Optimal. Leaf size=64 \[ \frac {\sqrt {-1+x^6} \left (1+2 x^6\right )}{6 x^6}-\frac {1}{6} \text {ArcTan}\left (\sqrt {-1+x^6}\right )-\frac {\text {ArcTan}\left (\frac {2 \sqrt {-1+x^6}}{\sqrt {3}}\right )}{2 \sqrt {3}} \]
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Rubi [A]
time = 0.06, antiderivative size = 70, normalized size of antiderivative = 1.09, number of steps
used = 9, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {587, 186, 43,
65, 209, 52} \begin {gather*} -\frac {1}{6} \text {ArcTan}\left (\sqrt {x^6-1}\right )-\frac {\text {ArcTan}\left (\frac {2 \sqrt {x^6-1}}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {\sqrt {x^6-1}}{6 x^6}+\frac {\sqrt {x^6-1}}{3} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 52
Rule 65
Rule 186
Rule 209
Rule 587
Rubi steps
\begin {align*} \int \frac {\sqrt {-1+x^6} \left (-1+2 x^6\right )^2}{x^7 \left (-1+4 x^6\right )} \, dx &=\frac {1}{6} \text {Subst}\left (\int \frac {\sqrt {-1+x} (-1+2 x)^2}{x^2 (-1+4 x)} \, dx,x,x^6\right )\\ &=\frac {1}{6} \text {Subst}\left (\int \left (-\frac {\sqrt {-1+x}}{x^2}+\frac {4 \sqrt {-1+x}}{-1+4 x}\right ) \, dx,x,x^6\right )\\ &=-\left (\frac {1}{6} \text {Subst}\left (\int \frac {\sqrt {-1+x}}{x^2} \, dx,x,x^6\right )\right )+\frac {2}{3} \text {Subst}\left (\int \frac {\sqrt {-1+x}}{-1+4 x} \, dx,x,x^6\right )\\ &=\frac {1}{3} \sqrt {-1+x^6}+\frac {\sqrt {-1+x^6}}{6 x^6}-\frac {1}{12} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^6\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} (-1+4 x)} \, dx,x,x^6\right )\\ &=\frac {1}{3} \sqrt {-1+x^6}+\frac {\sqrt {-1+x^6}}{6 x^6}-\frac {1}{6} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^6}\right )-\text {Subst}\left (\int \frac {1}{3+4 x^2} \, dx,x,\sqrt {-1+x^6}\right )\\ &=\frac {1}{3} \sqrt {-1+x^6}+\frac {\sqrt {-1+x^6}}{6 x^6}-\frac {1}{6} \tan ^{-1}\left (\sqrt {-1+x^6}\right )-\frac {\tan ^{-1}\left (\frac {2 \sqrt {-1+x^6}}{\sqrt {3}}\right )}{2 \sqrt {3}}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 61, normalized size = 0.95 \begin {gather*} \frac {1}{6} \left (\frac {\sqrt {-1+x^6} \left (1+2 x^6\right )}{x^6}-\text {ArcTan}\left (\sqrt {-1+x^6}\right )-\sqrt {3} \text {ArcTan}\left (\frac {2 \sqrt {-1+x^6}}{\sqrt {3}}\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 = 2.51, size = 107, normalized size = 1.67
method | result | size |
trager | \(\frac {\sqrt {x^{6}-1}\, \left (2 x^{6}+1\right )}{6 x^{6}}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{6}+\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) \ln \left (\frac {4 \RootOf \left (\textit {\_Z}^{2}+3\right ) x^{6}+12 \sqrt {x^{6}-1}-7 \RootOf \left (\textit {\_Z}^{2}+3\right )}{\left (2 x^{3}+1\right ) \left (2 x^{3}-1\right )}\right )}{12}\) | \(107\) |
risch | \(\frac {\sqrt {x^{6}-1}}{6 x^{6}}+\frac {\sqrt {x^{6}-1}}{3}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{6}-\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) \ln \left (\frac {-4 \RootOf \left (\textit {\_Z}^{2}+3\right ) x^{6}+12 \sqrt {x^{6}-1}+7 \RootOf \left (\textit {\_Z}^{2}+3\right )}{\left (2 x^{3}+1\right ) \left (2 x^{3}-1\right )}\right )}{12}\) | \(107\) |
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 = 0.35, size = 54, normalized size = 0.84 \begin {gather*} -\frac {\sqrt {3} x^{6} \arctan \left (\frac {2}{3} \, \sqrt {3} \sqrt {x^{6} - 1}\right ) + x^{6} \arctan \left (\sqrt {x^{6} - 1}\right ) - {\left (2 \, x^{6} + 1\right )} \sqrt {x^{6} - 1}}{6 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 25.23, size = 60, normalized size = 0.94 \begin {gather*} \frac {\sqrt {x^{6} - 1}}{3} - \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt {x^{6} - 1}}{3} \right )}}{6} - \frac {\operatorname {atan}{\left (\sqrt {x^{6} - 1} \right )}}{6} + \frac {\sqrt {x^{6} - 1}}{6 x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 50, normalized size = 0.78 \begin {gather*} -\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \sqrt {x^{6} - 1}\right ) + \frac {1}{3} \, \sqrt {x^{6} - 1} + \frac {\sqrt {x^{6} - 1}}{6 \, x^{6}} - \frac {1}{6} \, \arctan \left (\sqrt {x^{6} - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.98, size = 50, normalized size = 0.78 \begin {gather*} \frac {\sqrt {x^6-1}}{3}-\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {2\,\sqrt {3}\,\sqrt {x^6-1}}{3}\right )}{6}-\frac {\mathrm {atan}\left (\sqrt {x^6-1}\right )}{6}+\frac {\sqrt {x^6-1}}{6\,x^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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