\(\int \frac {e^{-x} (3 e^x x^2+e^{e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}}} (3 x^2+e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}+\frac {1}{3} e^{\frac {2-x^2}{x}}+\frac {2-x^2}{x}} (2+x^2)))}{3 x^2} \, dx\) [5926]

Optimal result

Integrand size = 107, antiderivative size = 30 \[ \int \frac {e^{-x} \left (3 e^x x^2+e^{e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}}} \left (3 x^2+e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}+\frac {1}{3} e^{\frac {2-x^2}{x}}+\frac {2-x^2}{x}} \left (2+x^2\right )\right )\right )}{3 x^2} \, dx=5-e^{e^{e^{\frac {1}{3} e^{\frac {2}{x}-x}}}-x}+x \]

[Out]

x-exp(exp(exp(1/3*exp(2/x-x))))/exp(x)+5

Rubi [F]

\[ \int \frac {e^{-x} \left (3 e^x x^2+e^{e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}}} \left (3 x^2+e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}+\frac {1}{3} e^{\frac {2-x^2}{x}}+\frac {2-x^2}{x}} \left (2+x^2\right )\right )\right )}{3 x^2} \, dx=\int \frac {e^{-x} \left (3 e^x x^2+e^{e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}}} \left (3 x^2+e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}+\frac {1}{3} e^{\frac {2-x^2}{x}}+\frac {2-x^2}{x}} \left (2+x^2\right )\right )\right )}{3 x^2} \, dx \]

[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{align*} \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{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {e^{-x} \left (3 e^x x^2+e^{e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}}} \left (3 x^2+e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}+\frac {1}{3} e^{\frac {2-x^2}{x}}+\frac {2-x^2}{x}} \left (2+x^2\right )\right )\right )}{3 x^2} \, dx=-e^{e^{e^{\frac {1}{3} e^{\frac {2}{x}-x}}}-x}+x \]

[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

Maple [A] (verified)

Time = 7.68 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83

method	result	size
risch	\(x -{\mathrm e}^{-x +{\mathrm e}^{{\mathrm e}^{\frac {{\mathrm e}^{-\frac {x^{2}-2}{x}}}{3}}}}\)	\(25\)
parallelrisch	\(\frac {\left (3 \,{\mathrm e}^{x} x -3 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{\frac {{\mathrm e}^{-\frac {x^{2}-2}{x}}}{3}}}}\right ) {\mathrm e}^{-x}}{3}\)	\(31\)

[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))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (24) = 48\).

Time = 0.24 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.67 \[ \int \frac {e^{-x} \left (3 e^x x^2+e^{e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}}} \left (3 x^2+e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}+\frac {1}{3} e^{\frac {2-x^2}{x}}+\frac {2-x^2}{x}} \left (2+x^2\right )\right )\right )}{3 x^2} \, dx={\left (x e^{x} - e^{\left (e^{\left (-\frac {3 \, x^{2} - 3 \, x e^{\left (\frac {1}{3} \, e^{\left (-\frac {x^{2} - 2}{x}\right )}\right )} - x e^{\left (-\frac {x^{2} - 2}{x}\right )} - 6}{3 \, x} + \frac {x^{2} - 2}{x} - \frac {1}{3} \, e^{\left (-\frac {x^{2} - 2}{x}\right )}\right )}\right )}\right )} e^{\left (-x\right )} \]

[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)

Sympy [A] (verification not implemented)

Time = 6.39 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.63 \[ \int \frac {e^{-x} \left (3 e^x x^2+e^{e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}}} \left (3 x^2+e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}+\frac {1}{3} e^{\frac {2-x^2}{x}}+\frac {2-x^2}{x}} \left (2+x^2\right )\right )\right )}{3 x^2} \, dx=x - e^{- x} e^{e^{e^{\frac {e^{\frac {2 - x^{2}}{x}}}{3}}}} \]

[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]

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

Maxima [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.77 \[ \int \frac {e^{-x} \left (3 e^x x^2+e^{e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}}} \left (3 x^2+e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}+\frac {1}{3} e^{\frac {2-x^2}{x}}+\frac {2-x^2}{x}} \left (2+x^2\right )\right )\right )}{3 x^2} \, dx=x - e^{\left (-x + e^{\left (e^{\left (\frac {1}{3} \, e^{\left (-x + \frac {2}{x}\right )}\right )}\right )}\right )} \]

[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))))

Giac [F]

\[ \int \frac {e^{-x} \left (3 e^x x^2+e^{e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}}} \left (3 x^2+e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}+\frac {1}{3} e^{\frac {2-x^2}{x}}+\frac {2-x^2}{x}} \left (2+x^2\right )\right )\right )}{3 x^2} \, dx=\int { \frac {{\left (3 \, x^{2} e^{x} + {\left (3 \, x^{2} + {\left (x^{2} + 2\right )} e^{\left (-\frac {x^{2} - 2}{x} + e^{\left (\frac {1}{3} \, e^{\left (-\frac {x^{2} - 2}{x}\right )}\right )} + \frac {1}{3} \, e^{\left (-\frac {x^{2} - 2}{x}\right )}\right )}\right )} e^{\left (e^{\left (e^{\left (\frac {1}{3} \, e^{\left (-\frac {x^{2} - 2}{x}\right )}\right )}\right )}\right )}\right )} e^{\left (-x\right )}}{3 \, x^{2}} \,d x } \]

[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)

Mupad [B] (verification not implemented)

Time = 12.18 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.90 \[ \int \frac {e^{-x} \left (3 e^x x^2+e^{e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}}} \left (3 x^2+e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}+\frac {1}{3} e^{\frac {2-x^2}{x}}+\frac {2-x^2}{x}} \left (2+x^2\right )\right )\right )}{3 x^2} \, dx=-{\mathrm {e}}^{-x}\,\left ({\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^{\frac {{\mathrm {e}}^{-x}\,{\mathrm {e}}^{2/x}}{3}}}}-x\,{\mathrm {e}}^x\right ) \]

[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))