\(\int \frac {e^{-e^{e^{-x} (e^x (-3-x)+x)}-x} (e^{e^{-x} (e^x (-3-x)+x)} (4 x^2-4 e^x x^2-4 x^3)+e^{e^{e^{-x} (e^x (-3-x)+x)}} (e^{2 x} (-1+x)+e^x (4-x^2)))}{x^2} \, dx\) [1380]

Optimal result

Integrand size = 114, antiderivative size = 37 \[ \int \frac {e^{-e^{e^{-x} \left (e^x (-3-x)+x\right )}-x} \left (e^{e^{-x} \left (e^x (-3-x)+x\right )} \left (4 x^2-4 e^x x^2-4 x^3\right )+e^{e^{e^{-x} \left (e^x (-3-x)+x\right )}} \left (e^{2 x} (-1+x)+e^x \left (4-x^2\right )\right )\right )}{x^2} \, dx=-4 e^{-e^{-3-x+e^{-x} x}}-x-\frac {4-e^x+x}{x} \] Output:

-x-(x-exp(x)+4)/x-4/exp(exp(x/exp(x)-3-x))
 

Mathematica [A] (verified)

Time = 0.67 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.97 \[ \int \frac {e^{-e^{e^{-x} \left (e^x (-3-x)+x\right )}-x} \left (e^{e^{-x} \left (e^x (-3-x)+x\right )} \left (4 x^2-4 e^x x^2-4 x^3\right )+e^{e^{e^{-x} \left (e^x (-3-x)+x\right )}} \left (e^{2 x} (-1+x)+e^x \left (4-x^2\right )\right )\right )}{x^2} \, dx=-4 e^{-e^{-3-x+e^{-x} x}}-\frac {4}{x}+\frac {e^x}{x}-x \] Input:

Integrate[(E^(-E^((E^x*(-3 - x) + x)/E^x) - x)*(E^((E^x*(-3 - x) + x)/E^x) 
*(4*x^2 - 4*E^x*x^2 - 4*x^3) + E^E^((E^x*(-3 - x) + x)/E^x)*(E^(2*x)*(-1 + 
 x) + E^x*(4 - x^2))))/x^2,x]
 

Output:

-4/E^E^(-3 - x + x/E^x) - 4/x + E^x/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.

\(\displaystyle \int \frac {e^{-x-e^{e^{-x} \left (e^x (-x-3)+x\right )}} \left (e^{e^{e^{-x} \left (e^x (-x-3)+x\right )}} \left (e^x \left (4-x^2\right )+e^{2 x} (x-1)\right )+e^{e^{-x} \left (e^x (-x-3)+x\right )} \left (-4 x^3-4 e^x x^2+4 x^2\right )\right )}{x^2} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \left (\frac {e^x (x-1)}{x^2}+\frac {4}{x^2}-4 e^{e^{-x} x-2 x-e^{\left (e^{-x}-1\right ) x-3}-3} \left (x+e^x-1\right )-1\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 4 \int e^{e^{-x} x-2 x-e^{\left (-1+e^{-x}\right ) x-3}-3}dx-4 \int e^{e^{-x} x-x-e^{\left (-1+e^{-x}\right ) x-3}-3}dx-4 \int e^{e^{-x} x-2 x-e^{\left (-1+e^{-x}\right ) x-3}-3} xdx-x+\frac {e^x}{x}-\frac {4}{x}\)

Input:

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

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.50 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.08

method	result	size
risch	\(-x -\frac {4}{x}+\frac {{\mathrm e}^{x}}{x}-4 \,{\mathrm e}^{-{\mathrm e}^{-\left ({\mathrm e}^{x} x +3 \,{\mathrm e}^{x}-x \right ) {\mathrm e}^{-x}}}\)	\(40\)
parallelrisch	\(\frac {\left (-{\mathrm e}^{{\mathrm e}^{\left (\left (-3-x \right ) {\mathrm e}^{x}+x \right ) {\mathrm e}^{-x}}} x^{2}+{\mathrm e}^{{\mathrm e}^{\left (\left (-3-x \right ) {\mathrm e}^{x}+x \right ) {\mathrm e}^{-x}}} {\mathrm e}^{x}-4 x -4 \,{\mathrm e}^{{\mathrm e}^{\left (\left (-3-x \right ) {\mathrm e}^{x}+x \right ) {\mathrm e}^{-x}}}\right ) {\mathrm e}^{-{\mathrm e}^{\left (\left (-3-x \right ) {\mathrm e}^{x}+x \right ) {\mathrm e}^{-x}}}}{x}\)	\(89\)

Input:

int((((-1+x)*exp(x)^2+(-x^2+4)*exp(x))*exp(exp(((-3-x)*exp(x)+x)/exp(x)))+ 
(-4*exp(x)*x^2-4*x^3+4*x^2)*exp(((-3-x)*exp(x)+x)/exp(x)))/x^2/exp(x)/exp( 
exp(((-3-x)*exp(x)+x)/exp(x))),x,method=_RETURNVERBOSE)
 

Output:

-x-4/x+exp(x)/x-4*exp(-exp(-(exp(x)*x+3*exp(x)-x)*exp(-x)))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.54 \[ \int \frac {e^{-e^{e^{-x} \left (e^x (-3-x)+x\right )}-x} \left (e^{e^{-x} \left (e^x (-3-x)+x\right )} \left (4 x^2-4 e^x x^2-4 x^3\right )+e^{e^{e^{-x} \left (e^x (-3-x)+x\right )}} \left (e^{2 x} (-1+x)+e^x \left (4-x^2\right )\right )\right )}{x^2} \, dx=-\frac {{\left ({\left (x^{2} - e^{x} + 4\right )} e^{\left (e^{\left (-{\left ({\left (x + 3\right )} e^{x} - x\right )} e^{\left (-x\right )}\right )}\right )} + 4 \, x\right )} e^{\left (-e^{\left (-{\left ({\left (x + 3\right )} e^{x} - x\right )} e^{\left (-x\right )}\right )}\right )}}{x} \] Input:

integrate((((-1+x)*exp(x)^2+(-x^2+4)*exp(x))*exp(exp(((-3-x)*exp(x)+x)/exp 
(x)))+(-4*exp(x)*x^2-4*x^3+4*x^2)*exp(((-3-x)*exp(x)+x)/exp(x)))/x^2/exp(x 
)/exp(exp(((-3-x)*exp(x)+x)/exp(x))),x, algorithm="fricas")
 

Output:

-((x^2 - e^x + 4)*e^(e^(-((x + 3)*e^x - x)*e^(-x))) + 4*x)*e^(-e^(-((x + 3 
)*e^x - x)*e^(-x)))/x
 

Sympy [A] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.73 \[ \int \frac {e^{-e^{e^{-x} \left (e^x (-3-x)+x\right )}-x} \left (e^{e^{-x} \left (e^x (-3-x)+x\right )} \left (4 x^2-4 e^x x^2-4 x^3\right )+e^{e^{e^{-x} \left (e^x (-3-x)+x\right )}} \left (e^{2 x} (-1+x)+e^x \left (4-x^2\right )\right )\right )}{x^2} \, dx=- x - 4 e^{- e^{\left (x + \left (- x - 3\right ) e^{x}\right ) e^{- x}}} + \frac {e^{x}}{x} - \frac {4}{x} \] Input:

integrate((((-1+x)*exp(x)**2+(-x**2+4)*exp(x))*exp(exp(((-3-x)*exp(x)+x)/e 
xp(x)))+(-4*exp(x)*x**2-4*x**3+4*x**2)*exp(((-3-x)*exp(x)+x)/exp(x)))/x**2 
/exp(x)/exp(exp(((-3-x)*exp(x)+x)/exp(x))),x)
 

Output:

-x - 4*exp(-exp((x + (-x - 3)*exp(x))*exp(-x))) + exp(x)/x - 4/x
 

Maxima [C] (verification not implemented)

Result contains higher order function than in optimal. Order 4 vs. order 3.

Time = 0.22 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.95 \[ \int \frac {e^{-e^{e^{-x} \left (e^x (-3-x)+x\right )}-x} \left (e^{e^{-x} \left (e^x (-3-x)+x\right )} \left (4 x^2-4 e^x x^2-4 x^3\right )+e^{e^{e^{-x} \left (e^x (-3-x)+x\right )}} \left (e^{2 x} (-1+x)+e^x \left (4-x^2\right )\right )\right )}{x^2} \, dx=-x - \frac {4}{x} + {\rm Ei}\left (x\right ) - 4 \, e^{\left (-e^{\left (x e^{\left (-x\right )} - x - 3\right )}\right )} - \Gamma \left (-1, -x\right ) \] Input:

integrate((((-1+x)*exp(x)^2+(-x^2+4)*exp(x))*exp(exp(((-3-x)*exp(x)+x)/exp 
(x)))+(-4*exp(x)*x^2-4*x^3+4*x^2)*exp(((-3-x)*exp(x)+x)/exp(x)))/x^2/exp(x 
)/exp(exp(((-3-x)*exp(x)+x)/exp(x))),x, algorithm="maxima")
 

Output:

-x - 4/x + Ei(x) - 4*e^(-e^(x*e^(-x) - x - 3)) - gamma(-1, -x)
 

Giac [F]

\[ \int \frac {e^{-e^{e^{-x} \left (e^x (-3-x)+x\right )}-x} \left (e^{e^{-x} \left (e^x (-3-x)+x\right )} \left (4 x^2-4 e^x x^2-4 x^3\right )+e^{e^{e^{-x} \left (e^x (-3-x)+x\right )}} \left (e^{2 x} (-1+x)+e^x \left (4-x^2\right )\right )\right )}{x^2} \, dx=\int { -\frac {{\left (4 \, {\left (x^{3} + x^{2} e^{x} - x^{2}\right )} e^{\left (-{\left ({\left (x + 3\right )} e^{x} - x\right )} e^{\left (-x\right )}\right )} - {\left ({\left (x - 1\right )} e^{\left (2 \, x\right )} - {\left (x^{2} - 4\right )} e^{x}\right )} e^{\left (e^{\left (-{\left ({\left (x + 3\right )} e^{x} - x\right )} e^{\left (-x\right )}\right )}\right )}\right )} e^{\left (-x - e^{\left (-{\left ({\left (x + 3\right )} e^{x} - x\right )} e^{\left (-x\right )}\right )}\right )}}{x^{2}} \,d x } \] Input:

integrate((((-1+x)*exp(x)^2+(-x^2+4)*exp(x))*exp(exp(((-3-x)*exp(x)+x)/exp 
(x)))+(-4*exp(x)*x^2-4*x^3+4*x^2)*exp(((-3-x)*exp(x)+x)/exp(x)))/x^2/exp(x 
)/exp(exp(((-3-x)*exp(x)+x)/exp(x))),x, algorithm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 1.71 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.89 \[ \int \frac {e^{-e^{e^{-x} \left (e^x (-3-x)+x\right )}-x} \left (e^{e^{-x} \left (e^x (-3-x)+x\right )} \left (4 x^2-4 e^x x^2-4 x^3\right )+e^{e^{e^{-x} \left (e^x (-3-x)+x\right )}} \left (e^{2 x} (-1+x)+e^x \left (4-x^2\right )\right )\right )}{x^2} \, dx=\frac {{\mathrm {e}}^x}{x}-4\,{\mathrm {e}}^{-{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-3}\,{\mathrm {e}}^{x\,{\mathrm {e}}^{-x}}}-x-\frac {4}{x} \] Input:

int(-(exp(-x)*exp(-exp(exp(-x)*(x - exp(x)*(x + 3))))*(exp(exp(-x)*(x - ex 
p(x)*(x + 3)))*(4*x^2*exp(x) - 4*x^2 + 4*x^3) - exp(exp(exp(-x)*(x - exp(x 
)*(x + 3))))*(exp(2*x)*(x - 1) - exp(x)*(x^2 - 4))))/x^2,x)
 

Output:

exp(x)/x - 4*exp(-exp(-x)*exp(-3)*exp(x*exp(-x))) - x - 4/x
 

Reduce [F]

\[ \int \frac {e^{-e^{e^{-x} \left (e^x (-3-x)+x\right )}-x} \left (e^{e^{-x} \left (e^x (-3-x)+x\right )} \left (4 x^2-4 e^x x^2-4 x^3\right )+e^{e^{e^{-x} \left (e^x (-3-x)+x\right )}} \left (e^{2 x} (-1+x)+e^x \left (4-x^2\right )\right )\right )}{x^2} \, dx=\frac {e^{x} e^{3}-4 \left (\int \frac {e^{\frac {x}{e^{x}}}}{e^{\frac {e^{\frac {x}{e^{x}}}+e^{x} e^{3} x}{e^{x} e^{3}}}}d x \right ) x +4 \left (\int \frac {e^{\frac {x}{e^{x}}}}{e^{\frac {e^{\frac {x}{e^{x}}}+2 e^{x} e^{3} x}{e^{x} e^{3}}}}d x \right ) x -4 \left (\int \frac {e^{\frac {x}{e^{x}}} x}{e^{\frac {e^{\frac {x}{e^{x}}}+2 e^{x} e^{3} x}{e^{x} e^{3}}}}d x \right ) x -e^{3} x^{2}-4 e^{3}}{e^{3} x} \] Input:

int((((-1+x)*exp(x)^2+(-x^2+4)*exp(x))*exp(exp(((-3-x)*exp(x)+x)/exp(x)))+ 
(-4*exp(x)*x^2-4*x^3+4*x^2)*exp(((-3-x)*exp(x)+x)/exp(x)))/x^2/exp(x)/exp( 
exp(((-3-x)*exp(x)+x)/exp(x))),x)
 

Output:

(e**x*e**3 - 4*int(e**(x/e**x)/e**((e**(x/e**x) + e**x*e**3*x)/(e**x*e**3) 
),x)*x + 4*int(e**(x/e**x)/e**((e**(x/e**x) + 2*e**x*e**3*x)/(e**x*e**3)), 
x)*x - 4*int((e**(x/e**x)*x)/e**((e**(x/e**x) + 2*e**x*e**3*x)/(e**x*e**3) 
),x)*x - e**3*x**2 - 4*e**3)/(e**3*x)