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 = 0.014687, 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, Rules used = {2246, 31} \[ \log \left (e^x+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[E^x/(1 + E^x),x]

[Out]

Log[1 + E^x]

Rule 2246

Int[((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)*((a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.))^(p_.),
x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int[(a + b*x)^p, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b,
c, d, e, n, p}, x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{e^x}{1+e^x} \, dx &=\operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,e^x\right )\\ &=\log \left (1+e^x\right )\\ \end{align*}

Mathematica [A]  time = 0.0040945, 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 = 0.001, size = 6, normalized size = 1. \begin{align*} \ln \left ( 1+{{\rm e}^{x}} \right ) \end{align*}

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 = 0.922343, size = 7, normalized size = 1.17 \begin{align*} \log \left (e^{x} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)/(1+exp(x)),x, algorithm="maxima")

[Out]

log(e^x + 1)

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Fricas [A]  time = 1.68215, size = 19, normalized size = 3.17 \begin{align*} \log \left (e^{x} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)/(1+exp(x)),x, algorithm="fricas")

[Out]

log(e^x + 1)

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Sympy [A]  time = 0.084821, size = 5, normalized size = 0.83 \begin{align*} \log{\left (e^{x} + 1 \right )} \end{align*}

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 [A]  time = 1.09375, size = 7, normalized size = 1.17 \begin{align*} \log \left (e^{x} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)/(1+exp(x)),x, algorithm="giac")

[Out]

log(e^x + 1)