Optimal. Leaf size=15 \[ \log \left (e^x+1\right )-\log \left (e^x+2\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 15, normalized size of antiderivative = 1.00, 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 31
Rule 616
Rule 2282
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*}
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Mathematica [A] time = 0.01, size = 10, normalized size = 0.67 \[ -2 \tanh ^{-1}\left (2 e^x+3\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 13, normalized size = 0.87 \[ -\log \left (e^{x} + 2\right ) + \log \left (e^{x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 13, normalized size = 0.87 \[ -\log \left (e^{x} + 2\right ) + \log \left (e^{x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 14, normalized size = 0.93 \[ \ln \left ({\mathrm e}^{x}+1\right )-\ln \left ({\mathrm e}^{x}+2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.03, size = 13, normalized size = 0.87 \[ -\log \left (e^{x} + 2\right ) + \log \left (e^{x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.49, size = 13, normalized size = 0.87 \[ \ln \left ({\mathrm {e}}^x+1\right )-\ln \left ({\mathrm {e}}^x+2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 12, normalized size = 0.80 \[ \log {\left (e^{x} + 1 \right )} - \log {\left (e^{x} + 2 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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