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

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

[Out]

ln(x+exp(x))

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Rubi [A]  time = 0.02, antiderivative size = 6, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.03, size = 6, normalized size = 1.00 \[ \log \left (x+e^x\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + E^x)/(E^x + x),x]

[Out]

Log[E^x + x]

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fricas [A]  time = 0.40, size = 5, normalized size = 0.83 \[ \log \left (x + e^{x}\right ) \]

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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giac [A]  time = 0.21, size = 5, normalized size = 0.83 \[ \log \left (x + e^{x}\right ) \]

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)

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maple [A]  time = 0.02, size = 6, normalized size = 1.00 \[ \ln \left (x +{\mathrm e}^{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(x+exp(x))

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maxima [A]  time = 1.00, size = 5, normalized size = 0.83 \[ \log \left (x + e^{x}\right ) \]

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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mupad [B]  time = 0.03, size = 5, normalized size = 0.83 \[ \ln \left (x+{\mathrm {e}}^x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x) + 1)/(x + exp(x)),x)

[Out]

log(x + exp(x))

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sympy [A]  time = 0.09, size = 5, normalized size = 0.83 \[ \log {\left (x + e^{x} \right )} \]

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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