Optimal. Leaf size=24 \[ \frac{x}{2}-\log \left (e^x+1\right )+\frac{1}{2} \log \left (e^x+2\right ) \]
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Rubi [A] time = 0.0198411, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {2282, 705, 29, 632, 31} \[ \frac{x}{2}-\log \left (e^x+1\right )+\frac{1}{2} \log \left (e^x+2\right ) \]
Antiderivative was successfully verified.
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Rule 2282
Rule 705
Rule 29
Rule 632
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{2+3 e^x+e^{2 x}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{x \left (2+3 x+x^2\right )} \, dx,x,e^x\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,e^x\right )+\frac{1}{2} \operatorname{Subst}\left (\int \frac{-3-x}{2+3 x+x^2} \, dx,x,e^x\right )\\ &=\frac{x}{2}+\frac{1}{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 )\\ &=\frac{x}{2}-\log \left (1+e^x\right )+\frac{1}{2} \log \left (2+e^x\right )\\ \end{align*}
Mathematica [A] time = 0.010407, size = 24, normalized size = 1. \[ \frac{x}{2}-\log \left (e^x+1\right )+\frac{1}{2} \log \left (e^x+2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 21, normalized size = 0.9 \begin{align*}{\frac{\ln \left ({{\rm e}^{x}} \right ) }{2}}+{\frac{\ln \left ( 2+{{\rm e}^{x}} \right ) }{2}}-\ln \left ( 1+{{\rm e}^{x}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.962841, size = 24, normalized size = 1. \begin{align*} \frac{1}{2} \, x + \frac{1}{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 = 1.4946, size = 55, normalized size = 2.29 \begin{align*} \frac{1}{2} \, x + \frac{1}{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 = 0.105901, size = 17, normalized size = 0.71 \begin{align*} \frac{x}{2} - \log{\left (e^{x} + 1 \right )} + \frac{\log{\left (e^{x} + 2 \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27243, size = 24, normalized size = 1. \begin{align*} \frac{1}{2} \, x + \frac{1}{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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