3.32.55 \(\int \frac {e^{-2-e^x} (-2-2 e^x x)}{x^2} \, dx\)

Optimal. Leaf size=14 \[ \frac {2 e^{-2-e^x}}{x} \]

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Rubi [A]  time = 0.04, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2288} \begin {gather*} \frac {2 e^{-e^x-2}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*E^(-2 - E^x))/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^{-2-e^x}}{x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 14, normalized size = 1.00 \begin {gather*} \frac {2 e^{-2-e^x}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*E^(-2 - E^x))/x

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fricas [A]  time = 0.66, size = 12, normalized size = 0.86 \begin {gather*} \frac {2 \, e^{\left (-e^{x} - 2\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*e^(-e^x - 2)/x

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giac [A]  time = 0.33, size = 12, normalized size = 0.86 \begin {gather*} \frac {2 \, e^{\left (-e^{x} - 2\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*e^(-e^x - 2)/x

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maple [A]  time = 0.08, size = 13, normalized size = 0.93

method	result	size
risch	\(\frac {2 \,{\mathrm e}^{-2-{\mathrm e}^{x}}}{x}\)	\(13\)
norman	\(\frac {2 \,{\mathrm e} \,{\mathrm e}^{x} {\mathrm e}^{-{\mathrm e}^{x}-3-x}}{x}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/x*exp(-2-exp(x))

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maxima [A]  time = 0.45, size = 12, normalized size = 0.86 \begin {gather*} \frac {2 \, e^{\left (-e^{x} - 2\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*e^(-e^x - 2)/x

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mupad [B]  time = 1.81, size = 12, normalized size = 0.86 \begin {gather*} \frac {2\,{\mathrm {e}}^{-2}\,{\mathrm {e}}^{-{\mathrm {e}}^x}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*exp(-2)*exp(-exp(x)))/x

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sympy [A]  time = 0.14, size = 19, normalized size = 1.36 \begin {gather*} \frac {2 e e^{x} e^{- x - e^{x} - 3}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*E*exp(x)*exp(-x - exp(x) - 3)/x

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