Optimal. Leaf size=64 \[ \frac {5}{6} \log \left (\sqrt {x^6-1}+x^3\right )-\frac {\tan ^{-1}\left (-\frac {4 x^6}{\sqrt {3}}-\frac {4 \sqrt {x^6-1} x^3}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt {3}} \]
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Rubi [A] time = 0.09, antiderivative size = 47, normalized size of antiderivative = 0.73, number of steps used = 7, number of rules used = 7, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {1593, 575, 523, 217, 206, 377, 204} \begin {gather*} \frac {5}{6} \tanh ^{-1}\left (\frac {x^3}{\sqrt {x^6-1}}\right )-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x^3}{\sqrt {x^6-1}}\right )}{2 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 206
Rule 217
Rule 377
Rule 523
Rule 575
Rule 1593
Rubi steps
\begin {align*} \int \frac {-x^2+10 x^8}{\sqrt {-1+x^6} \left (-1+4 x^6\right )} \, dx &=\int \frac {x^2 \left (-1+10 x^6\right )}{\sqrt {-1+x^6} \left (-1+4 x^6\right )} \, dx\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {-1+10 x^2}{\sqrt {-1+x^2} \left (-1+4 x^2\right )} \, dx,x,x^3\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^2} \left (-1+4 x^2\right )} \, dx,x,x^3\right )+\frac {5}{6} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^2}} \, dx,x,x^3\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{-1-3 x^2} \, dx,x,\frac {x^3}{\sqrt {-1+x^6}}\right )+\frac {5}{6} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^3}{\sqrt {-1+x^6}}\right )\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x^3}{\sqrt {-1+x^6}}\right )}{2 \sqrt {3}}+\frac {5}{6} \tanh ^{-1}\left (\frac {x^3}{\sqrt {-1+x^6}}\right )\\ \end {align*}
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Mathematica [A] time = 0.06, size = 47, normalized size = 0.73 \begin {gather*} \frac {5}{6} \tanh ^{-1}\left (\frac {x^3}{\sqrt {x^6-1}}\right )-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x^3}{\sqrt {x^6-1}}\right )}{2 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.10, size = 66, normalized size = 1.03 \begin {gather*} \frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {4 x^6}{\sqrt {3}}+\frac {4 x^3 \sqrt {-1+x^6}}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {5}{6} \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 = 51, normalized size = 0.80 \begin {gather*} \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {4}{3} \, \sqrt {3} \sqrt {x^{6} - 1} x^{3} - \frac {1}{3} \, \sqrt {3} {\left (4 \, x^{6} - 1\right )}\right ) - \frac {5}{6} \, \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 [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.33, size = 74, normalized size = 1.16
method | result | size |
trager | \(\frac {5 \ln \left (x^{3}+\sqrt {x^{6}-1}\right )}{6}-\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{2}+3\right ) x^{6}-6 x^{3} \sqrt {x^{6}-1}+\RootOf \left (\textit {\_Z}^{2}+3\right )}{\left (2 x^{3}-1\right ) \left (2 x^{3}+1\right )}\right )}{12}\) | \(74\) |
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 {10 \, x^{8} - x^{2}}{{\left (4 \, x^{6} - 1\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^2-10\,x^8}{\sqrt {x^6-1}\,\left (4\,x^6-1\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 (10 x^{6} - 1\right )}{\sqrt {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )} \left (2 x^{3} - 1\right ) \left (2 x^{3} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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