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

Optimal. Leaf size=8 \[ e \log \left (e^x+1\right ) \]

[Out]

E*Log[1 + E^x]

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Rubi [A]  time = 0.0340458, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2247, 2246, 31} \[ e \log \left (e^x+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[E^(1 + x)/(1 + E^x),x]

[Out]

E*Log[1 + E^x]

Rule 2247

Int[((a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.))^(p_.)*((G_)^((h_.)*((f_.) + (g_.)*(x_))))^(m_.),
x_Symbol] :> Dist[(G^(h*(f + g*x)))^m/(F^(e*(c + d*x)))^n, Int[(F^(e*(c + d*x)))^n*(a + b*(F^(e*(c + d*x)))^n)
^p, x], x] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, m, n, p}, x] && EqQ[d*e*n*Log[F], g*h*m*Log[G]]

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

Mathematica [A]  time = 0.0052211, size = 8, normalized size = 1. \[ e \log \left (e^x+1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^(1 + x)/(1 + E^x),x]

[Out]

E*Log[1 + E^x]

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Maple [A]  time = 0.001, size = 9, normalized size = 1.1 \begin{align*}{\rm e}\ln \left ( 1+{{\rm e}^{x}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(1+x)/(1+exp(x)),x)

[Out]

exp(1)*ln(1+exp(x))

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Maxima [A]  time = 0.931826, size = 11, normalized size = 1.38 \begin{align*} e \log \left (e^{x} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e*log(e^x + 1)

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Fricas [A]  time = 1.94737, size = 30, normalized size = 3.75 \begin{align*} e \log \left (e + e^{\left (x + 1\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e*log(e + e^(x + 1))

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Sympy [A]  time = 0.096068, size = 7, normalized size = 0.88 \begin{align*} e \log{\left (e^{x} + 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(1+x)/(1+exp(x)),x)

[Out]

E*log(exp(x) + 1)

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Giac [A]  time = 1.0825, size = 11, normalized size = 1.38 \begin{align*} e \log \left (e^{x} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e*log(e^x + 1)