Optimal. Leaf size=63 \[ \frac {1}{3} \log \left (\sqrt {x^6-1}+x^3\right )+\sqrt {\frac {2}{3}} \tanh ^{-1}\left (\frac {x^6}{\sqrt {6}}+\frac {\sqrt {x^6-1} x^3}{\sqrt {6}}+\sqrt {\frac {2}{3}}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 49, normalized size of antiderivative = 0.78, number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {575, 523, 217, 206, 377} \begin {gather*} \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
Rubi steps
\begin {align*} \int \frac {x^2 \left (-4+x^6\right )}{\sqrt {-1+x^6} \left (2+x^6\right )} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {-4+x^2}{\sqrt {-1+x^2} \left (2+x^2\right )} \, dx,x,x^3\right )\\ &=\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 {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 {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 [A] time = 0.07, size = 47, normalized size = 0.75 \begin {gather*} \frac {1}{3} \left (\tanh ^{-1}\left (\frac {x^3}{\sqrt {x^6-1}}\right )-\sqrt {6} \tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x^3}{\sqrt {x^6-1}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.15, size = 67, normalized size = 1.06 \begin {gather*} -\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 = 0.47, size = 82, normalized size = 1.30 \begin {gather*} \frac {1}{6} \, \sqrt {3} \sqrt {2} \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 ) - \frac {1}{3} \, \log \left (-x^{3} + \sqrt {x^{6} - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 72, normalized size = 1.14 \begin {gather*} \frac {1}{6} \, \sqrt {6} \log \left (\frac {{\left (x^{3} - \sqrt {x^{6} - 1}\right )}^{2} - 2 \, \sqrt {6} + 5}{{\left (x^{3} - \sqrt {x^{6} - 1}\right )}^{2} + 2 \, \sqrt {6} + 5}\right ) - \frac {1}{6} \, \log \left ({\left (x^{3} - \sqrt {x^{6} - 1}\right )}^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.31, size = 65, normalized size = 1.03
method | result | size |
trager | \(\frac {\ln \left (x^{3}+\sqrt {x^{6}-1}\right )}{3}+\frac {\RootOf \left (\textit {\_Z}^{2}-6\right ) \ln \left (\frac {-5 \RootOf \left (\textit {\_Z}^{2}-6\right ) x^{6}+12 x^{3} \sqrt {x^{6}-1}+2 \RootOf \left (\textit {\_Z}^{2}-6\right )}{x^{6}+2}\right )}{6}\) | \(65\) |
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 (x^{6} - 4\right )} x^{2}}{{\left (x^{6} + 2\right )} \sqrt {x^{6} - 1}}\,{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 = -0.02 \begin {gather*} \int \frac {x^2\,\left (x^6-4\right )}{\sqrt {x^6-1}\,\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 {x^{2} \left (x^{3} - 2\right ) \left (x^{3} + 2\right )}{\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 )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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