Optimal. Leaf size=10 \[ x-\log \left (e^x+1\right ) \]
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Rubi [A] time = 0.0061695, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {2282, 36, 29, 31} \[ x-\log \left (e^x+1\right ) \]
Antiderivative was successfully verified.
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Rule 2282
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{1+e^x} \, dx &=\operatorname{Subst}\left (\int \frac{1}{x (1+x)} \, dx,x,e^x\right )\\ &=\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,e^x\right )-\operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,e^x\right )\\ &=x-\log \left (1+e^x\right )\\ \end{align*}
Mathematica [A] time = 0.0025529, size = 10, normalized size = 1. \[ x-\log \left (e^x+1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 12, normalized size = 1.2 \begin{align*} \ln \left ({{\rm e}^{x}} \right ) -\ln \left ( 1+{{\rm e}^{x}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.960611, size = 12, normalized size = 1.2 \begin{align*} x - \log \left (e^{x} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.478029, size = 24, normalized size = 2.4 \begin{align*} x - \log \left (e^{x} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.070752, size = 7, normalized size = 0.7 \begin{align*} x - \log{\left (e^{x} + 1 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07975, size = 12, normalized size = 1.2 \begin{align*} x - \log \left (e^{x} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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