3.67.19 \(\int \frac {e^{5+e^{\frac {-40+8 x-96 x^2}{x}}+\frac {-40+8 x-96 x^2}{x}} (40-96 x^2)}{x^2} \, dx\)

Optimal. Leaf size=19 \[ e^{5+e^{8 \left (\frac {-5+x}{x}-12 x\right )}} \]

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Rubi [F]  time = 0.72, 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 {\exp \left (5+e^{\frac {-40+8 x-96 x^2}{x}}+\frac {-40+8 x-96 x^2}{x}\right ) \left (40-96 x^2\right )}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(5 + E^((-40 + 8*x - 96*x^2)/x) + (-40 + 8*x - 96*x^2)/x)*(40 - 96*x^2))/x^2,x]

[Out]

-96*Defer[Int][E^(13 + E^(8 - 40/x - 96*x) - 40/x - 96*x), x] + 40*Defer[Int][E^(13 + E^(8 - 40/x - 96*x) - 40
/x - 96*x)/x^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-96 \exp \left (5+e^{\frac {-40+8 x-96 x^2}{x}}+\frac {-40+8 x-96 x^2}{x}\right )+\frac {40 \exp \left (5+e^{\frac {-40+8 x-96 x^2}{x}}+\frac {-40+8 x-96 x^2}{x}\right )}{x^2}\right ) \, dx\\ &=40 \int \frac {\exp \left (5+e^{\frac {-40+8 x-96 x^2}{x}}+\frac {-40+8 x-96 x^2}{x}\right )}{x^2} \, dx-96 \int \exp \left (5+e^{\frac {-40+8 x-96 x^2}{x}}+\frac {-40+8 x-96 x^2}{x}\right ) \, dx\\ &=40 \int \frac {e^{13+e^{8-\frac {40}{x}-96 x}-\frac {40}{x}-96 x}}{x^2} \, dx-96 \int e^{13+e^{8-\frac {40}{x}-96 x}-\frac {40}{x}-96 x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.19, size = 16, normalized size = 0.84 \begin {gather*} e^{5+e^{8-\frac {40}{x}-96 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(5 + E^((-40 + 8*x - 96*x^2)/x) + (-40 + 8*x - 96*x^2)/x)*(40 - 96*x^2))/x^2,x]

[Out]

E^(5 + E^(8 - 40/x - 96*x))

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fricas [B]  time = 0.54, size = 51, normalized size = 2.68 \begin {gather*} e^{\left (-\frac {96 \, x^{2} - x e^{\left (-\frac {8 \, {\left (12 \, x^{2} - x + 5\right )}}{x}\right )} - 13 \, x + 40}{x} + \frac {8 \, {\left (12 \, x^{2} - x + 5\right )}}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-96*x^2+40)*exp((-96*x^2+8*x-40)/x)*exp(exp((-96*x^2+8*x-40)/x)+5)/x^2,x, algorithm="fricas")

[Out]

e^(-(96*x^2 - x*e^(-8*(12*x^2 - x + 5)/x) - 13*x + 40)/x + 8*(12*x^2 - x + 5)/x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {8 \, {\left (12 \, x^{2} - 5\right )} e^{\left (-\frac {8 \, {\left (12 \, x^{2} - x + 5\right )}}{x} + e^{\left (-\frac {8 \, {\left (12 \, x^{2} - x + 5\right )}}{x}\right )} + 5\right )}}{x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-96*x^2+40)*exp((-96*x^2+8*x-40)/x)*exp(exp((-96*x^2+8*x-40)/x)+5)/x^2,x, algorithm="giac")

[Out]

integrate(-8*(12*x^2 - 5)*e^(-8*(12*x^2 - x + 5)/x + e^(-8*(12*x^2 - x + 5)/x) + 5)/x^2, x)

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maple [A]  time = 0.23, size = 19, normalized size = 1.00

method	result	size
norman	\({\mathrm e}^{{\mathrm e}^{\frac {-96 x^{2}+8 x -40}{x}}+5}\)	\(19\)
risch	\({\mathrm e}^{{\mathrm e}^{-\frac {8 \left (12 x^{2}-x +5\right )}{x}}+5}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-96*x^2+40)*exp((-96*x^2+8*x-40)/x)*exp(exp((-96*x^2+8*x-40)/x)+5)/x^2,x,method=_RETURNVERBOSE)

[Out]

exp(exp((-96*x^2+8*x-40)/x)+5)

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maxima [A]  time = 1.77, size = 14, normalized size = 0.74 \begin {gather*} e^{\left (e^{\left (-96 \, x - \frac {40}{x} + 8\right )} + 5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-96*x^2+40)*exp((-96*x^2+8*x-40)/x)*exp(exp((-96*x^2+8*x-40)/x)+5)/x^2,x, algorithm="maxima")

[Out]

e^(e^(-96*x - 40/x + 8) + 5)

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mupad [B]  time = 4.41, size = 17, normalized size = 0.89 \begin {gather*} {\mathrm {e}}^{{\mathrm {e}}^{-96\,x}\,{\mathrm {e}}^8\,{\mathrm {e}}^{-\frac {40}{x}}}\,{\mathrm {e}}^5 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-(96*x^2 - 8*x + 40)/x)*exp(exp(-(96*x^2 - 8*x + 40)/x) + 5)*(96*x^2 - 40))/x^2,x)

[Out]

exp(exp(-96*x)*exp(8)*exp(-40/x))*exp(5)

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sympy [A]  time = 0.34, size = 15, normalized size = 0.79 \begin {gather*} e^{e^{\frac {- 96 x^{2} + 8 x - 40}{x}} + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-96*x**2+40)*exp((-96*x**2+8*x-40)/x)*exp(exp((-96*x**2+8*x-40)/x)+5)/x**2,x)

[Out]

exp(exp((-96*x**2 + 8*x - 40)/x) + 5)

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