Optimal. Leaf size=29 \[ -\frac {\text {ArcTan}\left (\frac {\sqrt {2} x}{\sqrt {-1-2 x^2+x^6}}\right )}{\sqrt {2}} \]
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Rubi [F]
time = 0.78, 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 {1+2 x^6}{\left (-1+x^6\right ) \sqrt {-1-2 x^2+x^6}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1+2 x^6}{\left (-1+x^6\right ) \sqrt {-1-2 x^2+x^6}} \, dx &=\int \left (\frac {2}{\sqrt {-1-2 x^2+x^6}}+\frac {3}{\left (-1+x^6\right ) \sqrt {-1-2 x^2+x^6}}\right ) \, dx\\ &=2 \int \frac {1}{\sqrt {-1-2 x^2+x^6}} \, dx+3 \int \frac {1}{\left (-1+x^6\right ) \sqrt {-1-2 x^2+x^6}} \, dx\\ &=2 \int \frac {1}{\sqrt {-1-2 x^2+x^6}} \, dx+3 \int \left (\frac {1}{3 \left (-1+x^2\right ) \sqrt {-1-2 x^2+x^6}}+\frac {-2+x}{6 \left (1-x+x^2\right ) \sqrt {-1-2 x^2+x^6}}+\frac {-2-x}{6 \left (1+x+x^2\right ) \sqrt {-1-2 x^2+x^6}}\right ) \, dx\\ &=\frac {1}{2} \int \frac {-2+x}{\left (1-x+x^2\right ) \sqrt {-1-2 x^2+x^6}} \, dx+\frac {1}{2} \int \frac {-2-x}{\left (1+x+x^2\right ) \sqrt {-1-2 x^2+x^6}} \, dx+2 \int \frac {1}{\sqrt {-1-2 x^2+x^6}} \, dx+\int \frac {1}{\left (-1+x^2\right ) \sqrt {-1-2 x^2+x^6}} \, dx\\ &=\frac {1}{2} \int \left (\frac {1+i \sqrt {3}}{\left (-1-i \sqrt {3}+2 x\right ) \sqrt {-1-2 x^2+x^6}}+\frac {1-i \sqrt {3}}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt {-1-2 x^2+x^6}}\right ) \, dx+\frac {1}{2} \int \left (\frac {-1+i \sqrt {3}}{\left (1-i \sqrt {3}+2 x\right ) \sqrt {-1-2 x^2+x^6}}+\frac {-1-i \sqrt {3}}{\left (1+i \sqrt {3}+2 x\right ) \sqrt {-1-2 x^2+x^6}}\right ) \, dx+2 \int \frac {1}{\sqrt {-1-2 x^2+x^6}} \, dx+\int \left (\frac {1}{2 (-1+x) \sqrt {-1-2 x^2+x^6}}-\frac {1}{2 (1+x) \sqrt {-1-2 x^2+x^6}}\right ) \, dx\\ &=\frac {1}{2} \int \frac {1}{(-1+x) \sqrt {-1-2 x^2+x^6}} \, dx-\frac {1}{2} \int \frac {1}{(1+x) \sqrt {-1-2 x^2+x^6}} \, dx+2 \int \frac {1}{\sqrt {-1-2 x^2+x^6}} \, dx+\frac {1}{2} \left (-1-i \sqrt {3}\right ) \int \frac {1}{\left (1+i \sqrt {3}+2 x\right ) \sqrt {-1-2 x^2+x^6}} \, dx+\frac {1}{2} \left (1-i \sqrt {3}\right ) \int \frac {1}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt {-1-2 x^2+x^6}} \, dx+\frac {1}{2} \left (-1+i \sqrt {3}\right ) \int \frac {1}{\left (1-i \sqrt {3}+2 x\right ) \sqrt {-1-2 x^2+x^6}} \, dx+\frac {1}{2} \left (1+i \sqrt {3}\right ) \int \frac {1}{\left (-1-i \sqrt {3}+2 x\right ) \sqrt {-1-2 x^2+x^6}} \, dx\\ \end {align*}
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Mathematica [A]
time = 0.67, size = 29, normalized size = 1.00 \begin {gather*} -\frac {\text {ArcTan}\left (\frac {\sqrt {2} x}{\sqrt {-1-2 x^2+x^6}}\right )}{\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 = 0.60, size = 85, normalized size = 2.93
method | result | size |
trager | \(-\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) x^{6}-4 \RootOf \left (\textit {\_Z}^{2}+2\right ) x^{2}+4 \sqrt {x^{6}-2 x^{2}-1}\, x -\RootOf \left (\textit {\_Z}^{2}+2\right )}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right )}\right )}{4}\) | \(85\) |
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 [A]
time = 0.38, size = 36, normalized size = 1.24 \begin {gather*} -\frac {1}{4} \, \sqrt {2} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {x^{6} - 2 \, x^{2} - 1} x}{x^{6} - 4 \, x^{2} - 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 {2 x^{6} + 1}{\sqrt {\left (x^{2} + 1\right ) \left (x^{4} - x^{2} - 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )}\, 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 = -0.03 \begin {gather*} \int \frac {2\,x^6+1}{\left (x^6-1\right )\,\sqrt {x^6-2\,x^2-1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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