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