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

Optimal. Leaf size=10 \[ x-\log \left (e^x+1\right ) \]

[Out]

x - Log[1 + E^x]

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Rubi [A]  time = 0.0134802, 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 \[ x-\log \left (e^x+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 + E^x)^(-1),x]

[Out]

x - Log[1 + E^x]

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Rubi in Sympy [A]  time = 1.18841, size = 10, normalized size = 1. \[ - \log{\left (e^{x} + 1 \right )} + \log{\left (e^{x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(1+exp(x)),x)

[Out]

-log(exp(x) + 1) + log(exp(x))

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Mathematica [A]  time = 0.00310863, size = 10, normalized size = 1. \[ x-\log \left (e^x+1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + E^x)^(-1),x]

[Out]

x - Log[1 + E^x]

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Maple [A]  time = 0.008, size = 12, normalized size = 1.2 \[ -\ln \left ( 1+{{\rm e}^{x}} \right ) +\ln \left ({{\rm e}^{x}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1+exp(x)),x)

[Out]

-ln(1+exp(x))+ln(exp(x))

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Maxima [A]  time = 1.34933, size = 12, normalized size = 1.2 \[ x - \log \left (e^{x} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x - log(e^x + 1)

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Fricas [A]  time = 0.218086, size = 12, normalized size = 1.2 \[ x - \log \left (e^{x} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x - log(e^x + 1)

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Sympy [A]  time = 0.055553, size = 7, normalized size = 0.7 \[ x - \log{\left (e^{x} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(1+exp(x)),x)

[Out]

x - log(exp(x) + 1)

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GIAC/XCAS [A]  time = 0.219186, size = 12, normalized size = 1.2 \[ x -{\rm ln}\left (e^{x} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x - ln(e^x + 1)