3.19.35 \(\int \frac {e^{-e^{2/x}+e^{-e^{2/x}} (4-2 e^{2 x})} (480 e^{2/x}-60 e^{e^{2/x}} x+e^{2 x} (-240 e^{2/x}-240 x^2))}{x^3} \, dx\)

Optimal. Leaf size=29 \[ \frac {60 e^{2 e^{-e^{2/x}} \left (2-e^{2 x}\right )}}{x} \]

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Rubi [F]  time = 2.70, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {e^{-e^{2/x}+e^{-e^{2/x}} \left (4-2 e^{2 x}\right )} \left (480 e^{2/x}-60 e^{e^{2/x}} x+e^{2 x} \left (-240 e^{2/x}-240 x^2\right )\right )}{x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(-E^(2/x) + (4 - 2*E^(2*x))/E^E^(2/x))*(480*E^(2/x) - 60*E^E^(2/x)*x + E^(2*x)*(-240*E^(2/x) - 240*x^2)
))/x^3,x]

[Out]

480*Defer[Int][E^(-E^(2/x) + (4 - 2*E^(2*x))/E^E^(2/x) + 2/x)/x^3, x] - 240*Defer[Int][E^(-E^(2/x) + (4 - 2*E^
(2*x))/E^E^(2/x) + 2/x + 2*x)/x^3, x] - 60*Defer[Int][1/(E^((2*(-2 + E^(2*x)))/E^E^(2/x))*x^2), x] - 240*Defer
[Int][E^(-E^(2/x) + (4 - 2*E^(2*x))/E^E^(2/x) + 2*x)/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {60 e^{-e^{2/x}+e^{-e^{2/x}} \left (4-2 e^{2 x}\right )} \left (-8 e^{2/x}+e^{e^{2/x}} x\right )}{x^3}-\frac {240 \exp \left (-e^{2/x}+e^{-e^{2/x}} \left (4-2 e^{2 x}\right )+2 x\right ) \left (e^{2/x}+x^2\right )}{x^3}\right ) \, dx\\ &=-\left (60 \int \frac {e^{-e^{2/x}+e^{-e^{2/x}} \left (4-2 e^{2 x}\right )} \left (-8 e^{2/x}+e^{e^{2/x}} x\right )}{x^3} \, dx\right )-240 \int \frac {\exp \left (-e^{2/x}+e^{-e^{2/x}} \left (4-2 e^{2 x}\right )+2 x\right ) \left (e^{2/x}+x^2\right )}{x^3} \, dx\\ &=-\left (60 \int \left (-\frac {8 \exp \left (-e^{2/x}+e^{-e^{2/x}} \left (4-2 e^{2 x}\right )+\frac {2}{x}\right )}{x^3}+\frac {e^{e^{-e^{2/x}} \left (4-2 e^{2 x}\right )}}{x^2}\right ) \, dx\right )-240 \int \left (\frac {\exp \left (-e^{2/x}+e^{-e^{2/x}} \left (4-2 e^{2 x}\right )+\frac {2}{x}+2 x\right )}{x^3}+\frac {\exp \left (-e^{2/x}+e^{-e^{2/x}} \left (4-2 e^{2 x}\right )+2 x\right )}{x}\right ) \, dx\\ &=-\left (60 \int \frac {e^{e^{-e^{2/x}} \left (4-2 e^{2 x}\right )}}{x^2} \, dx\right )-240 \int \frac {\exp \left (-e^{2/x}+e^{-e^{2/x}} \left (4-2 e^{2 x}\right )+\frac {2}{x}+2 x\right )}{x^3} \, dx-240 \int \frac {\exp \left (-e^{2/x}+e^{-e^{2/x}} \left (4-2 e^{2 x}\right )+2 x\right )}{x} \, dx+480 \int \frac {\exp \left (-e^{2/x}+e^{-e^{2/x}} \left (4-2 e^{2 x}\right )+\frac {2}{x}\right )}{x^3} \, dx\\ &=-\left (60 \int \frac {e^{-2 e^{-e^{2/x}} \left (-2+e^{2 x}\right )}}{x^2} \, dx\right )-240 \int \frac {\exp \left (-e^{2/x}+e^{-e^{2/x}} \left (4-2 e^{2 x}\right )+\frac {2}{x}+2 x\right )}{x^3} \, dx-240 \int \frac {\exp \left (-e^{2/x}+e^{-e^{2/x}} \left (4-2 e^{2 x}\right )+2 x\right )}{x} \, dx+480 \int \frac {\exp \left (-e^{2/x}+e^{-e^{2/x}} \left (4-2 e^{2 x}\right )+\frac {2}{x}\right )}{x^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 27, normalized size = 0.93 \begin {gather*} \frac {60 e^{-2 e^{-e^{2/x}} \left (-2+e^{2 x}\right )}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-E^(2/x) + (4 - 2*E^(2*x))/E^E^(2/x))*(480*E^(2/x) - 60*E^E^(2/x)*x + E^(2*x)*(-240*E^(2/x) - 24
0*x^2)))/x^3,x]

[Out]

60/(E^((2*(-2 + E^(2*x)))/E^E^(2/x))*x)

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fricas [A]  time = 0.85, size = 45, normalized size = 1.55 \begin {gather*} \frac {60 \, e^{\left (-{\left (2 \, e^{\left (2 \, x\right )} + e^{\left (\frac {2}{x} + e^{\frac {2}{x}}\right )} - 4\right )} e^{\left (-e^{\frac {2}{x}}\right )} + e^{\frac {2}{x}}\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-60*x*exp(exp(2/x))+(-240*exp(2/x)-240*x^2)*exp(2*x)+480*exp(2/x))*exp((-2*exp(2*x)+4)/exp(exp(2/x)
))/x^3/exp(exp(2/x)),x, algorithm="fricas")

[Out]

60*e^(-(2*e^(2*x) + e^(2/x + e^(2/x)) - 4)*e^(-e^(2/x)) + e^(2/x))/x

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {60 \, {\left (4 \, {\left (x^{2} + e^{\frac {2}{x}}\right )} e^{\left (2 \, x\right )} + x e^{\left (e^{\frac {2}{x}}\right )} - 8 \, e^{\frac {2}{x}}\right )} e^{\left (-2 \, {\left (e^{\left (2 \, x\right )} - 2\right )} e^{\left (-e^{\frac {2}{x}}\right )} - e^{\frac {2}{x}}\right )}}{x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-60*x*exp(exp(2/x))+(-240*exp(2/x)-240*x^2)*exp(2*x)+480*exp(2/x))*exp((-2*exp(2*x)+4)/exp(exp(2/x)
))/x^3/exp(exp(2/x)),x, algorithm="giac")

[Out]

integrate(-60*(4*(x^2 + e^(2/x))*e^(2*x) + x*e^(e^(2/x)) - 8*e^(2/x))*e^(-2*(e^(2*x) - 2)*e^(-e^(2/x)) - e^(2/
x))/x^3, x)

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maple [A]  time = 0.17, size = 24, normalized size = 0.83

method	result	size
risch	\(\frac {60 \,{\mathrm e}^{-2 \left ({\mathrm e}^{2 x}-2\right ) {\mathrm e}^{-{\mathrm e}^{\frac {2}{x}}}}}{x}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-60*x*exp(exp(2/x))+(-240*exp(2/x)-240*x^2)*exp(2*x)+480*exp(2/x))*exp((-2*exp(2*x)+4)/exp(exp(2/x)))/x^3
/exp(exp(2/x)),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -60 \, \int \frac {{\left (4 \, {\left (x^{2} + e^{\frac {2}{x}}\right )} e^{\left (2 \, x\right )} + x e^{\left (e^{\frac {2}{x}}\right )} - 8 \, e^{\frac {2}{x}}\right )} e^{\left (-2 \, {\left (e^{\left (2 \, x\right )} - 2\right )} e^{\left (-e^{\frac {2}{x}}\right )} - e^{\frac {2}{x}}\right )}}{x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-60*x*exp(exp(2/x))+(-240*exp(2/x)-240*x^2)*exp(2*x)+480*exp(2/x))*exp((-2*exp(2*x)+4)/exp(exp(2/x)
))/x^3/exp(exp(2/x)),x, algorithm="maxima")

[Out]

-60*integrate((4*(x^2 + e^(2/x))*e^(2*x) + x*e^(e^(2/x)) - 8*e^(2/x))*e^(-2*(e^(2*x) - 2)*e^(-e^(2/x)) - e^(2/
x))/x^3, x)

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mupad [B]  time = 1.27, size = 23, normalized size = 0.79 \begin {gather*} \frac {60\,{\mathrm {e}}^{-2\,{\mathrm {e}}^{-{\mathrm {e}}^{2/x}}\,\left ({\mathrm {e}}^{2\,x}-2\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-exp(-exp(2/x))*(2*exp(2*x) - 4))*exp(-exp(2/x))*(exp(2*x)*(240*exp(2/x) + 240*x^2) - 480*exp(2/x) +
60*x*exp(exp(2/x))))/x^3,x)

[Out]

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

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sympy [A]  time = 0.85, size = 19, normalized size = 0.66 \begin {gather*} \frac {60 e^{\left (4 - 2 e^{2 x}\right ) e^{- e^{\frac {2}{x}}}}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-60*x*exp(exp(2/x))+(-240*exp(2/x)-240*x**2)*exp(2*x)+480*exp(2/x))*exp((-2*exp(2*x)+4)/exp(exp(2/x
)))/x**3/exp(exp(2/x)),x)

[Out]

60*exp((4 - 2*exp(2*x))*exp(-exp(2/x)))/x

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