3.9.78 \(\int e^{-x+e^{-x} (-x-x^2+e^x (1+2 x))} (-1+2 e^x-x+x^2) \, dx\)

Optimal. Leaf size=18 \[ e^{1+2 x-e^{-x} x (1+x)} \]

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Rubi [F]  time = 0.64, 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+e^{-x} \left (-x-x^2+e^x (1+2 x)\right )} \left (-1+2 e^x-x+x^2\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[E^(-x + (-x - x^2 + E^x*(1 + 2*x))/E^x)*(-1 + 2*E^x - x + x^2),x]

[Out]

-Defer[Int][E^(((E^x - x)*(1 + x))/E^x), x] + 2*Defer[Int][E^(x + ((E^x - x)*(1 + x))/E^x), x] - Defer[Int][E^
(((E^x - x)*(1 + x))/E^x)*x, x] + Defer[Int][E^(((E^x - x)*(1 + x))/E^x)*x^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int e^{e^{-x} \left (e^x-x\right ) (1+x)} \left (-1+2 e^x-x+x^2\right ) \, dx\\ &=\int \left (-e^{e^{-x} \left (e^x-x\right ) (1+x)}+2 e^{x+e^{-x} \left (e^x-x\right ) (1+x)}-e^{e^{-x} \left (e^x-x\right ) (1+x)} x+e^{e^{-x} \left (e^x-x\right ) (1+x)} x^2\right ) \, dx\\ &=2 \int e^{x+e^{-x} \left (e^x-x\right ) (1+x)} \, dx-\int e^{e^{-x} \left (e^x-x\right ) (1+x)} \, dx-\int e^{e^{-x} \left (e^x-x\right ) (1+x)} x \, dx+\int e^{e^{-x} \left (e^x-x\right ) (1+x)} x^2 \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.21, size = 18, normalized size = 1.00 \begin {gather*} e^{1+2 x-e^{-x} x (1+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(-x + (-x - x^2 + E^x*(1 + 2*x))/E^x)*(-1 + 2*E^x - x + x^2),x]

[Out]

E^(1 + 2*x - (x*(1 + x))/E^x)

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fricas [A]  time = 0.75, size = 21, normalized size = 1.17 \begin {gather*} e^{\left (-{\left (x^{2} - {\left (x + 1\right )} e^{x} + x\right )} e^{\left (-x\right )} + x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(x)+x^2-x-1)*exp(((2*x+1)*exp(x)-x^2-x)/exp(x))/exp(x),x, algorithm="fricas")

[Out]

e^(-(x^2 - (x + 1)*e^x + x)*e^(-x) + x)

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giac [A]  time = 0.41, size = 22, normalized size = 1.22 \begin {gather*} e^{\left (-x^{2} e^{\left (-x\right )} - x e^{\left (-x\right )} + 2 \, x + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(x)+x^2-x-1)*exp(((2*x+1)*exp(x)-x^2-x)/exp(x))/exp(x),x, algorithm="giac")

[Out]

e^(-x^2*e^(-x) - x*e^(-x) + 2*x + 1)

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maple [A]  time = 0.07, size = 23, normalized size = 1.28

method	result	size
risch	\({\mathrm e}^{\left (2 \,{\mathrm e}^{x} x -x^{2}+{\mathrm e}^{x}-x \right ) {\mathrm e}^{-x}}\)	\(23\)
norman	\({\mathrm e}^{\left (\left (2 x +1\right ) {\mathrm e}^{x}-x^{2}-x \right ) {\mathrm e}^{-x}}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*exp(x)+x^2-x-1)*exp(((2*x+1)*exp(x)-x^2-x)/exp(x))/exp(x),x,method=_RETURNVERBOSE)

[Out]

exp((2*exp(x)*x-x^2+exp(x)-x)*exp(-x))

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maxima [A]  time = 0.49, size = 22, normalized size = 1.22 \begin {gather*} e^{\left (-x^{2} e^{\left (-x\right )} - x e^{\left (-x\right )} + 2 \, x + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(x)+x^2-x-1)*exp(((2*x+1)*exp(x)-x^2-x)/exp(x))/exp(x),x, algorithm="maxima")

[Out]

e^(-x^2*e^(-x) - x*e^(-x) + 2*x + 1)

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mupad [B]  time = 0.66, size = 25, normalized size = 1.39 \begin {gather*} {\mathrm {e}}^{2\,x}\,\mathrm {e}\,{\mathrm {e}}^{-x\,{\mathrm {e}}^{-x}}\,{\mathrm {e}}^{-x^2\,{\mathrm {e}}^{-x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-exp(-x)*(x - exp(x)*(2*x + 1) + x^2))*exp(-x)*(x - 2*exp(x) - x^2 + 1),x)

[Out]

exp(2*x)*exp(1)*exp(-x*exp(-x))*exp(-x^2*exp(-x))

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sympy [A]  time = 0.19, size = 17, normalized size = 0.94 \begin {gather*} e^{\left (- x^{2} - x + \left (2 x + 1\right ) e^{x}\right ) e^{- x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(x)+x**2-x-1)*exp(((2*x+1)*exp(x)-x**2-x)/exp(x))/exp(x),x)

[Out]

exp((-x**2 - x + (2*x + 1)*exp(x))*exp(-x))

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