Optimal. Leaf size=10 \[ x-\log \left (e^x+1\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 10, normalized size of antiderivative = 1.00, 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 29
Rule 31
Rule 36
Rule 2282
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*}
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Mathematica [A] time = 0.00, size = 10, normalized size = 1.00 \[ x-\log \left (e^x+1\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 9, normalized size = 0.90 \[ x - \log \left (e^{x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.01, size = 9, normalized size = 0.90 \[ x - \log \left (e^{x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 12, normalized size = 1.20 \[ -\ln \left ({\mathrm e}^{x}+1\right )+\ln \left ({\mathrm e}^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 9, normalized size = 0.90 \[ x - \log \left (e^{x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 9, normalized size = 0.90 \[ x-\ln \left ({\mathrm {e}}^x+1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 7, normalized size = 0.70 \[ x - \log {\left (e^{x} + 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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