Optimal. Leaf size=105 \[ \frac {1}{8} \sqrt {\frac {1}{2} \left (4-\sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {4-\sqrt {2}} x}{2 \sqrt {x^6-x^2-1}}\right )-\frac {1}{8} \sqrt {\frac {1}{2} \left (4+\sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {4+\sqrt {2}} x}{2 \sqrt {x^6-x^2-1}}\right ) \]
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Rubi [F] time = 0.61, 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 [F] time = 0.17, size = 0, normalized size = 0.00 \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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IntegrateAlgebraic [A] time = 2.22, size = 105, normalized size = 1.00 \begin {gather*} \frac {1}{8} \sqrt {\frac {1}{2} \left (4-\sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {4-\sqrt {2}} x}{2 \sqrt {x^6-x^2-1}}\right )-\frac {1}{8} \sqrt {\frac {1}{2} \left (4+\sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {4+\sqrt {2}} x}{2 \sqrt {x^6-x^2-1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.65, 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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giac [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}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 3.84, size = 580, normalized size = 5.52 \begin {gather*} -\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 (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 ) 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} \end {gather*}
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 (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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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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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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