Optimal. Leaf size=13 \[ \frac{1}{2} \log \left (x^2+2\right )+\tan ^{-1}(x) \]
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Rubi [A] time = 0.0283439, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {1673, 1149, 203, 1247, 626, 31} \[ \frac{1}{2} \log \left (x^2+2\right )+\tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 1673
Rule 1149
Rule 203
Rule 1247
Rule 626
Rule 31
Rubi steps
\begin{align*} \int \frac{2+x+x^2+x^3}{2+3 x^2+x^4} \, dx &=\int \frac{x \left (1+x^2\right )}{2+3 x^2+x^4} \, dx+\int \frac{2+x^2}{2+3 x^2+x^4} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1+x}{2+3 x+x^2} \, dx,x,x^2\right )+\int \frac{1}{1+x^2} \, dx\\ &=\tan ^{-1}(x)+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{2+x} \, dx,x,x^2\right )\\ &=\tan ^{-1}(x)+\frac{1}{2} \log \left (2+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0074211, size = 13, normalized size = 1. \[ \frac{1}{2} \log \left (x^2+2\right )+\tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 12, normalized size = 0.9 \begin{align*} \arctan \left ( x \right ) +{\frac{\ln \left ({x}^{2}+2 \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.5041, size = 15, normalized size = 1.15 \begin{align*} \arctan \left (x\right ) + \frac{1}{2} \, \log \left (x^{2} + 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.34474, size = 41, normalized size = 3.15 \begin{align*} \arctan \left (x\right ) + \frac{1}{2} \, \log \left (x^{2} + 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.104464, size = 10, normalized size = 0.77 \begin{align*} \frac{\log{\left (x^{2} + 2 \right )}}{2} + \operatorname{atan}{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11024, size = 15, normalized size = 1.15 \begin{align*} \arctan \left (x\right ) + \frac{1}{2} \, \log \left (x^{2} + 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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