Optimal. Leaf size=17 \[ 2 \log \left (e^x+2\right )-\log \left (e^x+1\right ) \]
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Rubi [A] time = 0.0318288, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {2282, 632, 31} \[ 2 \log \left (e^x+2\right )-\log \left (e^x+1\right ) \]
Antiderivative was successfully verified.
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Rule 2282
Rule 632
Rule 31
Rubi steps
\begin{align*} \int \frac{e^{2 x}}{2+3 e^x+e^{2 x}} \, dx &=\operatorname{Subst}\left (\int \frac{x}{2+3 x+x^2} \, dx,x,e^x\right )\\ &=2 \operatorname{Subst}\left (\int \frac{1}{2+x} \, dx,x,e^x\right )-\operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,e^x\right )\\ &=-\log \left (1+e^x\right )+2 \log \left (2+e^x\right )\\ \end{align*}
Mathematica [A] time = 0.0134487, size = 17, normalized size = 1. \[ 2 \log \left (e^x+2\right )-\log \left (e^x+1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 16, normalized size = 0.9 \begin{align*} -\ln \left ( 1+{{\rm e}^{x}} \right ) +2\,\ln \left ( 2+{{\rm e}^{x}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.949909, size = 20, normalized size = 1.18 \begin{align*} 2 \, \log \left (e^{x} + 2\right ) - \log \left (e^{x} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.25475, size = 42, normalized size = 2.47 \begin{align*} 2 \, \log \left (e^{x} + 2\right ) - \log \left (e^{x} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.112225, size = 14, normalized size = 0.82 \begin{align*} - \log{\left (e^{x} + 1 \right )} + 2 \log{\left (e^{x} + 2 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.0665, size = 20, normalized size = 1.18 \begin{align*} 2 \, \log \left (e^{x} + 2\right ) - \log \left (e^{x} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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