Optimal. Leaf size=41 \[ \frac {1}{12} \log \left (3 x^4-2 x^2+1\right )-\frac {\tan ^{-1}\left (\frac {1-3 x^2}{\sqrt {2}}\right )}{6 \sqrt {2}} \]
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Rubi [A] time = 0.04, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {1114, 634, 618, 204, 628} \[ \frac {1}{12} \log \left (3 x^4-2 x^2+1\right )-\frac {\tan ^{-1}\left (\frac {1-3 x^2}{\sqrt {2}}\right )}{6 \sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 1114
Rubi steps
\begin {align*} \int \frac {x^3}{1-2 x^2+3 x^4} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x}{1-2 x+3 x^2} \, dx,x,x^2\right )\\ &=\frac {1}{12} \operatorname {Subst}\left (\int \frac {-2+6 x}{1-2 x+3 x^2} \, dx,x,x^2\right )+\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{1-2 x+3 x^2} \, dx,x,x^2\right )\\ &=\frac {1}{12} \log \left (1-2 x^2+3 x^4\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{-8-x^2} \, dx,x,2 \left (-1+3 x^2\right )\right )\\ &=-\frac {\tan ^{-1}\left (\frac {1-3 x^2}{\sqrt {2}}\right )}{6 \sqrt {2}}+\frac {1}{12} \log \left (1-2 x^2+3 x^4\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 38, normalized size = 0.93 \[ \frac {1}{12} \left (\sqrt {2} \tan ^{-1}\left (\frac {3 x^2-1}{\sqrt {2}}\right )+\log \left (3 x^4-2 x^2+1\right )\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 44, normalized size = 1.07 \[ \frac {1}{12} \log \left (3 x^4-2 x^2+1\right )-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {2}}-\frac {3 x^2}{\sqrt {2}}\right )}{6 \sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.15, size = 34, normalized size = 0.83 \[ \frac {1}{12} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (3 \, x^{2} - 1\right )}\right ) + \frac {1}{12} \, \log \left (3 \, x^{4} - 2 \, x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.51, size = 34, normalized size = 0.83 \[ \frac {1}{12} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (3 \, x^{2} - 1\right )}\right ) + \frac {1}{12} \, \log \left (3 \, x^{4} - 2 \, x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 35, normalized size = 0.85
method | result | size |
default | \(\frac {\ln \left (3 x^{4}-2 x^{2}+1\right )}{12}+\frac {\sqrt {2}\, \arctan \left (\frac {\left (6 x^{2}-2\right ) \sqrt {2}}{4}\right )}{12}\) | \(35\) |
risch | \(\frac {\ln \left (9 x^{4}-6 x^{2}+3\right )}{12}+\frac {\sqrt {2}\, \arctan \left (\frac {\left (3 x^{2}-1\right ) \sqrt {2}}{2}\right )}{12}\) | \(35\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 34, normalized size = 0.83 \[ \frac {1}{12} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (3 \, x^{2} - 1\right )}\right ) + \frac {1}{12} \, \log \left (3 \, x^{4} - 2 \, x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.20, size = 34, normalized size = 0.83 \[ \frac {\ln \left (x^4-\frac {2\,x^2}{3}+\frac {1}{3}\right )}{12}-\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}}{2}-\frac {3\,\sqrt {2}\,x^2}{2}\right )}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 42, normalized size = 1.02 \[ \frac {\log {\left (x^{4} - \frac {2 x^{2}}{3} + \frac {1}{3} \right )}}{12} + \frac {\sqrt {2} \operatorname {atan}{\left (\frac {3 \sqrt {2} x^{2}}{2} - \frac {\sqrt {2}}{2} \right )}}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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