\(\int \frac {e^{-e^{\frac {-4-x+6 x^2-5 x^3}{x}}} (-x+e^{e^{\frac {-4-x+6 x^2-5 x^3}{x}}} x+e^{\frac {-4-x+6 x^2-5 x^3}{x}} (4+6 x^2-10 x^3))}{x} \, dx\) [2169]

Optimal result

Integrand size = 92, antiderivative size = 32 \[ \int \frac {e^{-e^{\frac {-4-x+6 x^2-5 x^3}{x}}} \left (-x+e^{e^{\frac {-4-x+6 x^2-5 x^3}{x}}} x+e^{\frac {-4-x+6 x^2-5 x^3}{x}} \left (4+6 x^2-10 x^3\right )\right )}{x} \, dx=x-e^{-e^{\frac {-4-x+x^2+(5-5 x) x^2}{x}}} x \] Output:

x-x/exp(exp(((-5*x+5)*x^2-4-x+x^2)/x))
                                                                                    
                                                                                    
 

Mathematica [F]

\[ \int \frac {e^{-e^{\frac {-4-x+6 x^2-5 x^3}{x}}} \left (-x+e^{e^{\frac {-4-x+6 x^2-5 x^3}{x}}} x+e^{\frac {-4-x+6 x^2-5 x^3}{x}} \left (4+6 x^2-10 x^3\right )\right )}{x} \, dx=\int \frac {e^{-e^{\frac {-4-x+6 x^2-5 x^3}{x}}} \left (-x+e^{e^{\frac {-4-x+6 x^2-5 x^3}{x}}} x+e^{\frac {-4-x+6 x^2-5 x^3}{x}} \left (4+6 x^2-10 x^3\right )\right )}{x} \, dx \] Input:

Integrate[(-x + E^E^((-4 - x + 6*x^2 - 5*x^3)/x)*x + E^((-4 - x + 6*x^2 - 
5*x^3)/x)*(4 + 6*x^2 - 10*x^3))/(E^E^((-4 - x + 6*x^2 - 5*x^3)/x)*x),x]
 

Output:

Integrate[(-x + E^E^((-4 - x + 6*x^2 - 5*x^3)/x)*x + E^((-4 - x + 6*x^2 - 
5*x^3)/x)*(4 + 6*x^2 - 10*x^3))/(E^E^((-4 - x + 6*x^2 - 5*x^3)/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.

\(\displaystyle \int \frac {e^{-e^{\frac {-5 x^3+6 x^2-x-4}{x}}} \left (e^{e^{\frac {-5 x^3+6 x^2-x-4}{x}}} x+e^{\frac {-5 x^3+6 x^2-x-4}{x}} \left (-10 x^3+6 x^2+4\right )-x\right )}{x} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {e^{-e^{-5 x^2+6 x-\frac {4}{x}-1}} \left (e^{e^{\frac {-5 x^3+6 x^2-x-4}{x}}} x+e^{\frac {-5 x^3+6 x^2-x-4}{x}} \left (-10 x^3+6 x^2+4\right )-x\right )}{x}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {2 \left (5 x^3-3 x^2-2\right ) \exp \left (-5 x^2-e^{-5 x^2+6 x-\frac {4}{x}-1}+6 x-\frac {4}{x}-1\right )}{x}-e^{-e^{-5 x^2+6 x-\frac {4}{x}-1}}+1\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 4 \int \frac {\exp \left (-5 x^2+6 x-e^{-5 x^2+6 x-1-\frac {4}{x}}-1-\frac {4}{x}\right )}{x}dx+6 \int \exp \left (-5 x^2+6 x-e^{-5 x^2+6 x-1-\frac {4}{x}}-1-\frac {4}{x}\right ) xdx-10 \int \exp \left (-5 x^2+6 x-e^{-5 x^2+6 x-1-\frac {4}{x}}-1-\frac {4}{x}\right ) x^2dx-\int e^{-e^{-5 x^2+6 x-1-\frac {4}{x}}}dx+x\)

Input:

Int[(-x + E^E^((-4 - x + 6*x^2 - 5*x^3)/x)*x + E^((-4 - x + 6*x^2 - 5*x^3) 
/x)*(4 + 6*x^2 - 10*x^3))/(E^E^((-4 - x + 6*x^2 - 5*x^3)/x)*x),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88

method	result	size
risch	\(x -x \,{\mathrm e}^{-{\mathrm e}^{-\frac {5 x^{3}-6 x^{2}+x +4}{x}}}\)	\(28\)
parallelrisch	\(-\left (-x \,{\mathrm e}^{{\mathrm e}^{-\frac {5 x^{3}-6 x^{2}+x +4}{x}}}+x \right ) {\mathrm e}^{-{\mathrm e}^{-\frac {5 x^{3}-6 x^{2}+x +4}{x}}}\)	\(50\)
norman	\(\left (x \,{\mathrm e}^{{\mathrm e}^{\frac {-5 x^{3}+6 x^{2}-x -4}{x}}}-x \right ) {\mathrm e}^{-{\mathrm e}^{\frac {-5 x^{3}+6 x^{2}-x -4}{x}}}\)	\(52\)

Input:

int((x*exp(exp((-5*x^3+6*x^2-x-4)/x))+(-10*x^3+6*x^2+4)*exp((-5*x^3+6*x^2- 
x-4)/x)-x)/x/exp(exp((-5*x^3+6*x^2-x-4)/x)),x,method=_RETURNVERBOSE)
 

Output:

x-x*exp(-exp(-(5*x^3-6*x^2+x+4)/x))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.53 \[ \int \frac {e^{-e^{\frac {-4-x+6 x^2-5 x^3}{x}}} \left (-x+e^{e^{\frac {-4-x+6 x^2-5 x^3}{x}}} x+e^{\frac {-4-x+6 x^2-5 x^3}{x}} \left (4+6 x^2-10 x^3\right )\right )}{x} \, dx={\left (x e^{\left (e^{\left (-\frac {5 \, x^{3} - 6 \, x^{2} + x + 4}{x}\right )}\right )} - x\right )} e^{\left (-e^{\left (-\frac {5 \, x^{3} - 6 \, x^{2} + x + 4}{x}\right )}\right )} \] Input:

integrate((x*exp(exp((-5*x^3+6*x^2-x-4)/x))+(-10*x^3+6*x^2+4)*exp((-5*x^3+ 
6*x^2-x-4)/x)-x)/x/exp(exp((-5*x^3+6*x^2-x-4)/x)),x, algorithm="fricas")
 

Output:

(x*e^(e^(-(5*x^3 - 6*x^2 + x + 4)/x)) - x)*e^(-e^(-(5*x^3 - 6*x^2 + x + 4) 
/x))
 

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{-e^{\frac {-4-x+6 x^2-5 x^3}{x}}} \left (-x+e^{e^{\frac {-4-x+6 x^2-5 x^3}{x}}} x+e^{\frac {-4-x+6 x^2-5 x^3}{x}} \left (4+6 x^2-10 x^3\right )\right )}{x} \, dx=\text {Timed out} \] Input:

integrate((x*exp(exp((-5*x**3+6*x**2-x-4)/x))+(-10*x**3+6*x**2+4)*exp((-5* 
x**3+6*x**2-x-4)/x)-x)/x/exp(exp((-5*x**3+6*x**2-x-4)/x)),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {e^{-e^{\frac {-4-x+6 x^2-5 x^3}{x}}} \left (-x+e^{e^{\frac {-4-x+6 x^2-5 x^3}{x}}} x+e^{\frac {-4-x+6 x^2-5 x^3}{x}} \left (4+6 x^2-10 x^3\right )\right )}{x} \, dx=\int { -\frac {{\left (2 \, {\left (5 \, x^{3} - 3 \, x^{2} - 2\right )} e^{\left (-\frac {5 \, x^{3} - 6 \, x^{2} + x + 4}{x}\right )} - x e^{\left (e^{\left (-\frac {5 \, x^{3} - 6 \, x^{2} + x + 4}{x}\right )}\right )} + x\right )} e^{\left (-e^{\left (-\frac {5 \, x^{3} - 6 \, x^{2} + x + 4}{x}\right )}\right )}}{x} \,d x } \] Input:

integrate((x*exp(exp((-5*x^3+6*x^2-x-4)/x))+(-10*x^3+6*x^2+4)*exp((-5*x^3+ 
6*x^2-x-4)/x)-x)/x/exp(exp((-5*x^3+6*x^2-x-4)/x)),x, algorithm="maxima")
 

Output:

x - integrate((x*e^(5*x^2 + 4/x + 1) + 2*(5*x^3 - 3*x^2 - 2)*e^(6*x))*e^(- 
5*x^2 - 4/x - e^(-5*x^2 + 6*x - 4/x - 1) - 1)/x, x)
 

Giac [F]

\[ \int \frac {e^{-e^{\frac {-4-x+6 x^2-5 x^3}{x}}} \left (-x+e^{e^{\frac {-4-x+6 x^2-5 x^3}{x}}} x+e^{\frac {-4-x+6 x^2-5 x^3}{x}} \left (4+6 x^2-10 x^3\right )\right )}{x} \, dx=\int { -\frac {{\left (2 \, {\left (5 \, x^{3} - 3 \, x^{2} - 2\right )} e^{\left (-\frac {5 \, x^{3} - 6 \, x^{2} + x + 4}{x}\right )} - x e^{\left (e^{\left (-\frac {5 \, x^{3} - 6 \, x^{2} + x + 4}{x}\right )}\right )} + x\right )} e^{\left (-e^{\left (-\frac {5 \, x^{3} - 6 \, x^{2} + x + 4}{x}\right )}\right )}}{x} \,d x } \] Input:

integrate((x*exp(exp((-5*x^3+6*x^2-x-4)/x))+(-10*x^3+6*x^2+4)*exp((-5*x^3+ 
6*x^2-x-4)/x)-x)/x/exp(exp((-5*x^3+6*x^2-x-4)/x)),x, algorithm="giac")
 

Output:

integrate(-(2*(5*x^3 - 3*x^2 - 2)*e^(-(5*x^3 - 6*x^2 + x + 4)/x) - x*e^(e^ 
(-(5*x^3 - 6*x^2 + x + 4)/x)) + x)*e^(-e^(-(5*x^3 - 6*x^2 + x + 4)/x))/x, 
x)
 

Mupad [B] (verification not implemented)

Time = 3.21 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int \frac {e^{-e^{\frac {-4-x+6 x^2-5 x^3}{x}}} \left (-x+e^{e^{\frac {-4-x+6 x^2-5 x^3}{x}}} x+e^{\frac {-4-x+6 x^2-5 x^3}{x}} \left (4+6 x^2-10 x^3\right )\right )}{x} \, dx=-x\,\left ({\mathrm {e}}^{-{\mathrm {e}}^{6\,x}\,{\mathrm {e}}^{-1}\,{\mathrm {e}}^{-\frac {4}{x}}\,{\mathrm {e}}^{-5\,x^2}}-1\right ) \] Input:

int((exp(-exp(-(x - 6*x^2 + 5*x^3 + 4)/x))*(exp(-(x - 6*x^2 + 5*x^3 + 4)/x 
)*(6*x^2 - 10*x^3 + 4) - x + x*exp(exp(-(x - 6*x^2 + 5*x^3 + 4)/x))))/x,x)
 

Output:

-x*(exp(-exp(6*x)*exp(-1)*exp(-4/x)*exp(-5*x^2)) - 1)
 

Reduce [F]

\[ \int \frac {e^{-e^{\frac {-4-x+6 x^2-5 x^3}{x}}} \left (-x+e^{e^{\frac {-4-x+6 x^2-5 x^3}{x}}} x+e^{\frac {-4-x+6 x^2-5 x^3}{x}} \left (4+6 x^2-10 x^3\right )\right )}{x} \, dx=\int \frac {x \,{\mathrm e}^{{\mathrm e}^{\frac {-5 x^{3}+6 x^{2}-x -4}{x}}}+\left (-10 x^{3}+6 x^{2}+4\right ) {\mathrm e}^{\frac {-5 x^{3}+6 x^{2}-x -4}{x}}-x}{x \,{\mathrm e}^{{\mathrm e}^{\frac {-5 x^{3}+6 x^{2}-x -4}{x}}}}d x \] Input:

int((x*exp(exp((-5*x^3+6*x^2-x-4)/x))+(-10*x^3+6*x^2+4)*exp((-5*x^3+6*x^2- 
x-4)/x)-x)/x/exp(exp((-5*x^3+6*x^2-x-4)/x)),x)
 

Output:

int((x*exp(exp((-5*x^3+6*x^2-x-4)/x))+(-10*x^3+6*x^2+4)*exp((-5*x^3+6*x^2- 
x-4)/x)-x)/x/exp(exp((-5*x^3+6*x^2-x-4)/x)),x)