\(\int \frac {e^{-x} (3 e^x x^2+e^{e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}}} (3 x^2+e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}+\frac {1}{3} e^{\frac {2-x^2}{x}}+\frac {2-x^2}{x}} (2+x^2)))}{3 x^2} \, dx\) [2926]

Optimal result

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

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

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {e^{-x} \left (3 e^x x^2+e^{e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}}} \left (3 x^2+e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}+\frac {1}{3} e^{\frac {2-x^2}{x}}+\frac {2-x^2}{x}} \left (2+x^2\right )\right )\right )}{3 x^2} \, dx=-e^{e^{e^{\frac {1}{3} e^{\frac {2}{x}-x}}}-x}+x \] Input:

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]
 

Output:

-E^(E^E^(E^(2/x - x)/3) - x) + x
 

Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \int \frac {e^{-x} \left (3 e^x x^2+e^{e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}}} \left (3 x^2+e^{\frac {2-x^2}{x}+e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}+\frac {1}{3} e^{\frac {2-x^2}{x}}} \left (x^2+2\right )\right )\right )}{x^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {1}{3} \int \left (\frac {\exp \left (-2 x+e^{e^{\frac {1}{3} e^{\frac {2}{x}-x}}}+e^{\frac {1}{3} e^{\frac {2}{x}-x}}+\frac {1}{3} e^{\frac {2}{x}-x}+\frac {2}{x}\right ) \left (x^2+2\right )}{x^2}+3 e^{e^{e^{\frac {1}{3} e^{\frac {2}{x}-x}}}-x}+3\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{3} \left (2 \int \frac {\exp \left (-2 x+e^{e^{\frac {1}{3} e^{\frac {2}{x}-x}}}+e^{\frac {1}{3} e^{\frac {2}{x}-x}}+\frac {1}{3} e^{\frac {2}{x}-x}+\frac {2}{x}\right )}{x^2}dx+\int \exp \left (-2 x+e^{e^{\frac {1}{3} e^{\frac {2}{x}-x}}}+e^{\frac {1}{3} e^{\frac {2}{x}-x}}+\frac {1}{3} e^{\frac {2}{x}-x}+\frac {2}{x}\right )dx+3 \int e^{e^{e^{\frac {1}{3} e^{\frac {2}{x}-x}}}-x}dx+3 x\right )\)

Input:

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]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 6.76 (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	\(\frac {\left (3 \,{\mathrm e}^{x} x -3 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{\frac {{\mathrm e}^{-\frac {x^{2}-2}{x}}}{3}}}}\right ) {\mathrm e}^{-x}}{3}\)	\(31\)

Input:

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)
 

Output:

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 = 0.11 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.67 \[ \int \frac {e^{-x} \left (3 e^x x^2+e^{e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}}} \left (3 x^2+e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}+\frac {1}{3} e^{\frac {2-x^2}{x}}+\frac {2-x^2}{x}} \left (2+x^2\right )\right )\right )}{3 x^2} \, dx={\left (x e^{x} - e^{\left (e^{\left (-\frac {3 \, x^{2} - 3 \, x e^{\left (\frac {1}{3} \, e^{\left (-\frac {x^{2} - 2}{x}\right )}\right )} - x e^{\left (-\frac {x^{2} - 2}{x}\right )} - 6}{3 \, x} + \frac {x^{2} - 2}{x} - \frac {1}{3} \, e^{\left (-\frac {x^{2} - 2}{x}\right )}\right )}\right )}\right )} e^{\left (-x\right )} \] Input:

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")
 

Output:

(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 [F(-1)]

Timed out. \[ \int \frac {e^{-x} \left (3 e^x x^2+e^{e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}}} \left (3 x^2+e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}+\frac {1}{3} e^{\frac {2-x^2}{x}}+\frac {2-x^2}{x}} \left (2+x^2\right )\right )\right )}{3 x^2} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.77 \[ \int \frac {e^{-x} \left (3 e^x x^2+e^{e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}}} \left (3 x^2+e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}+\frac {1}{3} e^{\frac {2-x^2}{x}}+\frac {2-x^2}{x}} \left (2+x^2\right )\right )\right )}{3 x^2} \, dx=x - e^{\left (-x + e^{\left (e^{\left (\frac {1}{3} \, e^{\left (-x + \frac {2}{x}\right )}\right )}\right )}\right )} \] Input:

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")
 

Output:

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

Giac [F]

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

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")
 

Output:

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 = 2.77 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.90 \[ \int \frac {e^{-x} \left (3 e^x x^2+e^{e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}}} \left (3 x^2+e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}+\frac {1}{3} e^{\frac {2-x^2}{x}}+\frac {2-x^2}{x}} \left (2+x^2\right )\right )\right )}{3 x^2} \, dx=-{\mathrm {e}}^{-x}\,\left ({\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^{\frac {{\mathrm {e}}^{-x}\,{\mathrm {e}}^{2/x}}{3}}}}-x\,{\mathrm {e}}^x\right ) \] Input:

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

Output:

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

Reduce [F]

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

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)
 

Output:

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)