sqrt {1+x^2-2 x^6}}{-1-4 x^2+2 x^6}+\frac {\left (1-3 x^4\right ) \sqrt {1+x^2-2 x^6}}{-1-2 x^2+2 x^6}\right ) \, dx\\ &=\int \frac {\left (-2+3 x^4\right ) \sqrt {1+x^2-2 x^6}}{-1-4 x^2+2 x^6} \, dx+\int \frac {\left (1-3 x^4\right ) \sqrt {1+x^2-2 x^6}}{-1-2 x^2+2 x^6} \, dx\\ &=\int \left (-\frac {2 \sqrt {1+x^2-2 x^6}}{-1-4 x^2+2 x^6}+\frac {3 x^4 \sqrt {1+x^2-2 x^6}}{-1-4 x^2+2 x^6}\right ) \, dx+\int \left (\frac {\sqrt {1+x^2-2 x^6}}{-1-2 x^2+2 x^6}-\frac {3 x^4 \sqrt {1+x^2-2 x^6}}{-1-2 x^2+2 x^6}\right ) \, dx\\ &=-\left (2 \int \frac {\sqrt {1+x^2-2 x^6}}{-1-4 x^2+2 x^6} \, dx\right )+3 \int \frac {x^4 \sqrt {1+x^2-2 x^6}}{-1-4 x^2+2 x^6} \, dx-3 \int \frac {x^4 \sqrt {1+x^2-2 x^6}}{-1-2 x^2+2 x^6} \, dx+\int \frac {\sqrt {1+x^2-2 x^6}}{-1-2 x^2+2 x^6} \, dx\\ \end {align*}
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Mathematica [F] time = 0.31, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {1+x^2-2 x^6} \left (1+4 x^6\right )}{\left (-1-4 x^2+2 x^6\right ) \left (-1-2 x^2+2 x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 2.80, size = 67, normalized size = 1.00 \begin {gather*} -\frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt {1+x^2-2 x^6}}\right )-\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x \sqrt {1+x^2-2 x^6}}{-1-x^2+2 x^6}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 66, normalized size = 0.99 \begin {gather*} -\frac {1}{4} \, \sqrt {3} \arctan \left (\frac {2 \, \sqrt {3} \sqrt {-2 \, x^{6} + x^{2} + 1} x}{2 \, x^{6} + 2 \, x^{2} - 1}\right ) + \frac {1}{4} \, \arctan \left (\frac {2 \, \sqrt {-2 \, x^{6} + x^{2} + 1} x}{2 \, x^{6} - 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 {{\left (4 \, x^{6} + 1\right )} \sqrt {-2 \, x^{6} + x^{2} + 1}}{{\left (2 \, x^{6} - 2 \, x^{2} - 1\right )} {\left (2 \, x^{6} - 4 \, x^{2} - 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.58, size = 132, normalized size = 1.97
method | result | size |
trager | \(-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{6}-2 \sqrt {-2 x^{6}+x^{2}+1}\, x -\RootOf \left (\textit {\_Z}^{2}+1\right )}{2 x^{6}-2 x^{2}-1}\right )}{4}-\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) \ln \left (-\frac {2 \RootOf \left (\textit {\_Z}^{2}+3\right ) x^{6}+2 \RootOf \left (\textit {\_Z}^{2}+3\right ) x^{2}+6 \sqrt {-2 x^{6}+x^{2}+1}\, x -\RootOf \left (\textit {\_Z}^{2}+3\right )}{2 x^{6}-4 x^{2}-1}\right )}{4}\) | \(132\) |
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 (4 \, x^{6} + 1\right )} \sqrt {-2 \, x^{6} + x^{2} + 1}}{{\left (2 \, x^{6} - 2 \, x^{2} - 1\right )} {\left (2 \, x^{6} - 4 \, x^{2} - 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 {\left (4\,x^6+1\right )\,\sqrt {-2\,x^6+x^2+1}}{\left (-2\,x^6+2\,x^2+1\right )\,\left (-2\,x^6+4\,x^2+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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