Optimal. Leaf size=15 \[ \log \left (1+e^x\right )-\log \left (2+e^x\right ) \]
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Rubi [A]
time = 0.02, 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 = {2320, 630, 31}
\begin {gather*} \log \left (e^x+1\right )-\log \left (e^x+2\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 630
Rule 2320
Rubi steps
\begin {align*} \int \frac {e^x}{2+3 e^x+e^{2 x}} \, dx &=\text {Subst}\left (\int \frac {1}{2+3 x+x^2} \, dx,x,e^x\right )\\ &=\text {Subst}\left (\int \frac {1}{1+x} \, dx,x,e^x\right )-\text {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 = 0.02, size = 10, normalized size = 0.67 \begin {gather*} -2 \tanh ^{-1}\left (3+2 e^x\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 14, normalized size = 0.93
method | result | size |
default | \(\ln \left (1+{\mathrm e}^{x}\right )-\ln \left (2+{\mathrm e}^{x}\right )\) | \(14\) |
norman | \(\ln \left (1+{\mathrm e}^{x}\right )-\ln \left (2+{\mathrm e}^{x}\right )\) | \(14\) |
risch | \(\ln \left (1+{\mathrm e}^{x}\right )-\ln \left (2+{\mathrm e}^{x}\right )\) | \(14\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 13, normalized size = 0.87 \begin {gather*} -\log \left (e^{x} + 2\right ) + \log \left (e^{x} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.44, size = 13, normalized size = 0.87 \begin {gather*} -\log \left (e^{x} + 2\right ) + \log \left (e^{x} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 12, normalized size = 0.80 \begin {gather*} \log {\left (e^{x} + 1 \right )} - \log {\left (e^{x} + 2 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.27, size = 13, normalized size = 0.87 \begin {gather*} -\log \left (e^{x} + 2\right ) + \log \left (e^{x} + 1\right ) \end {gather*}
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 \begin {gather*} \ln \left ({\mathrm {e}}^x+1\right )-\ln \left ({\mathrm {e}}^x+2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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