\(\int e^{-2 e^{1-5 e^{1-x}+5 x}} x^{-2+\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}} (1+e^{2 e^{1-5 e^{1-x}+5 x}} x^2-2 \log (x)+e^{-5 e^{1-x}+5 x} (-10 e x-10 e^{2-x} x) \log (x)) \, dx\) [1328]

Optimal result

Integrand size = 112, antiderivative size = 29 \[ \int e^{-2 e^{1-5 e^{1-x}+5 x}} x^{-2+\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}} \left (1+e^{2 e^{1-5 e^{1-x}+5 x}} x^2-2 \log (x)+e^{-5 e^{1-x}+5 x} \left (-10 e x-10 e^{2-x} x\right ) \log (x)\right ) \, dx=x^{1+\frac {e^{-2 e^{1+5 \left (-e^{1-x}+x\right )}}}{x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97 \[ \int e^{-2 e^{1-5 e^{1-x}+5 x}} x^{-2+\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}} \left (1+e^{2 e^{1-5 e^{1-x}+5 x}} x^2-2 \log (x)+e^{-5 e^{1-x}+5 x} \left (-10 e x-10 e^{2-x} x\right ) \log (x)\right ) \, dx=x^{1+\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}} \] Input:

Integrate[(x^(-2 + 1/(E^(2*E^(1 - 5*E^(1 - x) + 5*x))*x^2))*(1 + E^(2*E^(1 
 - 5*E^(1 - x) + 5*x))*x^2 - 2*Log[x] + E^(-5*E^(1 - x) + 5*x)*(-10*E*x - 
10*E^(2 - x)*x)*Log[x]))/E^(2*E^(1 - 5*E^(1 - x) + 5*x)),x]
 

Output:

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

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.

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle \int e^{-2 e^{5 x-5 e^{1-x}+1}} x^{\frac {e^{-2 e^{5 x-5 e^{1-x}+1}}}{x^2}-2}dx+2 \int \frac {\int e^{-2 e^{5 x-5 e^{1-x}+1}} x^{\frac {e^{-2 e^{5 x-5 e^{1-x}+1}}}{x^2}-2}dx}{x}dx+10 \int \frac {\int e^{4 x-5 e^{1-x}-2 e^{5 x-5 e^{1-x}+1}+2} x^{\frac {e^{-2 e^{5 x-5 e^{1-x}+1}}}{x^2}-1}dx}{x}dx+10 \int \frac {\int e^{5 x-5 e^{1-x}-2 e^{5 x-5 e^{1-x}+1}+1} x^{\frac {e^{-2 e^{5 x-5 e^{1-x}+1}}}{x^2}-1}dx}{x}dx-2 \log (x) \int e^{-2 e^{5 x-5 e^{1-x}+1}} x^{\frac {e^{-2 e^{5 x-5 e^{1-x}+1}}}{x^2}-2}dx-10 \log (x) \int e^{4 x-5 e^{1-x}-2 e^{5 x-5 e^{1-x}+1}+2} x^{\frac {e^{-2 e^{5 x-5 e^{1-x}+1}}}{x^2}-1}dx-10 \log (x) \int e^{5 x-5 e^{1-x}-2 e^{5 x-5 e^{1-x}+1}+1} x^{\frac {e^{-2 e^{5 x-5 e^{1-x}+1}}}{x^2}-1}dx+\int x^{\frac {e^{-2 e^{5 x-5 e^{1-x}+1}}}{x^2}}dx\)

Input:

Int[(x^(-2 + 1/(E^(2*E^(1 - 5*E^(1 - x) + 5*x))*x^2))*(1 + E^(2*E^(1 - 5*E 
^(1 - x) + 5*x))*x^2 - 2*Log[x] + E^(-5*E^(1 - x) + 5*x)*(-10*E*x - 10*E^( 
2 - x)*x)*Log[x]))/E^(2*E^(1 - 5*E^(1 - x) + 5*x)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90

\[x^{\frac {{\mathrm e}^{-2 \,{\mathrm e}^{1-5 \,{\mathrm e}^{1-x}+5 x}}}{x^{2}}} x\]

Input:

int((x^2*exp(exp(1)*exp(-5*exp(1-x)+5*x))^2+(-10*x*exp(1)*exp(1-x)-10*x*ex 
p(1))*ln(x)*exp(-5*exp(1-x)+5*x)-2*ln(x)+1)*exp(ln(x)/x^2/exp(exp(1)*exp(- 
5*exp(1-x)+5*x))^2)/x^2/exp(exp(1)*exp(-5*exp(1-x)+5*x))^2,x)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.10 \[ \int e^{-2 e^{1-5 e^{1-x}+5 x}} x^{-2+\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}} \left (1+e^{2 e^{1-5 e^{1-x}+5 x}} x^2-2 \log (x)+e^{-5 e^{1-x}+5 x} \left (-10 e x-10 e^{2-x} x\right ) \log (x)\right ) \, dx=x x^{\frac {e^{\left (-2 \, e^{\left ({\left ({\left (5 \, x + 1\right )} e - 5 \, e^{\left (-x + 2\right )}\right )} e^{\left (-1\right )}\right )}\right )}}{x^{2}}} \] Input:

integrate((x^2*exp(exp(1)*exp(-5*exp(1-x)+5*x))^2+(-10*x*exp(1)*exp(1-x)-1 
0*exp(1)*x)*log(x)*exp(-5*exp(1-x)+5*x)-2*log(x)+1)*exp(log(x)/x^2/exp(exp 
(1)*exp(-5*exp(1-x)+5*x))^2)/x^2/exp(exp(1)*exp(-5*exp(1-x)+5*x))^2,x, alg 
orithm="fricas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int e^{-2 e^{1-5 e^{1-x}+5 x}} x^{-2+\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}} \left (1+e^{2 e^{1-5 e^{1-x}+5 x}} x^2-2 \log (x)+e^{-5 e^{1-x}+5 x} \left (-10 e x-10 e^{2-x} x\right ) \log (x)\right ) \, dx=\text {Timed out} \] Input:

integrate((x**2*exp(exp(1)*exp(-5*exp(1-x)+5*x))**2+(-10*x*exp(1)*exp(1-x) 
-10*exp(1)*x)*ln(x)*exp(-5*exp(1-x)+5*x)-2*ln(x)+1)*exp(ln(x)/x**2/exp(exp 
(1)*exp(-5*exp(1-x)+5*x))**2)/x**2/exp(exp(1)*exp(-5*exp(1-x)+5*x))**2,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int e^{-2 e^{1-5 e^{1-x}+5 x}} x^{-2+\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}} \left (1+e^{2 e^{1-5 e^{1-x}+5 x}} x^2-2 \log (x)+e^{-5 e^{1-x}+5 x} \left (-10 e x-10 e^{2-x} x\right ) \log (x)\right ) \, dx=\int { \frac {{\left (x^{2} e^{\left (2 \, e^{\left (5 \, x - 5 \, e^{\left (-x + 1\right )} + 1\right )}\right )} - 10 \, {\left (x e + x e^{\left (-x + 2\right )}\right )} e^{\left (5 \, x - 5 \, e^{\left (-x + 1\right )}\right )} \log \left (x\right ) - 2 \, \log \left (x\right ) + 1\right )} x^{\frac {e^{\left (-2 \, e^{\left (5 \, x - 5 \, e^{\left (-x + 1\right )} + 1\right )}\right )}}{x^{2}}} e^{\left (-2 \, e^{\left (5 \, x - 5 \, e^{\left (-x + 1\right )} + 1\right )}\right )}}{x^{2}} \,d x } \] Input:

integrate((x^2*exp(exp(1)*exp(-5*exp(1-x)+5*x))^2+(-10*x*exp(1)*exp(1-x)-1 
0*exp(1)*x)*log(x)*exp(-5*exp(1-x)+5*x)-2*log(x)+1)*exp(log(x)/x^2/exp(exp 
(1)*exp(-5*exp(1-x)+5*x))^2)/x^2/exp(exp(1)*exp(-5*exp(1-x)+5*x))^2,x, alg 
orithm="maxima")
 

Output:

integrate((x^2*e^(2*e^(5*x - 5*e^(-x + 1) + 1)) - 10*(x*e + x*e^(-x + 2))* 
e^(5*x - 5*e^(-x + 1))*log(x) - 2*log(x) + 1)*x^(e^(-2*e^(5*x - 5*e^(-x + 
1) + 1))/x^2 - 2)*e^(-2*e^(5*x - 5*e^(-x + 1) + 1)), x)
 

Giac [F]

\[ \int e^{-2 e^{1-5 e^{1-x}+5 x}} x^{-2+\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}} \left (1+e^{2 e^{1-5 e^{1-x}+5 x}} x^2-2 \log (x)+e^{-5 e^{1-x}+5 x} \left (-10 e x-10 e^{2-x} x\right ) \log (x)\right ) \, dx=\int { \frac {{\left (x^{2} e^{\left (2 \, e^{\left (5 \, x - 5 \, e^{\left (-x + 1\right )} + 1\right )}\right )} - 10 \, {\left (x e + x e^{\left (-x + 2\right )}\right )} e^{\left (5 \, x - 5 \, e^{\left (-x + 1\right )}\right )} \log \left (x\right ) - 2 \, \log \left (x\right ) + 1\right )} x^{\frac {e^{\left (-2 \, e^{\left (5 \, x - 5 \, e^{\left (-x + 1\right )} + 1\right )}\right )}}{x^{2}}} e^{\left (-2 \, e^{\left (5 \, x - 5 \, e^{\left (-x + 1\right )} + 1\right )}\right )}}{x^{2}} \,d x } \] Input:

integrate((x^2*exp(exp(1)*exp(-5*exp(1-x)+5*x))^2+(-10*x*exp(1)*exp(1-x)-1 
0*exp(1)*x)*log(x)*exp(-5*exp(1-x)+5*x)-2*log(x)+1)*exp(log(x)/x^2/exp(exp 
(1)*exp(-5*exp(1-x)+5*x))^2)/x^2/exp(exp(1)*exp(-5*exp(1-x)+5*x))^2,x, alg 
orithm="giac")
 

Output:

integrate((x^2*e^(2*e^(5*x - 5*e^(-x + 1) + 1)) - 10*(x*e + x*e^(-x + 2))* 
e^(5*x - 5*e^(-x + 1))*log(x) - 2*log(x) + 1)*x^(e^(-2*e^(5*x - 5*e^(-x + 
1) + 1))/x^2)*e^(-2*e^(5*x - 5*e^(-x + 1) + 1))/x^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int e^{-2 e^{1-5 e^{1-x}+5 x}} x^{-2+\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}} \left (1+e^{2 e^{1-5 e^{1-x}+5 x}} x^2-2 \log (x)+e^{-5 e^{1-x}+5 x} \left (-10 e x-10 e^{2-x} x\right ) \log (x)\right ) \, dx=\int -\frac {{\mathrm {e}}^{\frac {{\mathrm {e}}^{-2\,\mathrm {e}\,{\mathrm {e}}^{5\,x-5\,{\mathrm {e}}^{1-x}}}\,\ln \left (x\right )}{x^2}}\,{\mathrm {e}}^{-2\,\mathrm {e}\,{\mathrm {e}}^{5\,x-5\,{\mathrm {e}}^{1-x}}}\,\left (2\,\ln \left (x\right )-x^2\,{\mathrm {e}}^{2\,\mathrm {e}\,{\mathrm {e}}^{5\,x-5\,{\mathrm {e}}^{1-x}}}+{\mathrm {e}}^{5\,x-5\,{\mathrm {e}}^{1-x}}\,\ln \left (x\right )\,\left (10\,x\,\mathrm {e}+10\,x\,\mathrm {e}\,{\mathrm {e}}^{1-x}\right )-1\right )}{x^2} \,d x \] Input:

int(-(exp((exp(-2*exp(1)*exp(5*x - 5*exp(1 - x)))*log(x))/x^2)*exp(-2*exp( 
1)*exp(5*x - 5*exp(1 - x)))*(2*log(x) - x^2*exp(2*exp(1)*exp(5*x - 5*exp(1 
 - x))) + exp(5*x - 5*exp(1 - x))*log(x)*(10*x*exp(1) + 10*x*exp(1)*exp(1 
- x)) - 1))/x^2,x)
 

Output:

int(-(exp((exp(-2*exp(1)*exp(5*x - 5*exp(1 - x)))*log(x))/x^2)*exp(-2*exp( 
1)*exp(5*x - 5*exp(1 - x)))*(2*log(x) - x^2*exp(2*exp(1)*exp(5*x - 5*exp(1 
 - x))) + exp(5*x - 5*exp(1 - x))*log(x)*(10*x*exp(1) + 10*x*exp(1)*exp(1 
- x)) - 1))/x^2, x)
 

Reduce [F]

\[ \int e^{-2 e^{1-5 e^{1-x}+5 x}} x^{-2+\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}} \left (1+e^{2 e^{1-5 e^{1-x}+5 x}} x^2-2 \log (x)+e^{-5 e^{1-x}+5 x} \left (-10 e x-10 e^{2-x} x\right ) \log (x)\right ) \, dx=\int e^{\frac {\mathrm {log}\left (x \right )}{e^{\frac {2 e^{5 x} e}{e^{\frac {5 e}{e^{x}}}}} x^{2}}}d x +\int \frac {e^{\frac {\mathrm {log}\left (x \right )}{e^{\frac {2 e^{5 x} e}{e^{\frac {5 e}{e^{x}}}}} x^{2}}}}{e^{\frac {2 e^{5 x} e}{e^{\frac {5 e}{e^{x}}}}} x^{2}}d x -2 \left (\int \frac {e^{\frac {\mathrm {log}\left (x \right )}{e^{\frac {2 e^{5 x} e}{e^{\frac {5 e}{e^{x}}}}} x^{2}}} \mathrm {log}\left (x \right )}{e^{\frac {2 e^{5 x} e}{e^{\frac {5 e}{e^{x}}}}} x^{2}}d x \right )-10 \left (\int \frac {e^{\frac {5 e^{\frac {2 e^{5 x} e}{e^{\frac {5 e}{e^{x}}}}} x^{3}+\mathrm {log}\left (x \right )}{e^{\frac {2 e^{5 x} e}{e^{\frac {5 e}{e^{x}}}}} x^{2}}} \mathrm {log}\left (x \right )}{e^{\frac {5 e^{\frac {5 e}{e^{x}}} e +2 e^{6 x} e}{e^{\frac {e^{x} x +5 e}{e^{x}}}}} x}d x \right ) e -10 \left (\int \frac {e^{\frac {4 e^{\frac {2 e^{5 x} e}{e^{\frac {5 e}{e^{x}}}}} x^{3}+\mathrm {log}\left (x \right )}{e^{\frac {2 e^{5 x} e}{e^{\frac {5 e}{e^{x}}}}} x^{2}}} \mathrm {log}\left (x \right )}{e^{\frac {5 e^{\frac {5 e}{e^{x}}} e +2 e^{6 x} e}{e^{\frac {e^{x} x +5 e}{e^{x}}}}} x}d x \right ) e^{2} \] Input:

int((x^2*exp(exp(1)*exp(-5*exp(1-x)+5*x))^2+(-10*x*exp(1)*exp(1-x)-10*exp( 
1)*x)*log(x)*exp(-5*exp(1-x)+5*x)-2*log(x)+1)*exp(log(x)/x^2/exp(exp(1)*ex 
p(-5*exp(1-x)+5*x))^2)/x^2/exp(exp(1)*exp(-5*exp(1-x)+5*x))^2,x)
 

Output:

int(e**(log(x)/(e**((2*e**(5*x)*e)/e**((5*e)/e**x))*x**2)),x) + int(e**(lo 
g(x)/(e**((2*e**(5*x)*e)/e**((5*e)/e**x))*x**2))/(e**((2*e**(5*x)*e)/e**(( 
5*e)/e**x))*x**2),x) - 2*int((e**(log(x)/(e**((2*e**(5*x)*e)/e**((5*e)/e** 
x))*x**2))*log(x))/(e**((2*e**(5*x)*e)/e**((5*e)/e**x))*x**2),x) - 10*int( 
(e**((5*e**((2*e**(5*x)*e)/e**((5*e)/e**x))*x**3 + log(x))/(e**((2*e**(5*x 
)*e)/e**((5*e)/e**x))*x**2))*log(x))/(e**((5*e**((5*e)/e**x)*e + 2*e**(6*x 
)*e)/e**((e**x*x + 5*e)/e**x))*x),x)*e - 10*int((e**((4*e**((2*e**(5*x)*e) 
/e**((5*e)/e**x))*x**3 + log(x))/(e**((2*e**(5*x)*e)/e**((5*e)/e**x))*x**2 
))*log(x))/(e**((5*e**((5*e)/e**x)*e + 2*e**(6*x)*e)/e**((e**x*x + 5*e)/e* 
*x))*x),x)*e**2