3.19 \(\int \frac{e^x}{1+e^x} \, dx\)

Optimal. Leaf size=6 \[ \log \left (e^x+1\right ) \]

[Out]

Log[1 + E^x]

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Rubi [A]  time = 0.0258402, antiderivative size = 6, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \log \left (e^x+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[E^x/(1 + E^x),x]

[Out]

Log[1 + E^x]

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Rubi in Sympy [A]  time = 2.74219, size = 5, normalized size = 0.83 \[ \log{\left (e^{x} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(exp(x) + 1)

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Mathematica [A]  time = 0.00232404, size = 6, normalized size = 1. \[ \log \left (e^x+1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[E^x/(1 + E^x),x]

[Out]

Log[1 + E^x]

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Maple [A]  time = 0.003, size = 6, normalized size = 1. \[ \ln \left ( 1+{{\rm e}^{x}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(x)/(1+exp(x)),x)

[Out]

ln(1+exp(x))

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Maxima [A]  time = 1.32803, size = 7, normalized size = 1.17 \[ \log \left (e^{x} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(e^x + 1)

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Fricas [A]  time = 0.197977, size = 7, normalized size = 1.17 \[ \log \left (e^{x} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(e^x + 1)

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Sympy [A]  time = 0.057866, size = 5, normalized size = 0.83 \[ \log{\left (e^{x} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(x)/(1+exp(x)),x)

[Out]

log(exp(x) + 1)

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GIAC/XCAS [A]  time = 0.212002, size = 7, normalized size = 1.17 \[{\rm ln}\left (e^{x} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

ln(e^x + 1)