Optimal. Leaf size=37 \[ \frac{1}{4} \log \left (x^4+x^2+1\right )+\frac{1}{2} \sqrt{3} \tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right ) \]
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Rubi [A] time = 0.0417176, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {1593, 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 1593
Rule 1247
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{2 x+x^3}{1+x^2+x^4} \, dx &=\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*}
Mathematica [A] time = 0.0059184, size = 37, normalized size = 1. \[ \frac{1}{4} \log \left (x^4+x^2+1\right )+\frac{1}{2} \sqrt{3} \tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 31, normalized size = 0.8 \begin{align*}{\frac{\ln \left ({x}^{4}+{x}^{2}+1 \right ) }{4}}+{\frac{\sqrt{3}}{2}\arctan \left ({\frac{ \left ( 2\,{x}^{2}+1 \right ) \sqrt{3}}{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.48006, size = 72, normalized size = 1.95 \begin{align*} -\frac{1}{2} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{2} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{4} \, \log \left (x^{2} + x + 1\right ) + \frac{1}{4} \, \log \left (x^{2} - x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52006, size = 95, normalized size = 2.57 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.114476, size = 37, normalized size = 1. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09119, size = 41, normalized size = 1.11 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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