\(\int e^{-x} (e^x+e^{e^{-x} x} (3-3 x)) \, dx\) [5300]

Optimal result

Integrand size = 25, antiderivative size = 13 \[ \int e^{-x} \left (e^x+e^{e^{-x} x} (3-3 x)\right ) \, dx=3 e^{e^{-x} x}+x \]

[Out]

x+3*exp(x/exp(x))

Rubi [F]

\[ \int e^{-x} \left (e^x+e^{e^{-x} x} (3-3 x)\right ) \, dx=\int e^{-x} \left (e^x+e^{e^{-x} x} (3-3 x)\right ) \, dx \]

[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{align*} \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{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int e^{-x} \left (e^x+e^{e^{-x} x} (3-3 x)\right ) \, dx=3 e^{e^{-x} x}+x \]

[In]

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

[Out]

3*E^(x/E^x) + x

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int e^{-x} \left (e^x+e^{e^{-x} x} (3-3 x)\right ) \, dx=x + 3 \, e^{\left (x e^{\left (-x\right )}\right )} \]

[In]

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

[Out]

x + 3*e^(x*e^(-x))

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.62 \[ \int e^{-x} \left (e^x+e^{e^{-x} x} (3-3 x)\right ) \, dx=x + 3 e^{x e^{- x}} \]

[In]

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

[Out]

x + 3*exp(x*exp(-x))

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int e^{-x} \left (e^x+e^{e^{-x} x} (3-3 x)\right ) \, dx=x + 3 \, e^{\left (x e^{\left (-x\right )}\right )} \]

[In]

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

[Out]

x + 3*e^(x*e^(-x))

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int e^{-x} \left (e^x+e^{e^{-x} x} (3-3 x)\right ) \, dx=x + 3 \, e^{\left (x e^{\left (-x\right )}\right )} \]

[In]

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

[Out]

x + 3*e^(x*e^(-x))

Mupad [B] (verification not implemented)

Time = 13.12 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int e^{-x} \left (e^x+e^{e^{-x} x} (3-3 x)\right ) \, dx=x+3\,{\mathrm {e}}^{x\,{\mathrm {e}}^{-x}} \]

[In]

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

[Out]

x + 3*exp(x*exp(-x))