3.47.78 \(\int \frac {e^{-e^{-\frac {2}{x^2}}+x} (-8 e^{-\frac {2}{x^2}}+2 x^3)}{x^3} \, dx\)

Optimal. Leaf size=15 \[ 2 e^{-e^{-\frac {2}{x^2}}+x} \]

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Rubi [A]  time = 0.27, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {6706} \begin {gather*} 2 e^{x-e^{-\frac {2}{x^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(-E^(-2/x^2) + x)*(-8/E^(2/x^2) + 2*x^3))/x^3,x]

[Out]

2*E^(-E^(-2/x^2) + x)

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=2 e^{-e^{-\frac {2}{x^2}}+x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.14, size = 15, normalized size = 1.00 \begin {gather*} 2 e^{-e^{-\frac {2}{x^2}}+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-E^(-2/x^2) + x)*(-8/E^(2/x^2) + 2*x^3))/x^3,x]

[Out]

2*E^(-E^(-2/x^2) + x)

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fricas [A]  time = 0.94, size = 13, normalized size = 0.87 \begin {gather*} 2 \, e^{\left (x - e^{\left (-\frac {2}{x^{2}}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*exp(-1/x^2)^2+2*x^3)*exp(-exp(-1/x^2)^2+x)/x^3,x, algorithm="fricas")

[Out]

2*e^(x - e^(-2/x^2))

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giac [A]  time = 0.18, size = 13, normalized size = 0.87 \begin {gather*} 2 \, e^{\left (x - e^{\left (-\frac {2}{x^{2}}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*exp(-1/x^2)^2+2*x^3)*exp(-exp(-1/x^2)^2+x)/x^3,x, algorithm="giac")

[Out]

2*e^(x - e^(-2/x^2))

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maple [A]  time = 0.05, size = 14, normalized size = 0.93

method	result	size
risch	\(2 \,{\mathrm e}^{-{\mathrm e}^{-\frac {2}{x^{2}}}+x}\)	\(14\)
norman	\(2 \,{\mathrm e}^{-{\mathrm e}^{-\frac {2}{x^{2}}}+x}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-8*exp(-1/x^2)^2+2*x^3)*exp(-exp(-1/x^2)^2+x)/x^3,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} 2 \, \int \frac {{\left (x^{3} - 4 \, e^{\left (-\frac {2}{x^{2}}\right )}\right )} e^{\left (x - e^{\left (-\frac {2}{x^{2}}\right )}\right )}}{x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*exp(-1/x^2)^2+2*x^3)*exp(-exp(-1/x^2)^2+x)/x^3,x, algorithm="maxima")

[Out]

2*integrate((x^3 - 4*e^(-2/x^2))*e^(x - e^(-2/x^2))/x^3, x)

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mupad [B]  time = 3.22, size = 13, normalized size = 0.87 \begin {gather*} 2\,{\mathrm {e}}^{-{\mathrm {e}}^{-\frac {2}{x^2}}}\,{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x - exp(-2/x^2))*(8*exp(-2/x^2) - 2*x^3))/x^3,x)

[Out]

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

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sympy [A]  time = 0.32, size = 12, normalized size = 0.80 \begin {gather*} 2 e^{x - e^{- \frac {2}{x^{2}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*exp(-1/x**2)**2+2*x**3)*exp(-exp(-1/x**2)**2+x)/x**3,x)

[Out]

2*exp(x - exp(-2/x**2))

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