Optimal. Leaf size=17 \[ 2 \log \left (e^x+2\right )-\log \left (e^x+1\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, 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 31
Rule 632
Rule 2282
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*}
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Mathematica [A] time = 0.01, size = 17, normalized size = 1.00 \[ 2 \log \left (e^x+2\right )-\log \left (e^x+1\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 15, normalized size = 0.88 \[ 2 \, \log \left (e^{x} + 2\right ) - \log \left (e^{x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 15, normalized size = 0.88 \[ 2 \, \log \left (e^{x} + 2\right ) - \log \left (e^{x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 16, normalized size = 0.94 \[ -\ln \left ({\mathrm e}^{x}+1\right )+2 \ln \left ({\mathrm e}^{x}+2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.73, size = 15, normalized size = 0.88 \[ 2 \, \log \left (e^{x} + 2\right ) - \log \left (e^{x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.54, size = 15, normalized size = 0.88 \[ 2\,\ln \left ({\mathrm {e}}^x+2\right )-\ln \left ({\mathrm {e}}^x+1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 14, normalized size = 0.82 \[ - \log {\left (e^{x} + 1 \right )} + 2 \log {\left (e^{x} + 2 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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