3.81.82 \(\int e^{-4-e^x (-1+x)-x} (4 x-2 x^2-2 e^x x^3) \, dx\)

Optimal. Leaf size=21 \[ 2 e^{-4+e^x-x-e^x x} x^2 \]

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Rubi [B]  time = 0.06, antiderivative size = 46, normalized size of antiderivative = 2.19, number of steps used = 1, number of rules used = 1, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {2288} \begin {gather*} \frac {2 e^{e^x (1-x)-x-4} \left (e^x x^3+x^2\right )}{-e^x (1-x)+e^x+1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(-4 - E^x*(-1 + x) - x)*(4*x - 2*x^2 - 2*E^x*x^3),x]

[Out]

(2*E^(-4 + E^x*(1 - x) - x)*(x^2 + E^x*x^3))/(1 + E^x - E^x*(1 - 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 {2 e^{-4+e^x (1-x)-x} \left (x^2+e^x x^3\right )}{1+e^x-e^x (1-x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 20, normalized size = 0.95 \begin {gather*} 2 e^{-4-e^x (-1+x)-x} x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(-4 - E^x*(-1 + x) - x)*(4*x - 2*x^2 - 2*E^x*x^3),x]

[Out]

2*E^(-4 - E^x*(-1 + x) - x)*x^2

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fricas [A]  time = 0.59, size = 18, normalized size = 0.86 \begin {gather*} 2 \, x^{2} e^{\left (-{\left (x - 1\right )} e^{x} - x - 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*exp(x)*x^3-2*x^2+4*x)/exp(4)/exp((x-1)*exp(x)+x),x, algorithm="fricas")

[Out]

2*x^2*e^(-(x - 1)*e^x - x - 4)

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giac [A]  time = 0.14, size = 18, normalized size = 0.86 \begin {gather*} 2 \, x^{2} e^{\left (-x e^{x} - x + e^{x} - 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*exp(x)*x^3-2*x^2+4*x)/exp(4)/exp((x-1)*exp(x)+x),x, algorithm="giac")

[Out]

2*x^2*e^(-x*e^x - x + e^x - 4)

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maple [A]  time = 0.06, size = 19, normalized size = 0.90

method	result	size
risch	\(2 x^{2} {\mathrm e}^{-4-{\mathrm e}^{x} x +{\mathrm e}^{x}-x}\)	\(19\)
norman	\(2 x^{2} {\mathrm e}^{-4} {\mathrm e}^{-\left (x -1\right ) {\mathrm e}^{x}-x}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*exp(x)*x^3-2*x^2+4*x)/exp(4)/exp((x-1)*exp(x)+x),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.43, size = 18, normalized size = 0.86 \begin {gather*} 2 \, x^{2} e^{\left (-x e^{x} - x + e^{x} - 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*exp(x)*x^3-2*x^2+4*x)/exp(4)/exp((x-1)*exp(x)+x),x, algorithm="maxima")

[Out]

2*x^2*e^(-x*e^x - x + e^x - 4)

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mupad [B]  time = 6.74, size = 20, normalized size = 0.95 \begin {gather*} 2\,x^2\,{\mathrm {e}}^{-x\,{\mathrm {e}}^x}\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^{-4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-4)*exp(- x - exp(x)*(x - 1))*(2*x^3*exp(x) - 4*x + 2*x^2),x)

[Out]

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

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sympy [A]  time = 0.22, size = 19, normalized size = 0.90 \begin {gather*} \frac {2 x^{2} e^{- x - \left (x - 1\right ) e^{x}}}{e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*exp(x)*x**3-2*x**2+4*x)/exp(4)/exp((x-1)*exp(x)+x),x)

[Out]

2*x**2*exp(-4)*exp(-x - (x - 1)*exp(x))

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