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

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

[Out]

Log[E^x + x]

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Rubi [A]  time = 0.0195129, antiderivative size = 6, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {6684} \[ \log \left (x+e^x\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 + E^x)/(E^x + x),x]

[Out]

Log[E^x + x]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rubi steps

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

Mathematica [A]  time = 0.0298571, size = 6, normalized size = 1. \[ \log \left (x+e^x\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + E^x)/(E^x + x),x]

[Out]

Log[E^x + x]

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Maple [A]  time = 0.017, size = 6, normalized size = 1. \begin{align*} \ln \left ({{\rm e}^{x}}+x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(exp(x)+x)

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Maxima [A]  time = 0.980101, size = 7, normalized size = 1.17 \begin{align*} \log \left (x + e^{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x + e^x)

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Fricas [A]  time = 0.89915, size = 19, normalized size = 3.17 \begin{align*} \log \left (x + e^{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x + e^x)

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Sympy [A]  time = 0.095375, size = 5, normalized size = 0.83 \begin{align*} \log{\left (x + e^{x} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x + exp(x))

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Giac [A]  time = 1.2249, size = 7, normalized size = 1.17 \begin{align*} \log \left (x + e^{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x + e^x)