Optimal. Leaf size=21 \[ -2 x-e^{-x}+2 \log \left (2 e^x+1\right ) \]
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Rubi [A] time = 0.03, 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 = {2248, 44} \[ -2 x-e^{-x}+2 \log \left (2 e^x+1\right ) \]
Antiderivative was successfully verified.
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Rule 44
Rule 2248
Rubi steps
\begin {align*} \int \frac {e^{-x}}{1+2 e^x} \, dx &=\operatorname {Subst}\left (\int \frac {1}{x^2 (1+2 x)} \, dx,x,e^x\right )\\ &=\operatorname {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 = 21, normalized size = 1.00 \[ -2 x-e^{-x}+2 \log \left (2 e^x+1\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 24, normalized size = 1.14 \[ -{\left (2 \, x e^{x} - 2 \, e^{x} \log \left (2 \, e^{x} + 1\right ) + 1\right )} e^{\left (-x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 19, normalized size = 0.90 \[ -2 \, x - e^{\left (-x\right )} + 2 \, \log \left (2 \, e^{x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 22, normalized size = 1.05 \[ -{\mathrm e}^{-x}+2 \ln \left (2 \,{\mathrm e}^{x}+1\right )-2 \ln \left ({\mathrm e}^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 16, normalized size = 0.76 \[ -e^{\left (-x\right )} + 2 \, \log \left (e^{\left (-x\right )} + 2\right ) \]
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 \[ 2\,\ln \left (2\,{\mathrm {e}}^x+1\right )-2\,x-{\mathrm {e}}^{-x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 17, normalized size = 0.81 \[ - 2 x + 2 \log {\left (e^{x} + \frac {1}{2} \right )} - e^{- x} \]
Verification of antiderivative is not currently implemented for this CAS.
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