\(\int e^{-1-x} (-9 e^{1+x}+e^{e^{-1-x} x} (1-x)) \, dx\) [8792]

Optimal result

Integrand size = 33, antiderivative size = 25 \[ \int e^{-1-x} \left (-9 e^{1+x}+e^{e^{-1-x} x} (1-x)\right ) \, dx=e^{e^{-1-x} x}-x-\log \left (3 e^{8 x}\right ) \]

[Out]

exp(x/exp(1+x))-ln(3*exp(8*x))-x

Rubi [F]

\[ \int e^{-1-x} \left (-9 e^{1+x}+e^{e^{-1-x} x} (1-x)\right ) \, dx=\int e^{-1-x} \left (-9 e^{1+x}+e^{e^{-1-x} x} (1-x)\right ) \, dx \]

[In]

Int[E^(-1 - x)*(-9*E^(1 + x) + E^(E^(-1 - x)*x)*(1 - x)),x]

[Out]

-9*x + Defer[Int][E^(-1 + (-1 + E^(-1 - x))*x), x] - Defer[Int][E^(-1 + (-1 + E^(-1 - x))*x)*x, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (-9-e^{-1+\left (-1+e^{-1-x}\right ) x} (-1+x)\right ) \, dx \\ & = -9 x-\int e^{-1+\left (-1+e^{-1-x}\right ) x} (-1+x) \, dx \\ & = -9 x-\int \left (-e^{-1+\left (-1+e^{-1-x}\right ) x}+e^{-1+\left (-1+e^{-1-x}\right ) x} x\right ) \, dx \\ & = -9 x+\int e^{-1+\left (-1+e^{-1-x}\right ) x} \, dx-\int e^{-1+\left (-1+e^{-1-x}\right ) x} x \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.60 \[ \int e^{-1-x} \left (-9 e^{1+x}+e^{e^{-1-x} x} (1-x)\right ) \, dx=e^{e^{-1-x} x}-9 x \]

[In]

Integrate[E^(-1 - x)*(-9*E^(1 + x) + E^(E^(-1 - x)*x)*(1 - x)),x]

[Out]

E^(E^(-1 - x)*x) - 9*x

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.56

method	result	size
risch	\(-9 x +{\mathrm e}^{x \,{\mathrm e}^{-1-x}}\)	\(14\)
parallelrisch	\(-9 x +{\mathrm e}^{x \,{\mathrm e}^{-1-x}}\)	\(14\)
parts	\(-9 x +{\mathrm e}^{x \,{\mathrm e}^{-1-x}}\)	\(14\)
norman	\(\left ({\mathrm e}^{1+x} {\mathrm e}^{x \,{\mathrm e}^{-1-x}}-9 x \,{\mathrm e}^{1+x}\right ) {\mathrm e}^{-1-x}\)	\(30\)

[In]

int(((1-x)*exp(x/exp(1+x))-9*exp(1+x))/exp(1+x),x,method=_RETURNVERBOSE)

[Out]

-9*x+exp(x*exp(-1-x))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.52 \[ \int e^{-1-x} \left (-9 e^{1+x}+e^{e^{-1-x} x} (1-x)\right ) \, dx=-9 \, x + e^{\left (x e^{\left (-x - 1\right )}\right )} \]

[In]

integrate(((1-x)*exp(x/exp(1+x))-9*exp(1+x))/exp(1+x),x, algorithm="fricas")

[Out]

-9*x + e^(x*e^(-x - 1))

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.48 \[ \int e^{-1-x} \left (-9 e^{1+x}+e^{e^{-1-x} x} (1-x)\right ) \, dx=- 9 x + e^{x e^{- x - 1}} \]

[In]

integrate(((1-x)*exp(x/exp(1+x))-9*exp(1+x))/exp(1+x),x)

[Out]

-9*x + exp(x*exp(-x - 1))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.52 \[ \int e^{-1-x} \left (-9 e^{1+x}+e^{e^{-1-x} x} (1-x)\right ) \, dx=-9 \, x + e^{\left (x e^{\left (-x - 1\right )}\right )} \]

[In]

integrate(((1-x)*exp(x/exp(1+x))-9*exp(1+x))/exp(1+x),x, algorithm="maxima")

[Out]

-9*x + e^(x*e^(-x - 1))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int e^{-1-x} \left (-9 e^{1+x}+e^{e^{-1-x} x} (1-x)\right ) \, dx=-{\left (9 \, x e^{\left (-x\right )} - e^{\left (x e^{\left (-x - 1\right )} - x\right )}\right )} e^{x} \]

[In]

integrate(((1-x)*exp(x/exp(1+x))-9*exp(1+x))/exp(1+x),x, algorithm="giac")

[Out]

-(9*x*e^(-x) - e^(x*e^(-x - 1) - x))*e^x

Mupad [B] (verification not implemented)

Time = 13.69 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.52 \[ \int e^{-1-x} \left (-9 e^{1+x}+e^{e^{-1-x} x} (1-x)\right ) \, dx={\mathrm {e}}^{x\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-1}}-9\,x \]

[In]

int(-exp(- x - 1)*(9*exp(x + 1) + exp(x*exp(- x - 1))*(x - 1)),x)

[Out]

exp(x*exp(-x)*exp(-1)) - 9*x