3.89.75 \(\int \frac {-10+5 \log (3)+e^{e^x+x} (-10+5 \log (3))}{e^{e^x}+x} \, dx\)

Optimal. Leaf size=24 \[ (-10+5 \log (3)) \left (2+\log \left (\frac {1}{3} e^5 \left (e^{e^x}+x\right )\right )\right ) \]

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Rubi [A]  time = 0.06, antiderivative size = 16, normalized size of antiderivative = 0.67, number of steps used = 1, number of rules used = 1, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {6684} \begin {gather*} -5 (2-\log (3)) \log \left (x+e^{e^x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-10 + 5*Log[3] + E^(E^x + x)*(-10 + 5*Log[3]))/(E^E^x + x),x]

[Out]

-5*(2 - Log[3])*Log[E^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 {gather*} \begin {aligned} \text {integral} &=-5 (2-\log (3)) \log \left (e^{e^x}+x\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 14, normalized size = 0.58 \begin {gather*} 5 (-2+\log (3)) \log \left (e^{e^x}+x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-10 + 5*Log[3] + E^(E^x + x)*(-10 + 5*Log[3]))/(E^E^x + x),x]

[Out]

5*(-2 + Log[3])*Log[E^E^x + x]

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fricas [A]  time = 0.45, size = 26, normalized size = 1.08 \begin {gather*} -5 \, x \log \relax (3) + 5 \, {\left (\log \relax (3) - 2\right )} \log \left (x e^{x} + e^{\left (x + e^{x}\right )}\right ) + 10 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*log(3)-10)*exp(x)*exp(exp(x))+5*log(3)-10)/(x+exp(exp(x))),x, algorithm="fricas")

[Out]

-5*x*log(3) + 5*(log(3) - 2)*log(x*e^x + e^(x + e^x)) + 10*x

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giac [B]  time = 0.22, size = 37, normalized size = 1.54 \begin {gather*} -5 \, x \log \relax (3) + 5 \, \log \relax (3) \log \left (x e^{x} + e^{\left (x + e^{x}\right )}\right ) + 10 \, x - 10 \, \log \left (x e^{x} + e^{\left (x + e^{x}\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*log(3)-10)*exp(x)*exp(exp(x))+5*log(3)-10)/(x+exp(exp(x))),x, algorithm="giac")

[Out]

-5*x*log(3) + 5*log(3)*log(x*e^x + e^(x + e^x)) + 10*x - 10*log(x*e^x + e^(x + e^x))

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maple [A]  time = 0.05, size = 14, normalized size = 0.58

method	result	size
norman	\(\left (5 \ln \relax (3)-10\right ) \ln \left (x +{\mathrm e}^{{\mathrm e}^{x}}\right )\)	\(14\)
risch	\(5 \ln \left (x +{\mathrm e}^{{\mathrm e}^{x}}\right ) \ln \relax (3)-10 \ln \left (x +{\mathrm e}^{{\mathrm e}^{x}}\right )\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((5*ln(3)-10)*exp(x)*exp(exp(x))+5*ln(3)-10)/(x+exp(exp(x))),x,method=_RETURNVERBOSE)

[Out]

(5*ln(3)-10)*ln(x+exp(exp(x)))

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maxima [A]  time = 0.49, size = 12, normalized size = 0.50 \begin {gather*} 5 \, {\left (\log \relax (3) - 2\right )} \log \left (x + e^{\left (e^{x}\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*log(3)-10)*exp(x)*exp(exp(x))+5*log(3)-10)/(x+exp(exp(x))),x, algorithm="maxima")

[Out]

5*(log(3) - 2)*log(x + e^(e^x))

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mupad [B]  time = 5.16, size = 13, normalized size = 0.54 \begin {gather*} \ln \left (x+{\mathrm {e}}^{{\mathrm {e}}^x}\right )\,\left (5\,\ln \relax (3)-10\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*log(3) + exp(exp(x))*exp(x)*(5*log(3) - 10) - 10)/(x + exp(exp(x))),x)

[Out]

log(x + exp(exp(x)))*(5*log(3) - 10)

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sympy [A]  time = 0.14, size = 14, normalized size = 0.58 \begin {gather*} 5 \left (-2 + \log {\relax (3 )}\right ) \log {\left (x + e^{e^{x}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*ln(3)-10)*exp(x)*exp(exp(x))+5*ln(3)-10)/(x+exp(exp(x))),x)

[Out]

5*(-2 + log(3))*log(x + exp(exp(x)))

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