Optimal. Leaf size=79 \[ \frac {\sqrt {x^6-1}}{3 x^3}+\frac {1}{3} \log \left (\sqrt {x^6-1}+x^3\right )+\sqrt {\frac {2}{3}} \tanh ^{-1}\left (\frac {x^6}{\sqrt {6}}+\frac {\sqrt {x^6-1} x^3}{\sqrt {6}}+\sqrt {\frac {2}{3}}\right ) \]
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Rubi [A] time = 0.09, antiderivative size = 65, normalized size of antiderivative = 0.82, number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {575, 580, 523, 217, 206, 377} \begin {gather*} \frac {\sqrt {x^6-1}}{3 x^3}+\frac {1}{3} \tanh ^{-1}\left (\frac {x^3}{\sqrt {x^6-1}}\right )-\sqrt {\frac {2}{3}} \tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x^3}{\sqrt {x^6-1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 377
Rule 523
Rule 575
Rule 580
Rubi steps
\begin {align*} \int \frac {\left (-2+x^6\right ) \sqrt {-1+x^6}}{x^4 \left (2+x^6\right )} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {\left (-2+x^2\right ) \sqrt {-1+x^2}}{x^2 \left (2+x^2\right )} \, dx,x,x^3\right )\\ &=\frac {\sqrt {-1+x^6}}{3 x^3}+\frac {1}{6} \operatorname {Subst}\left (\int \frac {-8+2 x^2}{\sqrt {-1+x^2} \left (2+x^2\right )} \, dx,x,x^3\right )\\ &=\frac {\sqrt {-1+x^6}}{3 x^3}+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^2}} \, dx,x,x^3\right )-2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^2} \left (2+x^2\right )} \, dx,x,x^3\right )\\ &=\frac {\sqrt {-1+x^6}}{3 x^3}+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^3}{\sqrt {-1+x^6}}\right )-2 \operatorname {Subst}\left (\int \frac {1}{2-3 x^2} \, dx,x,\frac {x^3}{\sqrt {-1+x^6}}\right )\\ &=\frac {\sqrt {-1+x^6}}{3 x^3}+\frac {1}{3} \tanh ^{-1}\left (\frac {x^3}{\sqrt {-1+x^6}}\right )-\sqrt {\frac {2}{3}} \tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x^3}{\sqrt {-1+x^6}}\right )\\ \end {align*}
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Mathematica [C] time = 0.10, size = 94, normalized size = 1.19 \begin {gather*} \frac {\sqrt {1-x^6} x^{12} F_1\left (\frac {3}{2};\frac {1}{2},1;\frac {5}{2};x^6,-\frac {x^6}{2}\right )+6 x^6-4 \sqrt {6-6 x^6} x^3 \sin ^{-1}\left (\frac {\sqrt {3} x^3}{\sqrt {x^6+2}}\right )-6}{18 x^3 \sqrt {x^6-1}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.36, size = 79, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-1+x^6}}{3 x^3}+\sqrt {\frac {2}{3}} \tanh ^{-1}\left (\sqrt {\frac {2}{3}}+\frac {x^6}{\sqrt {6}}+\frac {x^3 \sqrt {-1+x^6}}{\sqrt {6}}\right )+\frac {1}{3} \log \left (x^3+\sqrt {-1+x^6}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 106, normalized size = 1.34 \begin {gather*} \frac {\sqrt {3} \sqrt {2} x^{3} \log \left (\frac {25 \, x^{6} - 2 \, \sqrt {3} \sqrt {2} {\left (5 \, x^{6} - 2\right )} - 2 \, \sqrt {x^{6} - 1} {\left (5 \, \sqrt {3} \sqrt {2} x^{3} - 12 \, x^{3}\right )} - 10}{x^{6} + 2}\right ) - 2 \, x^{3} \log \left (-x^{3} + \sqrt {x^{6} - 1}\right ) + 2 \, x^{3} + 2 \, \sqrt {x^{6} - 1}}{6 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 97, normalized size = 1.23 \begin {gather*} \frac {\sqrt {6} \log \left (\frac {\sqrt {6} - 2 \, \sqrt {-\frac {1}{x^{6}} + 1}}{\sqrt {6} + 2 \, \sqrt {-\frac {1}{x^{6}} + 1}}\right )}{6 \, \mathrm {sgn}\relax (x)} + \frac {\log \left (\sqrt {-\frac {1}{x^{6}} + 1} + 1\right )}{6 \, \mathrm {sgn}\relax (x)} - \frac {\log \left (-\sqrt {-\frac {1}{x^{6}} + 1} + 1\right )}{6 \, \mathrm {sgn}\relax (x)} + \frac {\sqrt {-\frac {1}{x^{6}} + 1}}{3 \, \mathrm {sgn}\relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.98, size = 77, normalized size = 0.97
method | result | size |
trager | \(\frac {\sqrt {x^{6}-1}}{3 x^{3}}+\frac {\ln \left (x^{3}+\sqrt {x^{6}-1}\right )}{3}-\frac {\RootOf \left (\textit {\_Z}^{2}-6\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{2}-6\right ) x^{6}+12 x^{3} \sqrt {x^{6}-1}-2 \RootOf \left (\textit {\_Z}^{2}-6\right )}{x^{6}+2}\right )}{6}\) | \(77\) |
risch | \(\frac {\sqrt {x^{6}-1}}{3 x^{3}}+\frac {\ln \left (x^{3}+\sqrt {x^{6}-1}\right )}{3}+\frac {\RootOf \left (\textit {\_Z}^{2}-6\right ) \ln \left (\frac {-5 \RootOf \left (\textit {\_Z}^{2}-6\right ) x^{6}+12 x^{3} \sqrt {x^{6}-1}+2 \RootOf \left (\textit {\_Z}^{2}-6\right )}{x^{6}+2}\right )}{6}\) | \(77\) |
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 {\sqrt {x^{6} - 1} {\left (x^{6} - 2\right )}}{{\left (x^{6} + 2\right )} x^{4}}\,{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 {\sqrt {x^6-1}\,\left (x^6-2\right )}{x^4\,\left (x^6+2\right )} \,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 {\sqrt {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )} \left (x^{6} - 2\right )}{x^{4} \left (x^{6} + 2\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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