Optimal. Leaf size=37 \[ \frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {2 x^2+1}{\sqrt {3}}\right )+\frac {1}{4} \log \left (x^4+x^2+1\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {1247, 634, 618, 204, 628} \[ \frac {1}{4} \log \left (x^4+x^2+1\right )+\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {2 x^2+1}{\sqrt {3}}\right ) \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 1247
Rubi steps
\begin {align*} \int \frac {x \left (2+x^2\right )}{1+x^2+x^4} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {2+x}{1+x+x^2} \, dx,x,x^2\right )\\ &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,x^2\right )+\frac {3}{4} \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,x^2\right )\\ &=\frac {1}{4} \log \left (1+x^2+x^4\right )-\frac {3}{2} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x^2\right )\\ &=\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {1+2 x^2}{\sqrt {3}}\right )+\frac {1}{4} \log \left (1+x^2+x^4\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 37, normalized size = 1.00 \[ \frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {2 x^2+1}{\sqrt {3}}\right )+\frac {1}{4} \log \left (x^4+x^2+1\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 30, normalized size = 0.81 \[ \frac {1}{2} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} + 1\right )}\right ) + \frac {1}{4} \, \log \left (x^{4} + x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 30, normalized size = 0.81 \[ \frac {1}{2} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} + 1\right )}\right ) + \frac {1}{4} \, \log \left (x^{4} + x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 31, normalized size = 0.84 \[ \frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x^{2}+1\right ) \sqrt {3}}{3}\right )}{2}+\frac {\ln \left (x^{4}+x^{2}+1\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.40, size = 30, normalized size = 0.81 \[ \frac {1}{2} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} + 1\right )}\right ) + \frac {1}{4} \, \log \left (x^{4} + x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.21, size = 32, normalized size = 0.86 \[ \frac {\ln \left (x^4+x^2+1\right )}{4}+\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {2\,\sqrt {3}\,x^2}{3}+\frac {\sqrt {3}}{3}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 37, normalized size = 1.00 \[ \frac {\log {\left (x^{4} + x^{2} + 1 \right )}}{4} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x^{2}}{3} + \frac {\sqrt {3}}{3} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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