3.22.5 \(\int e^{-2+e^x-x} (-8 x+4 x^2-4 e^x x^2) \, dx\)

Optimal. Leaf size=19 \[ e^5-4 e^{-2+e^x-x} x^2 \]

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Rubi [A]  time = 0.05, antiderivative size = 33, normalized size of antiderivative = 1.74, number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {2288} \begin {gather*} -\frac {4 e^{-x+e^x-2} \left (x^2-e^x x^2\right )}{1-e^x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(-2 + E^x - x)*(-8*x + 4*x^2 - 4*E^x*x^2),x]

[Out]

(-4*E^(-2 + E^x - x)*(x^2 - E^x*x^2))/(1 - E^x)

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\frac {4 e^{-2+e^x-x} \left (x^2-e^x x^2\right )}{1-e^x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 15, normalized size = 0.79 \begin {gather*} -4 e^{-2+e^x-x} x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(-2 + E^x - x)*(-8*x + 4*x^2 - 4*E^x*x^2),x]

[Out]

-4*E^(-2 + E^x - x)*x^2

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fricas [A]  time = 0.71, size = 13, normalized size = 0.68 \begin {gather*} -4 \, x^{2} e^{\left (-x + e^{x} - 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*exp(x)*x^2+4*x^2-8*x)/exp(2)/exp(x-exp(x)),x, algorithm="fricas")

[Out]

-4*x^2*e^(-x + e^x - 2)

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giac [A]  time = 0.28, size = 13, normalized size = 0.68 \begin {gather*} -4 \, x^{2} e^{\left (-x + e^{x} - 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*exp(x)*x^2+4*x^2-8*x)/exp(2)/exp(x-exp(x)),x, algorithm="giac")

[Out]

-4*x^2*e^(-x + e^x - 2)

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maple [A]  time = 0.04, size = 14, normalized size = 0.74

method	result	size
risch	\(-4 x^{2} {\mathrm e}^{{\mathrm e}^{x}-2-x}\)	\(14\)
norman	\(-4 \,{\mathrm e}^{-2} x^{2} {\mathrm e}^{{\mathrm e}^{x}-x}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-4*exp(x)*x^2+4*x^2-8*x)/exp(2)/exp(x-exp(x)),x,method=_RETURNVERBOSE)

[Out]

-4*x^2*exp(exp(x)-2-x)

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maxima [A]  time = 0.49, size = 13, normalized size = 0.68 \begin {gather*} -4 \, x^{2} e^{\left (-x + e^{x} - 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*exp(x)*x^2+4*x^2-8*x)/exp(2)/exp(x-exp(x)),x, algorithm="maxima")

[Out]

-4*x^2*e^(-x + e^x - 2)

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mupad [B]  time = 1.14, size = 14, normalized size = 0.74 \begin {gather*} -4\,x^2\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^{-2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-2)*exp(exp(x) - x)*(8*x + 4*x^2*exp(x) - 4*x^2),x)

[Out]

-4*x^2*exp(-x)*exp(exp(x))*exp(-2)

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sympy [A]  time = 0.18, size = 15, normalized size = 0.79 \begin {gather*} - \frac {4 x^{2} e^{- x + e^{x}}}{e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*exp(x)*x**2+4*x**2-8*x)/exp(2)/exp(x-exp(x)),x)

[Out]

-4*x**2*exp(-2)*exp(-x + exp(x))

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