Optimal. Leaf size=15 \[ \log \left (e^x+1\right )-\log \left (e^x+2\right ) \]
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Rubi [A] time = 0.0289803, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2282, 616, 31} \[ \log \left (e^x+1\right )-\log \left (e^x+2\right ) \]
Antiderivative was successfully verified.
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Rule 2282
Rule 616
Rule 31
Rubi steps
\begin{align*} \int \frac{e^x}{2+3 e^x+e^{2 x}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{2+3 x+x^2} \, dx,x,e^x\right )\\ &=\operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,e^x\right )-\operatorname{Subst}\left (\int \frac{1}{2+x} \, dx,x,e^x\right )\\ &=\log \left (1+e^x\right )-\log \left (2+e^x\right )\\ \end{align*}
Mathematica [A] time = 0.0065474, size = 10, normalized size = 0.67 \[ -2 \tanh ^{-1}\left (2 e^x+3\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 14, normalized size = 0.9 \begin{align*} \ln \left ( 1+{{\rm e}^{x}} \right ) -\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 = 0.972564, size = 18, normalized size = 1.2 \begin{align*} -\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 = 0.896429, size = 41, normalized size = 2.73 \begin{align*} -\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 = 0.111125, size = 12, normalized size = 0.8 \begin{align*} \log{\left (e^{x} + 1 \right )} - \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 = 1.21116, size = 18, normalized size = 1.2 \begin{align*} -\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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