Optimal. Leaf size=21 \[ -e^{-x}-2 x+2 \log \left (1+2 e^x\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2280, 46}
\begin {gather*} -2 x-e^{-x}+2 \log \left (2 e^x+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 2280
Rubi steps
\begin {align*} \int \frac {e^{-x}}{1+2 e^x} \, dx &=\text {Subst}\left (\int \frac {1}{x^2 (1+2 x)} \, dx,x,e^x\right )\\ &=\text {Subst}\left (\int \left (\frac {1}{x^2}-\frac {2}{x}+\frac {4}{1+2 x}\right ) \, dx,x,e^x\right )\\ &=-e^{-x}-2 x+2 \log \left (1+2 e^x\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 24, normalized size = 1.14 \begin {gather*} -e^{-x}-2 \log \left (e^x\right )+2 \log \left (1+2 e^x\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 1.80, size = 19, normalized size = 0.90 \begin {gather*} -2 x-E^{-x}+2 \text {Log}\left [\frac {1}{2}+E^x\right ] \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 22, normalized size = 1.05
method | result | size |
risch | \(-{\mathrm e}^{-x}-2 x +2 \ln \left (\frac {1}{2}+{\mathrm e}^{x}\right )\) | \(18\) |
derivativedivides | \(2 \ln \left (1+2 \,{\mathrm e}^{x}\right )-{\mathrm e}^{-x}-2 \ln \left ({\mathrm e}^{x}\right )\) | \(22\) |
default | \(2 \ln \left (1+2 \,{\mathrm e}^{x}\right )-{\mathrm e}^{-x}-2 \ln \left ({\mathrm e}^{x}\right )\) | \(22\) |
norman | \(\left (-1-2 \,{\mathrm e}^{x} x \right ) {\mathrm e}^{-x}+2 \ln \left (1+2 \,{\mathrm e}^{x}\right )\) | \(23\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 16, normalized size = 0.76 \begin {gather*} -e^{\left (-x\right )} + 2 \, \log \left (e^{\left (-x\right )} + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 24, normalized size = 1.14 \begin {gather*} -{\left (2 \, x e^{x} - 2 \, e^{x} \log \left (2 \, e^{x} + 1\right ) + 1\right )} e^{\left (-x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 14, normalized size = 0.67 \begin {gather*} 2 \log {\left (2 + e^{- x} \right )} - e^{- x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 18, normalized size = 0.86 \begin {gather*} 2 \ln \left (2 \mathrm {e}^{x}+1\right )-2 x-\frac 1{\mathrm {e}^{x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 19, normalized size = 0.90 \begin {gather*} 2\,\ln \left (2\,{\mathrm {e}}^x+1\right )-2\,x-{\mathrm {e}}^{-x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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