3.60.26 \(\int \frac {e^{-x} (3 e^x x^2+e^{e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}}} (3 x^2+e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}+\frac {1}{3} e^{\frac {2-x^2}{x}}+\frac {2-x^2}{x}} (2+x^2)))}{3 x^2} \, dx\)

Optimal. Leaf size=30 \[ 5-e^{e^{e^{\frac {1}{3} e^{\frac {2}{x}-x}}}-x}+x \]

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Rubi [F]  time = 2.53, 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^{-x} \left (3 e^x x^2+e^{e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}}} \left (3 x^2+e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}+\frac {1}{3} e^{\frac {2-x^2}{x}}+\frac {2-x^2}{x}} \left (2+x^2\right )\right )\right )}{3 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int \frac {e^{-x} \left (3 e^x x^2+e^{e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}}} \left (3 x^2+e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}+\frac {1}{3} e^{\frac {2-x^2}{x}}+\frac {2-x^2}{x}} \left (2+x^2\right )\right )\right )}{x^2} \, dx\\ &=\frac {1}{3} \int \left (3+3 e^{e^{e^{\frac {1}{3} e^{\frac {2}{x}-x}}}-x}+\frac {\exp \left (e^{e^{\frac {1}{3} e^{\frac {2}{x}-x}}}+e^{\frac {1}{3} e^{\frac {2}{x}-x}}+\frac {1}{3} e^{\frac {2}{x}-x}+\frac {2}{x}-2 x\right ) \left (2+x^2\right )}{x^2}\right ) \, dx\\ &=x+\frac {1}{3} \int \frac {\exp \left (e^{e^{\frac {1}{3} e^{\frac {2}{x}-x}}}+e^{\frac {1}{3} e^{\frac {2}{x}-x}}+\frac {1}{3} e^{\frac {2}{x}-x}+\frac {2}{x}-2 x\right ) \left (2+x^2\right )}{x^2} \, dx+\int e^{e^{e^{\frac {1}{3} e^{\frac {2}{x}-x}}}-x} \, dx\\ &=x+\frac {1}{3} \int \left (\exp \left (e^{e^{\frac {1}{3} e^{\frac {2}{x}-x}}}+e^{\frac {1}{3} e^{\frac {2}{x}-x}}+\frac {1}{3} e^{\frac {2}{x}-x}+\frac {2}{x}-2 x\right )+\frac {2 \exp \left (e^{e^{\frac {1}{3} e^{\frac {2}{x}-x}}}+e^{\frac {1}{3} e^{\frac {2}{x}-x}}+\frac {1}{3} e^{\frac {2}{x}-x}+\frac {2}{x}-2 x\right )}{x^2}\right ) \, dx+\int e^{e^{e^{\frac {1}{3} e^{\frac {2}{x}-x}}}-x} \, dx\\ &=x+\frac {1}{3} \int \exp \left (e^{e^{\frac {1}{3} e^{\frac {2}{x}-x}}}+e^{\frac {1}{3} e^{\frac {2}{x}-x}}+\frac {1}{3} e^{\frac {2}{x}-x}+\frac {2}{x}-2 x\right ) \, dx+\frac {2}{3} \int \frac {\exp \left (e^{e^{\frac {1}{3} e^{\frac {2}{x}-x}}}+e^{\frac {1}{3} e^{\frac {2}{x}-x}}+\frac {1}{3} e^{\frac {2}{x}-x}+\frac {2}{x}-2 x\right )}{x^2} \, dx+\int e^{e^{e^{\frac {1}{3} e^{\frac {2}{x}-x}}}-x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.19, size = 29, normalized size = 0.97 \begin {gather*} -e^{e^{e^{\frac {1}{3} e^{\frac {2}{x}-x}}}-x}+x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-E^(E^E^(E^(2/x - x)/3) - x) + x

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fricas [B]  time = 0.72, size = 80, normalized size = 2.67 \begin {gather*} {\left (x e^{x} - e^{\left (e^{\left (-\frac {3 \, x^{2} - 3 \, x e^{\left (\frac {1}{3} \, e^{\left (-\frac {x^{2} - 2}{x}\right )}\right )} - x e^{\left (-\frac {x^{2} - 2}{x}\right )} - 6}{3 \, x} + \frac {x^{2} - 2}{x} - \frac {1}{3} \, e^{\left (-\frac {x^{2} - 2}{x}\right )}\right )}\right )}\right )} e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x*e^x - e^(e^(-1/3*(3*x^2 - 3*x*e^(1/3*e^(-(x^2 - 2)/x)) - x*e^(-(x^2 - 2)/x) - 6)/x + (x^2 - 2)/x - 1/3*e^(-
(x^2 - 2)/x))))*e^(-x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.30, size = 25, normalized size = 0.83

method	result	size
risch	\(x -{\mathrm e}^{-x +{\mathrm e}^{{\mathrm e}^{\frac {{\mathrm e}^{-\frac {x^{2}-2}{x}}}{3}}}}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.54, size = 23, normalized size = 0.77 \begin {gather*} x - e^{\left (-x + e^{\left (e^{\left (\frac {1}{3} \, e^{\left (-x + \frac {2}{x}\right )}\right )}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - e^(-x + e^(e^(1/3*e^(-x + 2/x))))

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mupad [B]  time = 4.38, size = 27, normalized size = 0.90 \begin {gather*} -{\mathrm {e}}^{-x}\,\left ({\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^{\frac {{\mathrm {e}}^{-x}\,{\mathrm {e}}^{2/x}}{3}}}}-x\,{\mathrm {e}}^x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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