3.29.47 \(\int \frac {e^{-x} (-3 e^x x+e^{\frac {5 e^{-x}}{x}} ((-5-5 x) \log (3)+e^x x \log (3)))}{x} \, dx\) [2847]

Optimal. Leaf size=20 \[ -3 x+e^{\frac {5 e^{-x}}{x}} x \log (3) \]

[Out]

ln(3)*x*exp(5/exp(x)/x)-3*x

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Rubi [F]
time = 0.53, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {e^{-x} \left (-3 e^x x+e^{\frac {5 e^{-x}}{x}} \left ((-5-5 x) \log (3)+e^x x \log (3)\right )\right )}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-3+\frac {e^{\frac {5 e^{-x}}{x}-x} \left (-5+\left (-5+e^x\right ) x\right ) \log (3)}{x}\right ) \, dx\\ &=-3 x+\log (3) \int \frac {e^{\frac {5 e^{-x}}{x}-x} \left (-5+\left (-5+e^x\right ) x\right )}{x} \, dx\\ &=-3 x+\log (3) \int \left (e^{\frac {5 e^{-x}}{x}}-\frac {5 e^{\frac {5 e^{-x}}{x}-x} (1+x)}{x}\right ) \, dx\\ &=-3 x+\log (3) \int e^{\frac {5 e^{-x}}{x}} \, dx-(5 \log (3)) \int \frac {e^{\frac {5 e^{-x}}{x}-x} (1+x)}{x} \, dx\\ &=-3 x+\log (3) \int e^{\frac {5 e^{-x}}{x}} \, dx-(5 \log (3)) \int \left (e^{\frac {5 e^{-x}}{x}-x}+\frac {e^{\frac {5 e^{-x}}{x}-x}}{x}\right ) \, dx\\ &=-3 x+\log (3) \int e^{\frac {5 e^{-x}}{x}} \, dx-(5 \log (3)) \int e^{\frac {5 e^{-x}}{x}-x} \, dx-(5 \log (3)) \int \frac {e^{\frac {5 e^{-x}}{x}-x}}{x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.08, size = 20, normalized size = 1.00 \begin {gather*} -3 x+e^{\frac {5 e^{-x}}{x}} x \log (3) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-3*x + E^(5/(E^x*x))*x*Log[3]

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Maple [A]
time = 0.24, size = 19, normalized size = 0.95

method	result	size
risch	\(\ln \left (3\right ) x \,{\mathrm e}^{\frac {5 \,{\mathrm e}^{-x}}{x}}-3 x\)	\(19\)
norman	\(\left (\ln \left (3\right ) {\mathrm e}^{x} {\mathrm e}^{\frac {5 \,{\mathrm e}^{-x}}{x}} x -3 \,{\mathrm e}^{x} x \right ) {\mathrm e}^{-x}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(3)*x*exp(5*exp(-x)/x)-3*x

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*x + integrate((x*e^x*log(3) - 5*x*log(3) - 5*log(3))*e^(-x + 5*e^(-x)/x)/x, x)

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Fricas [A]
time = 0.59, size = 18, normalized size = 0.90 \begin {gather*} x e^{\left (\frac {5 \, e^{\left (-x\right )}}{x}\right )} \log \left (3\right ) - 3 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*e^(5*e^(-x)/x)*log(3) - 3*x

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Sympy [A]
time = 2.26, size = 15, normalized size = 0.75 \begin {gather*} x e^{\frac {5 e^{- x}}{x}} \log {\left (3 \right )} - 3 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*exp(5*exp(-x)/x)*log(3) - 3*x

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Giac [A]
time = 0.38, size = 18, normalized size = 0.90 \begin {gather*} x e^{\left (\frac {5 \, e^{\left (-x\right )}}{x}\right )} \log \left (3\right ) - 3 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*e^(5*e^(-x)/x)*log(3) - 3*x

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Mupad [B]
time = 1.73, size = 17, normalized size = 0.85 \begin {gather*} x\,\left ({\mathrm {e}}^{\frac {5\,{\mathrm {e}}^{-x}}{x}}\,\ln \left (3\right )-3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*(exp((5*exp(-x))/x)*log(3) - 3)

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