3.53.100 \(\int e^{-x} (e^x+e^{e^{-x} x} (3-3 x)) \, dx\)

Optimal. Leaf size=13 \[ 3 e^{e^{-x} x}+x \]

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Rubi [F]  time = 0.26, 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 e^{-x} \left (e^x+e^{e^{-x} x} (3-3 x)\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

x + 3*Defer[Int][E^(-(((-1 + E^x)*x)/E^x)), x] - 3*Defer[Int][x/E^(((-1 + E^x)*x)/E^x), x]

Rubi steps

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

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Mathematica [A]  time = 0.08, size = 13, normalized size = 1.00 \begin {gather*} 3 e^{e^{-x} x}+x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

3*E^(x/E^x) + x

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fricas [A]  time = 0.62, size = 11, normalized size = 0.85 \begin {gather*} x + 3 \, e^{\left (x e^{\left (-x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + 3*e^(x*e^(-x))

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giac [A]  time = 0.12, size = 11, normalized size = 0.85 \begin {gather*} x + 3 \, e^{\left (x e^{\left (-x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + 3*e^(x*e^(-x))

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maple [A]  time = 0.04, size = 12, normalized size = 0.92

method	result	size
risch	\(x +3 \,{\mathrm e}^{x \,{\mathrm e}^{-x}}\)	\(12\)
norman	\(\left ({\mathrm e}^{x} x +3 \,{\mathrm e}^{x} {\mathrm e}^{x \,{\mathrm e}^{-x}}\right ) {\mathrm e}^{-x}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x+3*exp(x*exp(-x))

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maxima [A]  time = 0.63, size = 11, normalized size = 0.85 \begin {gather*} x + 3 \, e^{\left (x e^{\left (-x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + 3*e^(x*e^(-x))

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mupad [B]  time = 3.49, size = 11, normalized size = 0.85 \begin {gather*} x+3\,{\mathrm {e}}^{x\,{\mathrm {e}}^{-x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-x)*(exp(x) - exp(x*exp(-x))*(3*x - 3)),x)

[Out]

x + 3*exp(x*exp(-x))

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sympy [A]  time = 0.14, size = 8, normalized size = 0.62 \begin {gather*} x + 3 e^{x e^{- x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + 3*exp(x*exp(-x))

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