3.55.92 \(\int \frac {e^{-\frac {3 e^{-e^{x^2}+2 x}}{x}} (-2 x+e^{-e^{x^2}+2 x} (6-12 x+12 e^{x^2} x^2))}{x^3} \, dx\)

Optimal. Leaf size=25 \[ \frac {2 e^{-\frac {3 e^{-e^{x^2}+2 x}}{x}}}{x} \]

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Rubi [B]  time = 0.20, antiderivative size = 99, normalized size of antiderivative = 3.96, number of steps used = 1, number of rules used = 1, integrand size = 57, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.018, Rules used = {2288} \begin {gather*} \frac {2 e^{-e^{x^2}-\frac {3 e^{2 x-e^{x^2}}}{x}+2 x} \left (2 e^{x^2} x^2-2 x+1\right )}{x^3 \left (\frac {e^{2 x-e^{x^2}}}{x^2}-\frac {2 e^{2 x-e^{x^2}} \left (1-e^{x^2} x\right )}{x}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2*x + E^(-E^x^2 + 2*x)*(6 - 12*x + 12*E^x^2*x^2))/(E^((3*E^(-E^x^2 + 2*x))/x)*x^3),x]

[Out]

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

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Mathematica [A]  time = 1.78, size = 25, normalized size = 1.00 \begin {gather*} \frac {2 e^{-\frac {3 e^{-e^{x^2}+2 x}}{x}}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2*x + E^(-E^x^2 + 2*x)*(6 - 12*x + 12*E^x^2*x^2))/(E^((3*E^(-E^x^2 + 2*x))/x)*x^3),x]

[Out]

2/(E^((3*E^(-E^x^2 + 2*x))/x)*x)

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fricas [A]  time = 1.49, size = 22, normalized size = 0.88 \begin {gather*} \frac {2 \, e^{\left (-\frac {3 \, e^{\left (2 \, x - e^{\left (x^{2}\right )}\right )}}{x}\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x^2*exp(x^2)-12*x+6)*exp(-exp(x^2)+2*x)-2*x)/x^3/exp(3*exp(-exp(x^2)+2*x)/x),x, algorithm="fric
as")

[Out]

2*e^(-3*e^(2*x - e^(x^2))/x)/x

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, {\left (3 \, {\left (2 \, x^{2} e^{\left (x^{2}\right )} - 2 \, x + 1\right )} e^{\left (2 \, x - e^{\left (x^{2}\right )}\right )} - x\right )} e^{\left (-\frac {3 \, e^{\left (2 \, x - e^{\left (x^{2}\right )}\right )}}{x}\right )}}{x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x^2*exp(x^2)-12*x+6)*exp(-exp(x^2)+2*x)-2*x)/x^3/exp(3*exp(-exp(x^2)+2*x)/x),x, algorithm="giac
")

[Out]

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

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maple [A]  time = 0.06, size = 23, normalized size = 0.92

method	result	size
risch	\(\frac {2 \,{\mathrm e}^{-\frac {3 \,{\mathrm e}^{-{\mathrm e}^{x^{2}}+2 x}}{x}}}{x}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((12*x^2*exp(x^2)-12*x+6)*exp(-exp(x^2)+2*x)-2*x)/x^3/exp(3*exp(-exp(x^2)+2*x)/x),x,method=_RETURNVERBOSE)

[Out]

2/x*exp(-3*exp(-exp(x^2)+2*x)/x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} 2 \, \int \frac {{\left (3 \, {\left (2 \, x^{2} e^{\left (x^{2}\right )} - 2 \, x + 1\right )} e^{\left (2 \, x - e^{\left (x^{2}\right )}\right )} - x\right )} e^{\left (-\frac {3 \, e^{\left (2 \, x - e^{\left (x^{2}\right )}\right )}}{x}\right )}}{x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x^2*exp(x^2)-12*x+6)*exp(-exp(x^2)+2*x)-2*x)/x^3/exp(3*exp(-exp(x^2)+2*x)/x),x, algorithm="maxi
ma")

[Out]

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

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mupad [B]  time = 3.67, size = 22, normalized size = 0.88 \begin {gather*} \frac {2\,{\mathrm {e}}^{-\frac {3\,{\mathrm {e}}^{-{\mathrm {e}}^{x^2}}\,{\mathrm {e}}^{2\,x}}{x}}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-(3*exp(2*x - exp(x^2)))/x)*(2*x - exp(2*x - exp(x^2))*(12*x^2*exp(x^2) - 12*x + 6)))/x^3,x)

[Out]

(2*exp(-(3*exp(-exp(x^2))*exp(2*x))/x))/x

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sympy [A]  time = 0.53, size = 17, normalized size = 0.68 \begin {gather*} \frac {2 e^{- \frac {3 e^{2 x - e^{x^{2}}}}{x}}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x**2*exp(x**2)-12*x+6)*exp(-exp(x**2)+2*x)-2*x)/x**3/exp(3*exp(-exp(x**2)+2*x)/x),x)

[Out]

2*exp(-3*exp(2*x - exp(x**2))/x)/x

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