Optimal. Leaf size=18 \[ \frac{\tan ^{-1}\left (\frac{3 x+2}{\sqrt{2}}\right )}{\sqrt{2}} \]
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Rubi [A] time = 0.0112172, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {618, 204} \[ \frac{\tan ^{-1}\left (\frac{3 x+2}{\sqrt{2}}\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{2+4 x+3 x^2} \, dx &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{-8-x^2} \, dx,x,4+6 x\right )\right )\\ &=\frac{\tan ^{-1}\left (\frac{2+3 x}{\sqrt{2}}\right )}{\sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0049882, size = 18, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{3 x+2}{\sqrt{2}}\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 17, normalized size = 0.9 \begin{align*}{\frac{\sqrt{2}}{2}\arctan \left ({\frac{ \left ( 4+6\,x \right ) \sqrt{2}}{4}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.78149, size = 22, normalized size = 1.22 \begin{align*} \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (3 \, x + 2\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.13459, size = 58, normalized size = 3.22 \begin{align*} \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (3 \, x + 2\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.10425, size = 22, normalized size = 1.22 \begin{align*} \frac{\sqrt{2} \operatorname{atan}{\left (\frac{3 \sqrt{2} x}{2} + \sqrt{2} \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24023, size = 22, normalized size = 1.22 \begin{align*} \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (3 \, x + 2\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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