Optimal. Leaf size=79 \[ -\frac {\text {ArcTan}\left (\frac {\sqrt {2} x \sqrt {1-x^6}}{-1+x^2+x^6}\right )}{2 \sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {1-x^6}}{-1-x^2+x^6}\right )}{2 \sqrt {2}} \]
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Rubi [F]
time = 0.33, 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^6} \left (1+2 x^6\right )}{1+x^4-2 x^6+x^{12}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt {1-x^6} \left (1+2 x^6\right )}{1+x^4-2 x^6+x^{12}} \, dx &=\int \left (\frac {\sqrt {1-x^6}}{1+x^4-2 x^6+x^{12}}+\frac {2 x^6 \sqrt {1-x^6}}{1+x^4-2 x^6+x^{12}}\right ) \, dx\\ &=2 \int \frac {x^6 \sqrt {1-x^6}}{1+x^4-2 x^6+x^{12}} \, dx+\int \frac {\sqrt {1-x^6}}{1+x^4-2 x^6+x^{12}} \, dx\\ \end {align*}
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Mathematica [A]
time = 3.29, size = 60, normalized size = 0.76 \begin {gather*} -\frac {\text {ArcTan}\left (\frac {x \sqrt {2-2 x^6}}{-1+x^2+x^6}\right )+\tanh ^{-1}\left (\frac {-1-x^2+x^6}{x \sqrt {2-2 x^6}}\right )}{2 \sqrt {2}} \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 = 1.59, size = 153, normalized size = 1.94
method | result | size |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{6}+\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{2}-2 \sqrt {-x^{6}+1}\, x -\RootOf \left (\textit {\_Z}^{4}+1\right )}{-x^{6}+\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+1}\right )}{4}-\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{6}+\RootOf \left (\textit {\_Z}^{4}+1\right )^{5} x^{2}+\RootOf \left (\textit {\_Z}^{4}+1\right )^{3}-2 \sqrt {-x^{6}+1}\, x}{x^{6}+\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{2}-1}\right )}{4}\) | \(153\) |
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. 512 vs.
\(2 (63) = 126\).
time = 0.48, size = 512, normalized size = 6.48 \begin {gather*} \frac {1}{4} \, \sqrt {2} \arctan \left (-\frac {x^{12} - 2 \, x^{6} + x^{4} + 2 \, \sqrt {2} {\left (x^{7} + x^{3} - x\right )} \sqrt {-x^{6} + 1} - {\left (4 \, \sqrt {-x^{6} + 1} x^{3} - \sqrt {2} {\left (x^{12} + 2 \, x^{8} - 2 \, x^{6} - x^{4} - 2 \, x^{2} + 1\right )}\right )} \sqrt {\frac {x^{12} - 4 \, x^{8} - 2 \, x^{6} + x^{4} + 2 \, \sqrt {2} {\left (x^{7} - x^{3} - x\right )} \sqrt {-x^{6} + 1} + 4 \, x^{2} + 1}{x^{12} - 2 \, x^{6} + x^{4} + 1}} + 1}{x^{12} + 4 \, x^{8} - 2 \, x^{6} + x^{4} - 4 \, x^{2} + 1}\right ) - \frac {1}{4} \, \sqrt {2} \arctan \left (-\frac {x^{12} - 2 \, x^{6} + x^{4} - 2 \, \sqrt {2} {\left (x^{7} + x^{3} - x\right )} \sqrt {-x^{6} + 1} - {\left (4 \, \sqrt {-x^{6} + 1} x^{3} + \sqrt {2} {\left (x^{12} + 2 \, x^{8} - 2 \, x^{6} - x^{4} - 2 \, x^{2} + 1\right )}\right )} \sqrt {\frac {x^{12} - 4 \, x^{8} - 2 \, x^{6} + x^{4} - 2 \, \sqrt {2} {\left (x^{7} - x^{3} - x\right )} \sqrt {-x^{6} + 1} + 4 \, x^{2} + 1}{x^{12} - 2 \, x^{6} + x^{4} + 1}} + 1}{x^{12} + 4 \, x^{8} - 2 \, x^{6} + x^{4} - 4 \, x^{2} + 1}\right ) - \frac {1}{16} \, \sqrt {2} \log \left (\frac {4 \, {\left (x^{12} - 4 \, x^{8} - 2 \, x^{6} + x^{4} + 2 \, \sqrt {2} {\left (x^{7} - x^{3} - x\right )} \sqrt {-x^{6} + 1} + 4 \, x^{2} + 1\right )}}{x^{12} - 2 \, x^{6} + x^{4} + 1}\right ) + \frac {1}{16} \, \sqrt {2} \log \left (\frac {4 \, {\left (x^{12} - 4 \, x^{8} - 2 \, x^{6} + x^{4} - 2 \, \sqrt {2} {\left (x^{7} - x^{3} - x\right )} \sqrt {-x^{6} + 1} + 4 \, x^{2} + 1\right )}}{x^{12} - 2 \, x^{6} + x^{4} + 1}\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 {\sqrt {- \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )} \left (2 x^{6} + 1\right )}{x^{12} - 2 x^{6} + x^{4} + 1}\, 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 {\sqrt {1-x^6}\,\left (2\,x^6+1\right )}{x^{12}-2\,x^6+x^4+1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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