Optimal. Leaf size=53 \[ \frac {\sqrt {-1+x^6}}{6 x^6}-\frac {1}{2} \text {ArcTan}\left (\sqrt {-1+x^6}\right )+\frac {\text {ArcTan}\left (\frac {\sqrt {-1+x^6}}{\sqrt {3}}\right )}{\sqrt {3}} \]
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Rubi [A]
time = 0.03, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {587, 154, 162,
65, 209} \begin {gather*} -\frac {1}{2} \text {ArcTan}\left (\sqrt {x^6-1}\right )+\frac {\text {ArcTan}\left (\frac {\sqrt {x^6-1}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\sqrt {x^6-1}}{6 x^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 154
Rule 162
Rule 209
Rule 587
Rubi steps
\begin {align*} \int \frac {\left (-2+x^6\right ) \sqrt {-1+x^6}}{x^7 \left (2+x^6\right )} \, dx &=\frac {1}{6} \text {Subst}\left (\int \frac {(-2+x) \sqrt {-1+x}}{x^2 (2+x)} \, dx,x,x^6\right )\\ &=\frac {\sqrt {-1+x^6}}{6 x^6}+\frac {1}{12} \text {Subst}\left (\int \frac {-6+3 x}{\sqrt {-1+x} x (2+x)} \, dx,x,x^6\right )\\ &=\frac {\sqrt {-1+x^6}}{6 x^6}-\frac {1}{4} \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} (2+x)} \, dx,x,x^6\right )\\ &=\frac {\sqrt {-1+x^6}}{6 x^6}-\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^6}\right )+\text {Subst}\left (\int \frac {1}{3+x^2} \, dx,x,\sqrt {-1+x^6}\right )\\ &=\frac {\sqrt {-1+x^6}}{6 x^6}-\frac {1}{2} \tan ^{-1}\left (\sqrt {-1+x^6}\right )+\frac {\tan ^{-1}\left (\frac {\sqrt {-1+x^6}}{\sqrt {3}}\right )}{\sqrt {3}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 53, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-1+x^6}}{6 x^6}-\frac {1}{2} \text {ArcTan}\left (\sqrt {-1+x^6}\right )+\frac {\text {ArcTan}\left (\frac {\sqrt {-1+x^6}}{\sqrt {3}}\right )}{\sqrt {3}} \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 = 2.18, size = 87, normalized size = 1.64
method | result | size |
trager | \(\frac {\sqrt {x^{6}-1}}{6 x^{6}}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{2}+\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+3\right ) x^{6}+6 \sqrt {x^{6}-1}+4 \RootOf \left (\textit {\_Z}^{2}+3\right )}{x^{6}+2}\right )}{6}\) | \(87\) |
risch | \(\frac {\sqrt {x^{6}-1}}{6 x^{6}}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{2}+\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+3\right ) x^{6}+6 \sqrt {x^{6}-1}+4 \RootOf \left (\textit {\_Z}^{2}+3\right )}{x^{6}+2}\right )}{6}\) | \(89\) |
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 = 0.51, size = 47, normalized size = 0.89 \begin {gather*} \frac {2 \, \sqrt {3} x^{6} \arctan \left (\frac {1}{3} \, \sqrt {3} \sqrt {x^{6} - 1}\right ) - 3 \, x^{6} \arctan \left (\sqrt {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 = 14.85, size = 48, normalized size = 0.91 \begin {gather*} \frac {\sqrt {3} \operatorname {atan}{\left (\frac {\sqrt {3} \sqrt {x^{6} - 1}}{3} \right )}}{3} - \frac {\operatorname {atan}{\left (\sqrt {x^{6} - 1} \right )}}{2} + \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 = 41, normalized size = 0.77 \begin {gather*} \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} \sqrt {x^{6} - 1}\right ) + \frac {\sqrt {x^{6} - 1}}{6 \, x^{6}} - \frac {1}{2} \, \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 = 1.02, size = 41, normalized size = 0.77 \begin {gather*} \frac {\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}\,\sqrt {x^6-1}}{3}\right )}{3}-\frac {\mathrm {atan}\left (\sqrt {x^6-1}\right )}{2}+\frac {\sqrt {x^6-1}}{6\,x^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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