Optimal. Leaf size=105 \[ \frac {1}{8} \sqrt {\frac {1}{2} \left (4-\sqrt {2}\right )} \text {ArcTan}\left (\frac {\sqrt {4-\sqrt {2}} x}{2 \sqrt {-1-x^2+x^6}}\right )-\frac {1}{8} \sqrt {\frac {1}{2} \left (4+\sqrt {2}\right )} \text {ArcTan}\left (\frac {\sqrt {4+\sqrt {2}} x}{2 \sqrt {-1-x^2+x^6}}\right ) \]
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Rubi [F]
time = 0.38, 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+x^6} \left (1+2 x^6\right )}{8-x^4-16 x^6+8 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^2+x^6} \left (1+2 x^6\right )}{8-x^4-16 x^6+8 x^{12}} \, dx &=\int \left (\frac {\sqrt {-1-x^2+x^6}}{8-x^4-16 x^6+8 x^{12}}+\frac {2 x^6 \sqrt {-1-x^2+x^6}}{8-x^4-16 x^6+8 x^{12}}\right ) \, dx\\ &=2 \int \frac {x^6 \sqrt {-1-x^2+x^6}}{8-x^4-16 x^6+8 x^{12}} \, dx+\int \frac {\sqrt {-1-x^2+x^6}}{8-x^4-16 x^6+8 x^{12}} \, dx\\ \end {align*}
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Mathematica [A]
time = 1.33, size = 95, normalized size = 0.90 \begin {gather*} \frac {1}{16} \left (\sqrt {8-2 \sqrt {2}} \text {ArcTan}\left (\frac {x}{2 \sqrt {\frac {1+x^2-x^6}{-4+\sqrt {2}}}}\right )-\sqrt {2 \left (4+\sqrt {2}\right )} \text {ArcTan}\left (\frac {\sqrt {4+\sqrt {2}} x}{2 \sqrt {-1-x^2+x^6}}\right )\right ) \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 = 3.80, size = 578, normalized size = 5.50
method | result | size |
trager | \(\RootOf \left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right ) \ln \left (\frac {16384 \RootOf \left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{3} x^{6}+2097152 \RootOf \left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{5} x^{2}+112 \RootOf \left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right ) x^{6}-2048 \RootOf \left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{3} x^{2}-2048 \sqrt {x^{6}-x^{2}-1}\, \RootOf \left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{2} x -16384 \RootOf \left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{3}-112 \RootOf \left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right ) x^{2}-7 \sqrt {x^{6}-x^{2}-1}\, x -112 \RootOf \left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )}{-x^{6}+128 x^{2} \RootOf \left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{2}+x^{2}+1}\right )-\frac {\RootOf \left (\textit {\_Z}^{2}+64 \RootOf \left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{2}+1\right ) \ln \left (-\frac {-2048 \RootOf \left (\textit {\_Z}^{2}+64 \RootOf \left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{2}+1\right ) \RootOf \left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{2} x^{6}+262144 \RootOf \left (\textit {\_Z}^{2}+64 \RootOf \left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{2}+1\right ) \RootOf \left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{4} x^{2}-18 \RootOf \left (\textit {\_Z}^{2}+64 \RootOf \left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{2}+1\right ) x^{6}+8448 \RootOf \left (\textit {\_Z}^{2}+64 \RootOf \left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{2}+1\right ) \RootOf \left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{2} x^{2}-2048 \sqrt {x^{6}-x^{2}-1}\, \RootOf \left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{2} x +2048 \RootOf \left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{2} \RootOf \left (\textit {\_Z}^{2}+64 \RootOf \left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{2}+1\right )+54 \RootOf \left (\textit {\_Z}^{2}+64 \RootOf \left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{2}+1\right ) x^{2}-25 \sqrt {x^{6}-x^{2}-1}\, x +18 \RootOf \left (\textit {\_Z}^{2}+64 \RootOf \left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{2}+1\right )}{x^{6}+128 x^{2} \RootOf \left (131072 \textit {\_Z}^{4}+2048 \textit {\_Z}^{2}+7\right )^{2}+x^{2}-1}\right )}{8}\) | \(578\) |
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. 336 vs.
\(2 (71) = 142\).
time = 0.48, size = 336, normalized size = 3.20 \begin {gather*} -\frac {1}{16} \, \sqrt {2} \sqrt {\sqrt {2} + 4} \arctan \left (\frac {196 \, {\left (4 \, x^{7} + \sqrt {2} x^{3} - 8 \, x^{3} - 4 \, x\right )} \sqrt {x^{6} - x^{2} - 1} \sqrt {\sqrt {2} + 4} - {\left (72 \, x^{12} - 176 \, x^{8} - 144 \, x^{6} + 41 \, x^{4} + 176 \, x^{2} - 4 \, \sqrt {2} {\left (8 \, x^{12} - 25 \, x^{8} - 16 \, x^{6} + 10 \, x^{4} + 25 \, x^{2} + 8\right )} + 72\right )} \sqrt {50 \, \sqrt {2} + 88} \sqrt {\sqrt {2} + 4}}{98 \, {\left (8 \, x^{12} - 32 \, x^{8} - 16 \, x^{6} + 31 \, x^{4} + 32 \, x^{2} + 8\right )}}\right ) - \frac {1}{16} \, \sqrt {2} \sqrt {-\sqrt {2} + 4} \arctan \left (-\frac {196 \, {\left (4 \, x^{7} - \sqrt {2} x^{3} - 8 \, x^{3} - 4 \, x\right )} \sqrt {x^{6} - x^{2} - 1} \sqrt {-\sqrt {2} + 4} - {\left (72 \, x^{12} - 176 \, x^{8} - 144 \, x^{6} + 41 \, x^{4} + 176 \, x^{2} + 4 \, \sqrt {2} {\left (8 \, x^{12} - 25 \, x^{8} - 16 \, x^{6} + 10 \, x^{4} + 25 \, x^{2} + 8\right )} + 72\right )} \sqrt {-\sqrt {2} + 4} \sqrt {-50 \, \sqrt {2} + 88}}{98 \, {\left (8 \, x^{12} - 32 \, x^{8} - 16 \, x^{6} + 31 \, x^{4} + 32 \, x^{2} + 8\right )}}\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 (2 x^{6} + 1\right ) \sqrt {x^{6} - x^{2} - 1}}{8 x^{12} - 16 x^{6} - x^{4} + 8}\, 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 (2\,x^6+1\right )\,\sqrt {x^6-x^2-1}}{-8\,x^{12}+16\,x^6+x^4-8} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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