Optimal. Leaf size=95 \[ -\frac {1}{4} \sqrt {\frac {1}{3} \left (1+\sqrt {3}\right )} \text {ArcTan}\left (\frac {\sqrt {1+\sqrt {3}} x}{\sqrt {-2-x^2+x^6}}\right )-\frac {1}{4} \sqrt {\frac {1}{3} \left (-1+\sqrt {3}\right )} \tanh ^{-1}\left (\frac {\sqrt {-1+\sqrt {3}} x}{\sqrt {-2-x^2+x^6}}\right ) \]
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Rubi [F]
time = 0.35, 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 {\left (1+x^6\right ) \sqrt {-2-x^2+x^6}}{4-3 x^4-4 x^6+x^{12}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (1+x^6\right ) \sqrt {-2-x^2+x^6}}{4-3 x^4-4 x^6+x^{12}} \, dx &=\int \left (\frac {\sqrt {-2-x^2+x^6}}{4-3 x^4-4 x^6+x^{12}}+\frac {x^6 \sqrt {-2-x^2+x^6}}{4-3 x^4-4 x^6+x^{12}}\right ) \, dx\\ &=\int \frac {\sqrt {-2-x^2+x^6}}{4-3 x^4-4 x^6+x^{12}} \, dx+\int \frac {x^6 \sqrt {-2-x^2+x^6}}{4-3 x^4-4 x^6+x^{12}} \, dx\\ \end {align*}
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Mathematica [A]
time = 1.27, size = 90, normalized size = 0.95 \begin {gather*} -\frac {\sqrt {1+\sqrt {3}} \text {ArcTan}\left (\frac {\sqrt {1+\sqrt {3}} x}{\sqrt {-2-x^2+x^6}}\right )+\sqrt {-1+\sqrt {3}} \tanh ^{-1}\left (\frac {\sqrt {-1+\sqrt {3}} x}{\sqrt {-2-x^2+x^6}}\right )}{4 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 2.26, size = 579, normalized size = 6.09
method | result | size |
trager | \(-\RootOf \left (18432 \textit {\_Z}^{4}+192 \textit {\_Z}^{2}-1\right ) \ln \left (-\frac {384 \RootOf \left (18432 \textit {\_Z}^{4}+192 \textit {\_Z}^{2}-1\right )^{3} x^{6}+73728 \RootOf \left (18432 \textit {\_Z}^{4}+192 \textit {\_Z}^{2}-1\right )^{5} x^{2}-4 \RootOf \left (18432 \textit {\_Z}^{4}+192 \textit {\_Z}^{2}-1\right ) x^{6}-1152 \RootOf \left (18432 \textit {\_Z}^{4}+192 \textit {\_Z}^{2}-1\right )^{3} x^{2}+192 \sqrt {x^{6}-x^{2}-2}\, \RootOf \left (18432 \textit {\_Z}^{4}+192 \textit {\_Z}^{2}-1\right )^{2} x -768 \RootOf \left (18432 \textit {\_Z}^{4}+192 \textit {\_Z}^{2}-1\right )^{3}+4 \RootOf \left (18432 \textit {\_Z}^{4}+192 \textit {\_Z}^{2}-1\right ) x^{2}-\sqrt {x^{6}-x^{2}-2}\, x +8 \RootOf \left (18432 \textit {\_Z}^{4}+192 \textit {\_Z}^{2}-1\right )}{-x^{6}+192 x^{2} \RootOf \left (18432 \textit {\_Z}^{4}+192 \textit {\_Z}^{2}-1\right )^{2}+x^{2}+2}\right )+\frac {\RootOf \left (\textit {\_Z}^{2}+576 \RootOf \left (18432 \textit {\_Z}^{4}+192 \textit {\_Z}^{2}-1\right )^{2}+6\right ) \ln \left (\frac {-48 \RootOf \left (\textit {\_Z}^{2}+576 \RootOf \left (18432 \textit {\_Z}^{4}+192 \textit {\_Z}^{2}-1\right )^{2}+6\right ) \RootOf \left (18432 \textit {\_Z}^{4}+192 \textit {\_Z}^{2}-1\right )^{2} x^{6}+9216 \RootOf \left (18432 \textit {\_Z}^{4}+192 \textit {\_Z}^{2}-1\right )^{4} \RootOf \left (\textit {\_Z}^{2}+576 \RootOf \left (18432 \textit {\_Z}^{4}+192 \textit {\_Z}^{2}-1\right )^{2}+6\right ) x^{2}-\RootOf \left (\textit {\_Z}^{2}+576 \RootOf \left (18432 \textit {\_Z}^{4}+192 \textit {\_Z}^{2}-1\right )^{2}+6\right ) x^{6}+336 \RootOf \left (18432 \textit {\_Z}^{4}+192 \textit {\_Z}^{2}-1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+576 \RootOf \left (18432 \textit {\_Z}^{4}+192 \textit {\_Z}^{2}-1\right )^{2}+6\right ) x^{2}+576 \sqrt {x^{6}-x^{2}-2}\, \RootOf \left (18432 \textit {\_Z}^{4}+192 \textit {\_Z}^{2}-1\right )^{2} x +96 \RootOf \left (18432 \textit {\_Z}^{4}+192 \textit {\_Z}^{2}-1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+576 \RootOf \left (18432 \textit {\_Z}^{4}+192 \textit {\_Z}^{2}-1\right )^{2}+6\right )+3 \RootOf \left (\textit {\_Z}^{2}+576 \RootOf \left (18432 \textit {\_Z}^{4}+192 \textit {\_Z}^{2}-1\right )^{2}+6\right ) x^{2}+9 \sqrt {x^{6}-x^{2}-2}\, x +2 \RootOf \left (\textit {\_Z}^{2}+576 \RootOf \left (18432 \textit {\_Z}^{4}+192 \textit {\_Z}^{2}-1\right )^{2}+6\right )}{x^{6}+192 x^{2} \RootOf \left (18432 \textit {\_Z}^{4}+192 \textit {\_Z}^{2}-1\right )^{2}+x^{2}-2}\right )}{24}\) | \(579\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 438 vs.
\(2 (67) = 134\).
time = 0.54, size = 438, normalized size = 4.61 \begin {gather*} -\frac {1}{12} \, \sqrt {3} \sqrt {\sqrt {3} + 1} \arctan \left (\frac {4 \, {\left (x^{7} + \sqrt {3} x^{3} - 2 \, x^{3} - 2 \, x\right )} \sqrt {x^{6} - x^{2} - 2} \sqrt {\sqrt {3} + 1} - {\left (2 \, x^{12} - 10 \, x^{8} - 8 \, x^{6} + 12 \, x^{4} + 20 \, x^{2} - \sqrt {3} {\left (x^{12} - 6 \, x^{8} - 4 \, x^{6} + 7 \, x^{4} + 12 \, x^{2} + 4\right )} + 8\right )} \sqrt {6 \, \sqrt {3} + 10} \sqrt {\sqrt {3} + 1}}{2 \, {\left (x^{12} - 4 \, x^{8} - 4 \, x^{6} + x^{4} + 8 \, x^{2} + 4\right )}}\right ) + \frac {1}{48} \, \sqrt {3} \sqrt {\sqrt {3} - 1} \log \left (\frac {4 \, {\left (2 \, x^{7} - 3 \, x^{3} - \sqrt {3} {\left (x^{7} - 2 \, x^{3} - 2 \, x\right )} - 4 \, x\right )} \sqrt {x^{6} - x^{2} - 2} + {\left (x^{12} - 8 \, x^{8} - 4 \, x^{6} + 9 \, x^{4} + 16 \, x^{2} - \sqrt {3} {\left (x^{12} - 4 \, x^{8} - 4 \, x^{6} + 5 \, x^{4} + 8 \, x^{2} + 4\right )} + 4\right )} \sqrt {\sqrt {3} - 1}}{x^{12} - 4 \, x^{6} - 3 \, x^{4} + 4}\right ) - \frac {1}{48} \, \sqrt {3} \sqrt {\sqrt {3} - 1} \log \left (\frac {4 \, {\left (2 \, x^{7} - 3 \, x^{3} - \sqrt {3} {\left (x^{7} - 2 \, x^{3} - 2 \, x\right )} - 4 \, x\right )} \sqrt {x^{6} - x^{2} - 2} - {\left (x^{12} - 8 \, x^{8} - 4 \, x^{6} + 9 \, x^{4} + 16 \, x^{2} - \sqrt {3} {\left (x^{12} - 4 \, x^{8} - 4 \, x^{6} + 5 \, x^{4} + 8 \, x^{2} + 4\right )} + 4\right )} \sqrt {\sqrt {3} - 1}}{x^{12} - 4 \, x^{6} - 3 \, x^{4} + 4}\right ) \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 (x^{2} + 1\right ) \left (x^{4} - x^{2} + 1\right ) \sqrt {x^{6} - x^{2} - 2}}{x^{12} - 4 x^{6} - 3 x^{4} + 4}\, dx \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*} \text {could not integrate} \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^6+1\right )\,\sqrt {x^6-x^2-2}}{-x^{12}+4\,x^6+3\,x^4-4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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