Optimal. Leaf size=64 \[ \frac {\sqrt {x^6-1} \left (2 x^6+1\right )}{6 x^6}-\frac {1}{6} \tan ^{-1}\left (\sqrt {x^6-1}\right )-\frac {\tan ^{-1}\left (\frac {2 \sqrt {x^6-1}}{\sqrt {3}}\right )}{2 \sqrt {3}} \]
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Rubi [A] time = 0.10, 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 = {573, 180, 47, 63, 203, 50} \begin {gather*} \frac {\sqrt {x^6-1}}{6 x^6}+\frac {\sqrt {x^6-1}}{3}-\frac {1}{6} \tan ^{-1}\left (\sqrt {x^6-1}\right )-\frac {\tan ^{-1}\left (\frac {2 \sqrt {x^6-1}}{\sqrt {3}}\right )}{2 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 180
Rule 203
Rule 573
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} \operatorname {Subst}\left (\int \frac {\sqrt {-1+x} (-1+2 x)^2}{x^2 (-1+4 x)} \, dx,x,x^6\right )\\ &=\frac {1}{6} \operatorname {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} \operatorname {Subst}\left (\int \frac {\sqrt {-1+x}}{x^2} \, dx,x,x^6\right )\right )+\frac {2}{3} \operatorname {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} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^6\right )-\frac {1}{2} \operatorname {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} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^6}\right )-\operatorname {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 [B] time = 0.09, size = 135, normalized size = 2.11 \begin {gather*} \frac {4 x^{12}-2 x^6+2 \sqrt {1-x^6} x^6 \tanh ^{-1}\left (\sqrt {1-x^6}\right )-\sqrt {3} \sqrt {x^6-1} x^6 \tan ^{-1}\left (\frac {x^3-2}{\sqrt {3} \sqrt {x^6-1}}\right )+\sqrt {3} \sqrt {x^6-1} x^6 \tan ^{-1}\left (\frac {x^3+2}{\sqrt {3} \sqrt {x^6-1}}\right )-2}{12 x^6 \sqrt {x^6-1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.08, size = 64, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-1+x^6} \left (1+2 x^6\right )}{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 {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.47, 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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giac [A] time = 0.42, 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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maple [C] time = 1.56, size = 105, normalized size = 1.64
method | result | size |
trager | \(\frac {\sqrt {x^{6}-1}\, \left (2 x^{6}+1\right )}{6 x^{6}}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{6}+\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) \ln \left (\frac {4 \RootOf \left (\textit {\_Z}^{2}+3\right ) x^{6}-7 \RootOf \left (\textit {\_Z}^{2}+3\right )+12 \sqrt {x^{6}-1}}{\left (2 x^{3}-1\right ) \left (2 x^{3}+1\right )}\right )}{12}\) | \(105\) |
risch | \(\frac {\sqrt {x^{6}-1}}{6 x^{6}}+\frac {\sqrt {x^{6}-1}}{3}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{6}+\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) \ln \left (\frac {4 \RootOf \left (\textit {\_Z}^{2}+3\right ) x^{6}-7 \RootOf \left (\textit {\_Z}^{2}+3\right )+12 \sqrt {x^{6}-1}}{\left (2 x^{3}-1\right ) \left (2 x^{3}+1\right )}\right )}{12}\) | \(107\) |
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*} \int \frac {{\left (2 \, x^{6} - 1\right )}^{2} \sqrt {x^{6} - 1}}{{\left (4 \, x^{6} - 1\right )} x^{7}}\,{d x} \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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sympy [A] time = 50.60, 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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