3.358 \(\int \frac{e^{-x}}{1+2 e^x} \, dx\)

Optimal. Leaf size=21 \[ -2 x-e^{-x}+2 \log \left (2 e^x+1\right ) \]

[Out]

-E^(-x) - 2*x + 2*Log[1 + 2*E^x]

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Rubi [A]  time = 0.044486, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -2 x-e^{-x}+2 \log \left (2 e^x+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(E^x*(1 + 2*E^x)),x]

[Out]

-E^(-x) - 2*x + 2*Log[1 + 2*E^x]

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Rubi in Sympy [A]  time = 4.1642, size = 20, normalized size = 0.95 \[ 2 \log{\left (2 e^{x} + 1 \right )} - 2 \log{\left (e^{x} \right )} - e^{- x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/exp(x)/(1+2*exp(x)),x)

[Out]

2*log(2*exp(x) + 1) - 2*log(exp(x)) - exp(-x)

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Mathematica [A]  time = 0.0115379, size = 18, normalized size = 0.86 \[ 2 \log \left (e^{-x}+2\right )-e^{-x} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(E^x*(1 + 2*E^x)),x]

[Out]

-E^(-x) + 2*Log[2 + E^(-x)]

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Maple [A]  time = 0.017, size = 22, normalized size = 1.1 \[ - \left ({{\rm e}^{x}} \right ) ^{-1}-2\,\ln \left ({{\rm e}^{x}} \right ) +2\,\ln \left ( 1+2\,{{\rm e}^{x}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/exp(x)/(1+2*exp(x)),x)

[Out]

-1/exp(x)-2*ln(exp(x))+2*ln(1+2*exp(x))

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Maxima [A]  time = 1.33652, size = 22, normalized size = 1.05 \[ -e^{\left (-x\right )} + 2 \, \log \left (e^{\left (-x\right )} + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(-x)/(2*e^x + 1),x, algorithm="maxima")

[Out]

-e^(-x) + 2*log(e^(-x) + 2)

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Fricas [A]  time = 0.229665, size = 32, normalized size = 1.52 \[ -{\left (2 \, x e^{x} - 2 \, e^{x} \log \left (2 \, e^{x} + 1\right ) + 1\right )} e^{\left (-x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(-x)/(2*e^x + 1),x, algorithm="fricas")

[Out]

-(2*x*e^x - 2*e^x*log(2*e^x + 1) + 1)*e^(-x)

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Sympy [A]  time = 0.087837, size = 14, normalized size = 0.67 \[ 2 \log{\left (2 + e^{- x} \right )} - e^{- x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/exp(x)/(1+2*exp(x)),x)

[Out]

2*log(2 + exp(-x)) - exp(-x)

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GIAC/XCAS [A]  time = 0.200094, size = 26, normalized size = 1.24 \[ -2 \, x - e^{\left (-x\right )} + 2 \,{\rm ln}\left (2 \, e^{x} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(-x)/(2*e^x + 1),x, algorithm="giac")

[Out]

-2*x - e^(-x) + 2*ln(2*e^x + 1)