\(\int \frac {e^{-1/x} (e^{\frac {1}{x}+x} (-4 x+6 x^2+4 x^3-2 x^4)+e^{2 e^{-1/x}+x} (-8+8 x-2 x^2+e^{\frac {1}{x}} (-4 x^2+4 x^3-x^4))+e^{\frac {1}{x}+x} (-6 x^2+2 x^3) \log (\frac {x}{4}))}{8 x^2-8 x^3+2 x^4} \, dx\) [2764]

Optimal result

Integrand size = 125, antiderivative size = 36 \[ \int \frac {e^{-1/x} \left (e^{\frac {1}{x}+x} \left (-4 x+6 x^2+4 x^3-2 x^4\right )+e^{2 e^{-1/x}+x} \left (-8+8 x-2 x^2+e^{\frac {1}{x}} \left (-4 x^2+4 x^3-x^4\right )\right )+e^{\frac {1}{x}+x} \left (-6 x^2+2 x^3\right ) \log \left (\frac {x}{4}\right )\right )}{8 x^2-8 x^3+2 x^4} \, dx=e^x \left (-\frac {1}{2} e^{2 e^{-1/x}}+\frac {-x+\log \left (\frac {x}{4}\right )}{-2+x}\right ) \] Output:

((ln(1/4*x)-x)/(-2+x)-1/2*exp(2/exp(1/x)))*exp(x)
 

Mathematica [A] (verified)

Time = 5.11 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.03 \[ \int \frac {e^{-1/x} \left (e^{\frac {1}{x}+x} \left (-4 x+6 x^2+4 x^3-2 x^4\right )+e^{2 e^{-1/x}+x} \left (-8+8 x-2 x^2+e^{\frac {1}{x}} \left (-4 x^2+4 x^3-x^4\right )\right )+e^{\frac {1}{x}+x} \left (-6 x^2+2 x^3\right ) \log \left (\frac {x}{4}\right )\right )}{8 x^2-8 x^3+2 x^4} \, dx=-\frac {e^x \left (e^{2 e^{-1/x}} (-2+x)+2 x+\log (16)-2 \log (x)\right )}{2 (-2+x)} \] Input:

Integrate[(E^(x^(-1) + x)*(-4*x + 6*x^2 + 4*x^3 - 2*x^4) + E^(2/E^x^(-1) + 
 x)*(-8 + 8*x - 2*x^2 + E^x^(-1)*(-4*x^2 + 4*x^3 - x^4)) + E^(x^(-1) + x)* 
(-6*x^2 + 2*x^3)*Log[x/4])/(E^x^(-1)*(8*x^2 - 8*x^3 + 2*x^4)),x]
 

Output:

-1/2*(E^x*(E^(2/E^x^(-1))*(-2 + x) + 2*x + Log[16] - 2*Log[x]))/(-2 + 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.

\(\displaystyle \int \frac {e^{-1/x} \left (e^{x+\frac {1}{x}} \left (2 x^3-6 x^2\right ) \log \left (\frac {x}{4}\right )+e^{x+\frac {1}{x}} \left (-2 x^4+4 x^3+6 x^2-4 x\right )+e^{x+2 e^{-1/x}} \left (-2 x^2+e^{\frac {1}{x}} \left (-x^4+4 x^3-4 x^2\right )+8 x-8\right )\right )}{2 x^4-8 x^3+8 x^2} \, dx\)

\(\Big \downarrow \) 2026

\(\displaystyle \int \frac {e^{-1/x} \left (e^{x+\frac {1}{x}} \left (2 x^3-6 x^2\right ) \log \left (\frac {x}{4}\right )+e^{x+\frac {1}{x}} \left (-2 x^4+4 x^3+6 x^2-4 x\right )+e^{x+2 e^{-1/x}} \left (-2 x^2+e^{\frac {1}{x}} \left (-x^4+4 x^3-4 x^2\right )+8 x-8\right )\right )}{x^2 \left (2 x^2-8 x+8\right )}dx\)

\(\Big \downarrow \) 7277

\(\displaystyle 8 \int -\frac {e^{-1/x} \left (2 e^{x+\frac {1}{x}} \left (x^4-2 x^3-3 x^2+2 x\right )+e^{x+2 e^{-1/x}} \left (2 x^2-8 x+e^{\frac {1}{x}} \left (x^4-4 x^3+4 x^2\right )+8\right )+2 e^{x+\frac {1}{x}} \left (3 x^2-x^3\right ) \log \left (\frac {x}{4}\right )\right )}{16 (2-x)^2 x^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{2} \int \frac {e^{-1/x} \left (2 e^{x+\frac {1}{x}} \left (x^4-2 x^3-3 x^2+2 x\right )+e^{x+2 e^{-1/x}} \left (2 x^2-8 x+e^{\frac {1}{x}} \left (x^4-4 x^3+4 x^2\right )+8\right )+2 e^{x+\frac {1}{x}} \left (3 x^2-x^3\right ) \log \left (\frac {x}{4}\right )\right )}{(2-x)^2 x^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {1}{2} \int \left (-\frac {e^{x-\frac {1}{x}} \left (e^{2 e^{-1/x}+\frac {1}{x}} x^4+2 e^{\frac {1}{x}} x^4-4 e^{2 e^{-1/x}+\frac {1}{x}} x^3-4 e^{\frac {1}{x}} x^3-2 e^{\frac {1}{x}} \log \left (\frac {x}{4}\right ) x^3+2 e^{2 e^{-1/x}} x^2+4 e^{2 e^{-1/x}+\frac {1}{x}} x^2-6 e^{\frac {1}{x}} x^2+6 e^{\frac {1}{x}} \log \left (\frac {x}{4}\right ) x^2-8 e^{2 e^{-1/x}} x+4 e^{\frac {1}{x}} x+8 e^{2 e^{-1/x}}\right )}{4 (x-2)}+\frac {e^{x-\frac {1}{x}} \left (e^{2 e^{-1/x}+\frac {1}{x}} x^4+2 e^{\frac {1}{x}} x^4-4 e^{2 e^{-1/x}+\frac {1}{x}} x^3-4 e^{\frac {1}{x}} x^3-2 e^{\frac {1}{x}} \log \left (\frac {x}{4}\right ) x^3+2 e^{2 e^{-1/x}} x^2+4 e^{2 e^{-1/x}+\frac {1}{x}} x^2-6 e^{\frac {1}{x}} x^2+6 e^{\frac {1}{x}} \log \left (\frac {x}{4}\right ) x^2-8 e^{2 e^{-1/x}} x+4 e^{\frac {1}{x}} x+8 e^{2 e^{-1/x}}\right )}{4 x}+\frac {e^{x-\frac {1}{x}} \left (e^{2 e^{-1/x}+\frac {1}{x}} x^4+2 e^{\frac {1}{x}} x^4-4 e^{2 e^{-1/x}+\frac {1}{x}} x^3-4 e^{\frac {1}{x}} x^3-2 e^{\frac {1}{x}} \log \left (\frac {x}{4}\right ) x^3+2 e^{2 e^{-1/x}} x^2+4 e^{2 e^{-1/x}+\frac {1}{x}} x^2-6 e^{\frac {1}{x}} x^2+6 e^{\frac {1}{x}} \log \left (\frac {x}{4}\right ) x^2-8 e^{2 e^{-1/x}} x+4 e^{\frac {1}{x}} x+8 e^{2 e^{-1/x}}\right )}{4 (x-2)^2}+\frac {e^{x-\frac {1}{x}} \left (e^{2 e^{-1/x}+\frac {1}{x}} x^4+2 e^{\frac {1}{x}} x^4-4 e^{2 e^{-1/x}+\frac {1}{x}} x^3-4 e^{\frac {1}{x}} x^3-2 e^{\frac {1}{x}} \log \left (\frac {x}{4}\right ) x^3+2 e^{2 e^{-1/x}} x^2+4 e^{2 e^{-1/x}+\frac {1}{x}} x^2-6 e^{\frac {1}{x}} x^2+6 e^{\frac {1}{x}} \log \left (\frac {x}{4}\right ) x^2-8 e^{2 e^{-1/x}} x+4 e^{\frac {1}{x}} x+8 e^{2 e^{-1/x}}\right )}{4 x^2}\right )dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

Int[(E^(x^(-1) + x)*(-4*x + 6*x^2 + 4*x^3 - 2*x^4) + E^(2/E^x^(-1) + x)*(- 
8 + 8*x - 2*x^2 + E^x^(-1)*(-4*x^2 + 4*x^3 - x^4)) + E^(x^(-1) + x)*(-6*x^ 
2 + 2*x^3)*Log[x/4])/(E^x^(-1)*(8*x^2 - 8*x^3 + 2*x^4)),x]
 

Output:

$Aborted
 

Maple [F(-1)]

Timed out.

\[\int \frac {\left (\left (\left (-x^{4}+4 x^{3}-4 x^{2}\right ) {\mathrm e}^{\frac {1}{x}}-2 x^{2}+8 x -8\right ) {\mathrm e}^{x} {\mathrm e}^{2 \,{\mathrm e}^{-\frac {1}{x}}}+\left (2 x^{3}-6 x^{2}\right ) {\mathrm e}^{\frac {1}{x}} {\mathrm e}^{x} \ln \left (\frac {x}{4}\right )+\left (-2 x^{4}+4 x^{3}+6 x^{2}-4 x \right ) {\mathrm e}^{\frac {1}{x}} {\mathrm e}^{x}\right ) {\mathrm e}^{-\frac {1}{x}}}{2 x^{4}-8 x^{3}+8 x^{2}}d x\]

Input:

int((((-x^4+4*x^3-4*x^2)*exp(1/x)-2*x^2+8*x-8)*exp(x)*exp(2/exp(1/x))+(2*x 
^3-6*x^2)*exp(1/x)*exp(x)*ln(1/4*x)+(-2*x^4+4*x^3+6*x^2-4*x)*exp(1/x)*exp( 
x))/(2*x^4-8*x^3+8*x^2)/exp(1/x),x)
 

Output:

int((((-x^4+4*x^3-4*x^2)*exp(1/x)-2*x^2+8*x-8)*exp(x)*exp(2/exp(1/x))+(2*x 
^3-6*x^2)*exp(1/x)*exp(x)*ln(1/4*x)+(-2*x^4+4*x^3+6*x^2-4*x)*exp(1/x)*exp( 
x))/(2*x^4-8*x^3+8*x^2)/exp(1/x),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (29) = 58\).

Time = 0.08 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.86 \[ \int \frac {e^{-1/x} \left (e^{\frac {1}{x}+x} \left (-4 x+6 x^2+4 x^3-2 x^4\right )+e^{2 e^{-1/x}+x} \left (-8+8 x-2 x^2+e^{\frac {1}{x}} \left (-4 x^2+4 x^3-x^4\right )\right )+e^{\frac {1}{x}+x} \left (-6 x^2+2 x^3\right ) \log \left (\frac {x}{4}\right )\right )}{8 x^2-8 x^3+2 x^4} \, dx=-\frac {{\left ({\left (x - 2\right )} e^{\left ({\left (x e^{\frac {1}{x}} + 2\right )} e^{\left (-\frac {1}{x}\right )} + \frac {1}{x}\right )} + 2 \, x e^{\left (\frac {x^{2} + 1}{x}\right )} - 2 \, e^{\left (\frac {x^{2} + 1}{x}\right )} \log \left (\frac {1}{4} \, x\right )\right )} e^{\left (-\frac {1}{x}\right )}}{2 \, {\left (x - 2\right )}} \] Input:

integrate((((-x^4+4*x^3-4*x^2)*exp(1/x)-2*x^2+8*x-8)*exp(x)*exp(2/exp(1/x) 
)+(2*x^3-6*x^2)*exp(1/x)*exp(x)*log(1/4*x)+(-2*x^4+4*x^3+6*x^2-4*x)*exp(1/ 
x)*exp(x))/(2*x^4-8*x^3+8*x^2)/exp(1/x),x, algorithm="fricas")
 

Output:

-1/2*((x - 2)*e^((x*e^(1/x) + 2)*e^(-1/x) + 1/x) + 2*x*e^((x^2 + 1)/x) - 2 
*e^((x^2 + 1)/x)*log(1/4*x))*e^(-1/x)/(x - 2)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{-1/x} \left (e^{\frac {1}{x}+x} \left (-4 x+6 x^2+4 x^3-2 x^4\right )+e^{2 e^{-1/x}+x} \left (-8+8 x-2 x^2+e^{\frac {1}{x}} \left (-4 x^2+4 x^3-x^4\right )\right )+e^{\frac {1}{x}+x} \left (-6 x^2+2 x^3\right ) \log \left (\frac {x}{4}\right )\right )}{8 x^2-8 x^3+2 x^4} \, dx=\text {Timed out} \] Input:

integrate((((-x**4+4*x**3-4*x**2)*exp(1/x)-2*x**2+8*x-8)*exp(x)*exp(2/exp( 
1/x))+(2*x**3-6*x**2)*exp(1/x)*exp(x)*ln(1/4*x)+(-2*x**4+4*x**3+6*x**2-4*x 
)*exp(1/x)*exp(x))/(2*x**4-8*x**3+8*x**2)/exp(1/x),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.03 \[ \int \frac {e^{-1/x} \left (e^{\frac {1}{x}+x} \left (-4 x+6 x^2+4 x^3-2 x^4\right )+e^{2 e^{-1/x}+x} \left (-8+8 x-2 x^2+e^{\frac {1}{x}} \left (-4 x^2+4 x^3-x^4\right )\right )+e^{\frac {1}{x}+x} \left (-6 x^2+2 x^3\right ) \log \left (\frac {x}{4}\right )\right )}{8 x^2-8 x^3+2 x^4} \, dx=-\frac {{\left (x - 2\right )} e^{\left (x + 2 \, e^{\left (-\frac {1}{x}\right )}\right )} + 2 \, {\left (x + 2 \, \log \left (2\right ) - \log \left (x\right )\right )} e^{x}}{2 \, {\left (x - 2\right )}} \] Input:

integrate((((-x^4+4*x^3-4*x^2)*exp(1/x)-2*x^2+8*x-8)*exp(x)*exp(2/exp(1/x) 
)+(2*x^3-6*x^2)*exp(1/x)*exp(x)*log(1/4*x)+(-2*x^4+4*x^3+6*x^2-4*x)*exp(1/ 
x)*exp(x))/(2*x^4-8*x^3+8*x^2)/exp(1/x),x, algorithm="maxima")
 

Output:

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

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.06 \[ \int \frac {e^{-1/x} \left (e^{\frac {1}{x}+x} \left (-4 x+6 x^2+4 x^3-2 x^4\right )+e^{2 e^{-1/x}+x} \left (-8+8 x-2 x^2+e^{\frac {1}{x}} \left (-4 x^2+4 x^3-x^4\right )\right )+e^{\frac {1}{x}+x} \left (-6 x^2+2 x^3\right ) \log \left (\frac {x}{4}\right )\right )}{8 x^2-8 x^3+2 x^4} \, dx=-\frac {x e^{x} + 2 \, e^{x} \log \left (2\right ) - e^{x} \log \left (x\right )}{x - 2} - \frac {1}{2} \, e^{\left (x + 2 \, e^{\left (-\frac {1}{x}\right )}\right )} \] Input:

integrate((((-x^4+4*x^3-4*x^2)*exp(1/x)-2*x^2+8*x-8)*exp(x)*exp(2/exp(1/x) 
)+(2*x^3-6*x^2)*exp(1/x)*exp(x)*log(1/4*x)+(-2*x^4+4*x^3+6*x^2-4*x)*exp(1/ 
x)*exp(x))/(2*x^4-8*x^3+8*x^2)/exp(1/x),x, algorithm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 2.83 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.22 \[ \int \frac {e^{-1/x} \left (e^{\frac {1}{x}+x} \left (-4 x+6 x^2+4 x^3-2 x^4\right )+e^{2 e^{-1/x}+x} \left (-8+8 x-2 x^2+e^{\frac {1}{x}} \left (-4 x^2+4 x^3-x^4\right )\right )+e^{\frac {1}{x}+x} \left (-6 x^2+2 x^3\right ) \log \left (\frac {x}{4}\right )\right )}{8 x^2-8 x^3+2 x^4} \, dx=-\frac {{\mathrm {e}}^{x+2\,{\mathrm {e}}^{-\frac {1}{x}}}}{2}-\frac {x\,{\mathrm {e}}^x}{x-2}-\frac {x\,\ln \left (\frac {x}{4}\right )\,{\mathrm {e}}^x}{2\,x-x^2} \] Input:

int(-(exp(-1/x)*(exp(2*exp(-1/x))*exp(x)*(exp(1/x)*(4*x^2 - 4*x^3 + x^4) - 
 8*x + 2*x^2 + 8) + exp(1/x)*exp(x)*(4*x - 6*x^2 - 4*x^3 + 2*x^4) + log(x/ 
4)*exp(1/x)*exp(x)*(6*x^2 - 2*x^3)))/(8*x^2 - 8*x^3 + 2*x^4),x)
                                                                                    
                                                                                    
 

Output:

- exp(x + 2*exp(-1/x))/2 - (x*exp(x))/(x - 2) - (x*log(x/4)*exp(x))/(2*x - 
 x^2)
 

Reduce [B] (verification not implemented)

Time = 2.86 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.33 \[ \int \frac {e^{-1/x} \left (e^{\frac {1}{x}+x} \left (-4 x+6 x^2+4 x^3-2 x^4\right )+e^{2 e^{-1/x}+x} \left (-8+8 x-2 x^2+e^{\frac {1}{x}} \left (-4 x^2+4 x^3-x^4\right )\right )+e^{\frac {1}{x}+x} \left (-6 x^2+2 x^3\right ) \log \left (\frac {x}{4}\right )\right )}{8 x^2-8 x^3+2 x^4} \, dx=\frac {e^{x} \left (-e^{\frac {2}{e^{\frac {1}{x}}}} x +2 e^{\frac {2}{e^{\frac {1}{x}}}}+2 \,\mathrm {log}\left (\frac {x}{4}\right )-2 x \right )}{2 x -4} \] Input:

int((((-x^4+4*x^3-4*x^2)*exp(1/x)-2*x^2+8*x-8)*exp(x)*exp(2/exp(1/x))+(2*x 
^3-6*x^2)*exp(1/x)*exp(x)*log(1/4*x)+(-2*x^4+4*x^3+6*x^2-4*x)*exp(1/x)*exp 
(x))/(2*x^4-8*x^3+8*x^2)/exp(1/x),x)
 

Output:

(e**x*( - e**(2/e**(1/x))*x + 2*e**(2/e**(1/x)) + 2*log(x/4) - 2*x))/(2*(x 
 - 2))