Optimal. Leaf size=21 \[ -2 x-e^{-x}+2 \log \left (2 e^x+1\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0259144, 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, Rules used = {2248, 44} \[ -2 x-e^{-x}+2 \log \left (2 e^x+1\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2248
Rule 44
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*}
Mathematica [A] time = 0.0156914, size = 21, normalized size = 1. \[ -2 x-e^{-x}+2 \log \left (2 e^x+1\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.007, size = 22, normalized size = 1.1 \begin{align*} - \left ({{\rm e}^{x}} \right ) ^{-1}-2\,\ln \left ({{\rm e}^{x}} \right ) +2\,\ln \left ( 1+2\,{{\rm e}^{x}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.925633, size = 22, normalized size = 1.05 \begin{align*} -e^{\left (-x\right )} + 2 \, \log \left (e^{\left (-x\right )} + 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.02538, size = 62, normalized size = 2.95 \begin{align*} -{\left (2 \, x e^{x} - 2 \, e^{x} \log \left (2 \, e^{x} + 1\right ) + 1\right )} e^{\left (-x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.089545, size = 14, normalized size = 0.67 \begin{align*} 2 \log{\left (2 + e^{- x} \right )} - e^{- x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.052, size = 26, normalized size = 1.24 \begin{align*} -2 \, x - e^{\left (-x\right )} + 2 \, \log \left (2 \, e^{x} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]