\(\int \frac {e^{-x} (e^x x^2+e^{\frac {1-e^{e^{e^{-x} (-x+e^x x^6)}}+x^2}{x}} (e^x (-1+x^2)+e^{e^{e^{-x} (-x+e^x x^6)}} (e^x+e^{e^{-x} (-x+e^x x^6)} (x-x^2-6 e^x x^6))))}{x^2} \, dx\) [1318]

Optimal result

Integrand size = 123, antiderivative size = 29 \[ \int \frac {e^{-x} \left (e^x x^2+e^{\frac {1-e^{e^{e^{-x} \left (-x+e^x x^6\right )}}+x^2}{x}} \left (e^x \left (-1+x^2\right )+e^{e^{e^{-x} \left (-x+e^x x^6\right )}} \left (e^x+e^{e^{-x} \left (-x+e^x x^6\right )} \left (x-x^2-6 e^x x^6\right )\right )\right )\right )}{x^2} \, dx=e^{-\frac {-1+e^{e^{-e^{-x} x+x^6}}}{x}+x}+x \] Output:

x+exp(x-(exp(exp(x^6-x/exp(x)))-1)/x)
 

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03 \[ \int \frac {e^{-x} \left (e^x x^2+e^{\frac {1-e^{e^{e^{-x} \left (-x+e^x x^6\right )}}+x^2}{x}} \left (e^x \left (-1+x^2\right )+e^{e^{e^{-x} \left (-x+e^x x^6\right )}} \left (e^x+e^{e^{-x} \left (-x+e^x x^6\right )} \left (x-x^2-6 e^x x^6\right )\right )\right )\right )}{x^2} \, dx=e^{\frac {1}{x}-\frac {e^{e^{-e^{-x} x+x^6}}}{x}+x}+x \] Input:

Integrate[(E^x*x^2 + E^((1 - E^E^((-x + E^x*x^6)/E^x) + x^2)/x)*(E^x*(-1 + 
 x^2) + E^E^((-x + E^x*x^6)/E^x)*(E^x + E^((-x + E^x*x^6)/E^x)*(x - x^2 - 
6*E^x*x^6))))/(E^x*x^2),x]
 

Output:

E^(x^(-1) - E^E^(-(x/E^x) + x^6)/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.

\(\displaystyle \int \frac {e^{-x} \left (e^x x^2+e^{\frac {-e^{e^{e^{-x} \left (e^x x^6-x\right )}}+x^2+1}{x}} \left (e^x \left (x^2-1\right )+e^{e^{e^{-x} \left (e^x x^6-x\right )}} \left (e^{e^{-x} \left (e^x x^6-x\right )} \left (-6 e^x x^6-x^2+x\right )+e^x\right )\right )\right )}{x^2} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^{-\frac {e^{e^{x^6-e^{-x} x}}}{x}} \left (e^{e^{x^6-e^{-x} x}+x+\frac {1}{x}}+e^{x+\frac {1}{x}} x^2+e^{\frac {e^{e^{x^6-e^{-x} x}}}{x}} x^2-e^{x+\frac {1}{x}}\right )}{x^2}-\frac {\left (6 e^x x^5+x-1\right ) \exp \left (x^6+e^{x^6-e^{-x} x}-\frac {e^{e^{x^6-e^{-x} x}}}{x}-e^{-x} x+\frac {1}{x}\right )}{x}\right )dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \left (\frac {e^{-\frac {e^{e^{x^6-e^{-x} x}}}{x}} \left (e^{e^{x^6-e^{-x} x}+x+\frac {1}{x}}+e^{x+\frac {1}{x}} x^2+e^{\frac {e^{e^{x^6-e^{-x} x}}}{x}} x^2-e^{x+\frac {1}{x}}\right )}{x^2}-\frac {\left (6 e^x x^5+x-1\right ) \exp \left (x^6+e^{x^6-e^{-x} x}-\frac {e^{e^{x^6-e^{-x} x}}}{x}-e^{-x} x+\frac {1}{x}\right )}{x}\right )dx\)

Input:

Int[(E^x*x^2 + E^((1 - E^E^((-x + E^x*x^6)/E^x) + x^2)/x)*(E^x*(-1 + x^2) 
+ E^E^((-x + E^x*x^6)/E^x)*(E^x + E^((-x + E^x*x^6)/E^x)*(x - x^2 - 6*E^x* 
x^6))))/(E^x*x^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 60.48 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07

method	result	size
risch	\(x +{\mathrm e}^{\frac {-{\mathrm e}^{{\mathrm e}^{x \left (x^{5} {\mathrm e}^{x}-1\right ) {\mathrm e}^{-x}}}+x^{2}+1}{x}}\)	\(31\)
parallelrisch	\(\ln \left ({\mathrm e}^{x}\right )+{\mathrm e}^{\frac {-{\mathrm e}^{{\mathrm e}^{\left (x^{6} {\mathrm e}^{x}-x \right ) {\mathrm e}^{-x}}}+x^{2}+1}{x}}\)	\(34\)

Input:

int(((((-6*x^6*exp(x)-x^2+x)*exp((x^6*exp(x)-x)/exp(x))+exp(x))*exp(exp((x 
^6*exp(x)-x)/exp(x)))+(x^2-1)*exp(x))*exp((-exp(exp((x^6*exp(x)-x)/exp(x)) 
)+x^2+1)/x)+exp(x)*x^2)/exp(x)/x^2,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {e^{-x} \left (e^x x^2+e^{\frac {1-e^{e^{e^{-x} \left (-x+e^x x^6\right )}}+x^2}{x}} \left (e^x \left (-1+x^2\right )+e^{e^{e^{-x} \left (-x+e^x x^6\right )}} \left (e^x+e^{e^{-x} \left (-x+e^x x^6\right )} \left (x-x^2-6 e^x x^6\right )\right )\right )\right )}{x^2} \, dx=x + e^{\left (\frac {x^{2} - e^{\left (e^{\left ({\left (x^{6} e^{x} - x\right )} e^{\left (-x\right )}\right )}\right )} + 1}{x}\right )} \] Input:

integrate(((((-6*x^6*exp(x)-x^2+x)*exp((x^6*exp(x)-x)/exp(x))+exp(x))*exp( 
exp((x^6*exp(x)-x)/exp(x)))+(x^2-1)*exp(x))*exp((-exp(exp((x^6*exp(x)-x)/e 
xp(x)))+x^2+1)/x)+exp(x)*x^2)/exp(x)/x^2,x, algorithm="fricas")
 

Output:

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

Sympy [A] (verification not implemented)

Time = 9.59 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83 \[ \int \frac {e^{-x} \left (e^x x^2+e^{\frac {1-e^{e^{e^{-x} \left (-x+e^x x^6\right )}}+x^2}{x}} \left (e^x \left (-1+x^2\right )+e^{e^{e^{-x} \left (-x+e^x x^6\right )}} \left (e^x+e^{e^{-x} \left (-x+e^x x^6\right )} \left (x-x^2-6 e^x x^6\right )\right )\right )\right )}{x^2} \, dx=x + e^{\frac {x^{2} - e^{e^{\left (x^{6} e^{x} - x\right ) e^{- x}}} + 1}{x}} \] Input:

integrate(((((-6*x**6*exp(x)-x**2+x)*exp((x**6*exp(x)-x)/exp(x))+exp(x))*e 
xp(exp((x**6*exp(x)-x)/exp(x)))+(x**2-1)*exp(x))*exp((-exp(exp((x**6*exp(x 
)-x)/exp(x)))+x**2+1)/x)+exp(x)*x**2)/exp(x)/x**2,x)
 

Output:

x + exp((x**2 - exp(exp((x**6*exp(x) - x)*exp(-x))) + 1)/x)
 

Maxima [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {e^{-x} \left (e^x x^2+e^{\frac {1-e^{e^{e^{-x} \left (-x+e^x x^6\right )}}+x^2}{x}} \left (e^x \left (-1+x^2\right )+e^{e^{e^{-x} \left (-x+e^x x^6\right )}} \left (e^x+e^{e^{-x} \left (-x+e^x x^6\right )} \left (x-x^2-6 e^x x^6\right )\right )\right )\right )}{x^2} \, dx=x + e^{\left (x - \frac {e^{\left (e^{\left (x^{6} - x e^{\left (-x\right )}\right )}\right )}}{x} + \frac {1}{x}\right )} \] Input:

integrate(((((-6*x^6*exp(x)-x^2+x)*exp((x^6*exp(x)-x)/exp(x))+exp(x))*exp( 
exp((x^6*exp(x)-x)/exp(x)))+(x^2-1)*exp(x))*exp((-exp(exp((x^6*exp(x)-x)/e 
xp(x)))+x^2+1)/x)+exp(x)*x^2)/exp(x)/x^2,x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {e^{-x} \left (e^x x^2+e^{\frac {1-e^{e^{e^{-x} \left (-x+e^x x^6\right )}}+x^2}{x}} \left (e^x \left (-1+x^2\right )+e^{e^{e^{-x} \left (-x+e^x x^6\right )}} \left (e^x+e^{e^{-x} \left (-x+e^x x^6\right )} \left (x-x^2-6 e^x x^6\right )\right )\right )\right )}{x^2} \, dx=\int { \frac {{\left (x^{2} e^{x} + {\left ({\left (x^{2} - 1\right )} e^{x} - {\left ({\left (6 \, x^{6} e^{x} + x^{2} - x\right )} e^{\left ({\left (x^{6} e^{x} - x\right )} e^{\left (-x\right )}\right )} - e^{x}\right )} e^{\left (e^{\left ({\left (x^{6} e^{x} - x\right )} e^{\left (-x\right )}\right )}\right )}\right )} e^{\left (\frac {x^{2} - e^{\left (e^{\left ({\left (x^{6} e^{x} - x\right )} e^{\left (-x\right )}\right )}\right )} + 1}{x}\right )}\right )} e^{\left (-x\right )}}{x^{2}} \,d x } \] Input:

integrate(((((-6*x^6*exp(x)-x^2+x)*exp((x^6*exp(x)-x)/exp(x))+exp(x))*exp( 
exp((x^6*exp(x)-x)/exp(x)))+(x^2-1)*exp(x))*exp((-exp(exp((x^6*exp(x)-x)/e 
xp(x)))+x^2+1)/x)+exp(x)*x^2)/exp(x)/x^2,x, algorithm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 1.97 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-x} \left (e^x x^2+e^{\frac {1-e^{e^{e^{-x} \left (-x+e^x x^6\right )}}+x^2}{x}} \left (e^x \left (-1+x^2\right )+e^{e^{e^{-x} \left (-x+e^x x^6\right )}} \left (e^x+e^{e^{-x} \left (-x+e^x x^6\right )} \left (x-x^2-6 e^x x^6\right )\right )\right )\right )}{x^2} \, dx=x+{\mathrm {e}}^{-\frac {{\mathrm {e}}^{{\mathrm {e}}^{x^6}\,{\mathrm {e}}^{-x\,{\mathrm {e}}^{-x}}}}{x}}\,{\mathrm {e}}^{1/x}\,{\mathrm {e}}^x \] Input:

int((exp(-x)*(x^2*exp(x) + exp((x^2 - exp(exp(-exp(-x)*(x - x^6*exp(x)))) 
+ 1)/x)*(exp(exp(-exp(-x)*(x - x^6*exp(x))))*(exp(x) - exp(-exp(-x)*(x - x 
^6*exp(x)))*(6*x^6*exp(x) - x + x^2)) + exp(x)*(x^2 - 1))))/x^2,x)
 

Output:

x + exp(-exp(exp(x^6)*exp(-x*exp(-x)))/x)*exp(1/x)*exp(x)
 

Reduce [F]

\[ \int \frac {e^{-x} \left (e^x x^2+e^{\frac {1-e^{e^{e^{-x} \left (-x+e^x x^6\right )}}+x^2}{x}} \left (e^x \left (-1+x^2\right )+e^{e^{e^{-x} \left (-x+e^x x^6\right )}} \left (e^x+e^{e^{-x} \left (-x+e^x x^6\right )} \left (x-x^2-6 e^x x^6\right )\right )\right )\right )}{x^2} \, dx=\int \frac {\left (\left (\left (-6 x^{6} {\mathrm e}^{x}-x^{2}+x \right ) {\mathrm e}^{\frac {x^{6} {\mathrm e}^{x}-x}{{\mathrm e}^{x}}}+{\mathrm e}^{x}\right ) {\mathrm e}^{{\mathrm e}^{\frac {x^{6} {\mathrm e}^{x}-x}{{\mathrm e}^{x}}}}+\left (x^{2}-1\right ) {\mathrm e}^{x}\right ) {\mathrm e}^{\frac {-{\mathrm e}^{{\mathrm e}^{\frac {x^{6} {\mathrm e}^{x}-x}{{\mathrm e}^{x}}}}+x^{2}+1}{x}}+{\mathrm e}^{x} x^{2}}{{\mathrm e}^{x} x^{2}}d x \] Input:

int(((((-6*x^6*exp(x)-x^2+x)*exp((x^6*exp(x)-x)/exp(x))+exp(x))*exp(exp((x 
^6*exp(x)-x)/exp(x)))+(x^2-1)*exp(x))*exp((-exp(exp((x^6*exp(x)-x)/exp(x)) 
)+x^2+1)/x)+exp(x)*x^2)/exp(x)/x^2,x)
 

Output:

int(((((-6*x^6*exp(x)-x^2+x)*exp((x^6*exp(x)-x)/exp(x))+exp(x))*exp(exp((x 
^6*exp(x)-x)/exp(x)))+(x^2-1)*exp(x))*exp((-exp(exp((x^6*exp(x)-x)/exp(x)) 
)+x^2+1)/x)+exp(x)*x^2)/exp(x)/x^2,x)