\(\int e^{e^{-\frac {e^{25+e^x+x} (-4+4 x)}{x}} x-\frac {e^{25+e^x+x} (-4+4 x)}{x}} (1+\frac {e^{25+e^x+x} (-4+4 x-4 x^2+e^x (4 x-4 x^2))}{x}) \, dx\) [596]

Optimal result

Integrand size = 81, antiderivative size = 22 \[ \int e^{e^{-\frac {e^{25+e^x+x} (-4+4 x)}{x}} x-\frac {e^{25+e^x+x} (-4+4 x)}{x}} \left (1+\frac {e^{25+e^x+x} \left (-4+4 x-4 x^2+e^x \left (4 x-4 x^2\right )\right )}{x}\right ) \, dx=e^{e^{-\frac {4 e^{25+e^x+x} (-1+x)}{x}} x} \] Output:

exp(x/exp(4*exp(ln(exp(25)/x)+exp(x)+x)*(-1+x)))
 

Mathematica [A] (verified)

Time = 5.17 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int e^{e^{-\frac {e^{25+e^x+x} (-4+4 x)}{x}} x-\frac {e^{25+e^x+x} (-4+4 x)}{x}} \left (1+\frac {e^{25+e^x+x} \left (-4+4 x-4 x^2+e^x \left (4 x-4 x^2\right )\right )}{x}\right ) \, dx=e^{e^{-\frac {4 e^{25+e^x+x} (-1+x)}{x}} x} \] Input:

Integrate[E^(x/E^((E^(25 + E^x + x)*(-4 + 4*x))/x) - (E^(25 + E^x + x)*(-4 
 + 4*x))/x)*(1 + (E^(25 + E^x + x)*(-4 + 4*x - 4*x^2 + E^x*(4*x - 4*x^2))) 
/x),x]
 

Output:

E^(x/E^((4*E^(25 + E^x + x)*(-1 + 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.

\(\displaystyle \int \left (\frac {e^{x+e^x+25} \left (-4 x^2+e^x \left (4 x-4 x^2\right )+4 x-4\right )}{x}+1\right ) \exp \left (e^{-\frac {e^{x+e^x+25} (4 x-4)}{x}} x-\frac {e^{x+e^x+25} (4 x-4)}{x}\right ) \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\exp \left (e^{-\frac {e^{x+e^x+25} (4 x-4)}{x}} x-\frac {e^{x+e^x+25} (4 x-4)}{x}\right )-\frac {4 \left (e^x x^2+x^2-e^x x-x+1\right ) \exp \left (e^{-\frac {e^{x+e^x+25} (4 x-4)}{x}} x+x+e^x-\frac {e^{x+e^x+25} (4 x-4)}{x}+25\right )}{x}\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\exp \left (e^{-\frac {4 e^{x+e^x+25} (x-1)}{x}} x-\frac {4 e^{x+e^x+25} (x-1)}{x}\right )-\frac {4 \left (e^x x^2+x^2-e^x x-x+1\right ) \exp \left (-\frac {4 e^{x+e^x+25} (x-1)}{x}+e^x+e^{-\frac {4 e^{x+e^x+25} (x-1)}{x}} x+x+25\right )}{x}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \int \exp \left (e^{-\frac {4 e^{x+e^x+25} (x-1)}{x}} x-\frac {4 e^{x+e^x+25} (x-1)}{x}\right )dx+4 \int \exp \left (-\frac {4 e^{x+e^x+25} (x-1)}{x}+e^x+e^{-\frac {4 e^{x+e^x+25} (x-1)}{x}} x+x+25\right )dx+4 \int \exp \left (-\frac {4 e^{x+e^x+25} (x-1)}{x}+e^x+e^{-\frac {4 e^{x+e^x+25} (x-1)}{x}} x+2 x+25\right )dx-4 \int \frac {\exp \left (-\frac {4 e^{x+e^x+25} (x-1)}{x}+e^x+e^{-\frac {4 e^{x+e^x+25} (x-1)}{x}} x+x+25\right )}{x}dx-4 \int \exp \left (-\frac {4 e^{x+e^x+25} (x-1)}{x}+e^x+e^{-\frac {4 e^{x+e^x+25} (x-1)}{x}} x+x+25\right ) xdx-4 \int \exp \left (-\frac {4 e^{x+e^x+25} (x-1)}{x}+e^x+e^{-\frac {4 e^{x+e^x+25} (x-1)}{x}} x+2 x+25\right ) xdx\)

Input:

Int[E^(x/E^((E^(25 + E^x + x)*(-4 + 4*x))/x) - (E^(25 + E^x + x)*(-4 + 4*x 
))/x)*(1 + (E^(25 + E^x + x)*(-4 + 4*x - 4*x^2 + E^x*(4*x - 4*x^2)))/x),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 8.38 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86

method	result	size
risch	\({\mathrm e}^{x \,{\mathrm e}^{-\frac {4 \left (-1+x \right ) {\mathrm e}^{25+{\mathrm e}^{x}+x}}{x}}}\)	\(19\)
parallelrisch	\({\mathrm e}^{x \,{\mathrm e}^{-\frac {\left (-4+4 x \right ) {\mathrm e}^{25+{\mathrm e}^{x}+x}}{x}}}\)	\(25\)

Input:

int((((-4*x^2+4*x)*exp(x)-4*x^2+4*x-4)*exp(ln(exp(25)/x)+exp(x)+x)+1)*exp( 
x/exp((-4+4*x)*exp(ln(exp(25)/x)+exp(x)+x)))/exp((-4+4*x)*exp(ln(exp(25)/x 
)+exp(x)+x)),x,method=_RETURNVERBOSE)
 

Output:

exp(x*exp(-4*(-1+x)*exp(25+exp(x)+x)/x))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int e^{e^{-\frac {e^{25+e^x+x} (-4+4 x)}{x}} x-\frac {e^{25+e^x+x} (-4+4 x)}{x}} \left (1+\frac {e^{25+e^x+x} \left (-4+4 x-4 x^2+e^x \left (4 x-4 x^2\right )\right )}{x}\right ) \, dx=e^{\left (x e^{\left (-4 \, {\left (x - 1\right )} e^{\left (x + e^{x} + \log \left (\frac {e^{25}}{x}\right )\right )}\right )}\right )} \] Input:

integrate((((-4*x^2+4*x)*exp(x)-4*x^2+4*x-4)*exp(log(exp(25)/x)+exp(x)+x)+ 
1)*exp(x/exp((-4+4*x)*exp(log(exp(25)/x)+exp(x)+x)))/exp((-4+4*x)*exp(log( 
exp(25)/x)+exp(x)+x)),x, algorithm="fricas")
 

Output:

e^(x*e^(-4*(x - 1)*e^(x + e^x + log(e^25/x))))
 

Sympy [A] (verification not implemented)

Time = 20.58 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int e^{e^{-\frac {e^{25+e^x+x} (-4+4 x)}{x}} x-\frac {e^{25+e^x+x} (-4+4 x)}{x}} \left (1+\frac {e^{25+e^x+x} \left (-4+4 x-4 x^2+e^x \left (4 x-4 x^2\right )\right )}{x}\right ) \, dx=e^{x e^{- \frac {\left (4 x - 4\right ) e^{25} e^{x + e^{x}}}{x}}} \] Input:

integrate((((-4*x**2+4*x)*exp(x)-4*x**2+4*x-4)*exp(ln(exp(25)/x)+exp(x)+x) 
+1)*exp(x/exp((-4+4*x)*exp(ln(exp(25)/x)+exp(x)+x)))/exp((-4+4*x)*exp(ln(e 
xp(25)/x)+exp(x)+x)),x)
 

Output:

exp(x*exp(-(4*x - 4)*exp(25)*exp(x + exp(x))/x))
 

Maxima [A] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int e^{e^{-\frac {e^{25+e^x+x} (-4+4 x)}{x}} x-\frac {e^{25+e^x+x} (-4+4 x)}{x}} \left (1+\frac {e^{25+e^x+x} \left (-4+4 x-4 x^2+e^x \left (4 x-4 x^2\right )\right )}{x}\right ) \, dx=e^{\left (x e^{\left (\frac {4 \, e^{\left (x + e^{x} + 25\right )}}{x} - 4 \, e^{\left (x + e^{x} + 25\right )}\right )}\right )} \] Input:

integrate((((-4*x^2+4*x)*exp(x)-4*x^2+4*x-4)*exp(log(exp(25)/x)+exp(x)+x)+ 
1)*exp(x/exp((-4+4*x)*exp(log(exp(25)/x)+exp(x)+x)))/exp((-4+4*x)*exp(log( 
exp(25)/x)+exp(x)+x)),x, algorithm="maxima")
 

Output:

e^(x*e^(4*e^(x + e^x + 25)/x - 4*e^(x + e^x + 25)))
 

Giac [F]

\[ \int e^{e^{-\frac {e^{25+e^x+x} (-4+4 x)}{x}} x-\frac {e^{25+e^x+x} (-4+4 x)}{x}} \left (1+\frac {e^{25+e^x+x} \left (-4+4 x-4 x^2+e^x \left (4 x-4 x^2\right )\right )}{x}\right ) \, dx=\int { -{\left (4 \, {\left (x^{2} + {\left (x^{2} - x\right )} e^{x} - x + 1\right )} e^{\left (x + e^{x} + \log \left (\frac {e^{25}}{x}\right )\right )} - 1\right )} e^{\left (x e^{\left (-4 \, {\left (x - 1\right )} e^{\left (x + e^{x} + \log \left (\frac {e^{25}}{x}\right )\right )}\right )} - 4 \, {\left (x - 1\right )} e^{\left (x + e^{x} + \log \left (\frac {e^{25}}{x}\right )\right )}\right )} \,d x } \] Input:

integrate((((-4*x^2+4*x)*exp(x)-4*x^2+4*x-4)*exp(log(exp(25)/x)+exp(x)+x)+ 
1)*exp(x/exp((-4+4*x)*exp(log(exp(25)/x)+exp(x)+x)))/exp((-4+4*x)*exp(log( 
exp(25)/x)+exp(x)+x)),x, algorithm="giac")
 

Output:

integrate(-(4*(x^2 + (x^2 - x)*e^x - x + 1)*e^(x + e^x + log(e^25/x)) - 1) 
*e^(x*e^(-4*(x - 1)*e^(x + e^x + log(e^25/x))) - 4*(x - 1)*e^(x + e^x + lo 
g(e^25/x))), x)
 

Mupad [B] (verification not implemented)

Time = 3.89 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int e^{e^{-\frac {e^{25+e^x+x} (-4+4 x)}{x}} x-\frac {e^{25+e^x+x} (-4+4 x)}{x}} \left (1+\frac {e^{25+e^x+x} \left (-4+4 x-4 x^2+e^x \left (4 x-4 x^2\right )\right )}{x}\right ) \, dx={\mathrm {e}}^{x\,{\mathrm {e}}^{\frac {4\,{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^{25}\,{\mathrm {e}}^x}{x}}\,{\mathrm {e}}^{-4\,{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^{25}\,{\mathrm {e}}^x}} \] Input:

int(exp(-exp(x + log(exp(25)/x) + exp(x))*(4*x - 4))*exp(x*exp(-exp(x + lo 
g(exp(25)/x) + exp(x))*(4*x - 4)))*(exp(x + log(exp(25)/x) + exp(x))*(4*x 
+ exp(x)*(4*x - 4*x^2) - 4*x^2 - 4) + 1),x)
 

Output:

exp(x*exp((4*exp(exp(x))*exp(25)*exp(x))/x)*exp(-4*exp(exp(x))*exp(25)*exp 
(x)))
 

Reduce [F]

\[ \int e^{e^{-\frac {e^{25+e^x+x} (-4+4 x)}{x}} x-\frac {e^{25+e^x+x} (-4+4 x)}{x}} \left (1+\frac {e^{25+e^x+x} \left (-4+4 x-4 x^2+e^x \left (4 x-4 x^2\right )\right )}{x}\right ) \, dx=4 \left (\int \frac {e^{\frac {4 e^{4 e^{e^{x}+x} e^{25}+e^{x}+x} e^{25}+e^{\frac {4 e^{e^{x}+x} e^{25}}{x}} x^{2}+e^{4 e^{e^{x}+x} e^{25}+x} x +e^{4 e^{e^{x}+x} e^{25}} x^{2}}{e^{4 e^{e^{x}+x} e^{25}} x}}}{e^{4 e^{e^{x}+x} e^{25}}}d x \right ) e^{25}-4 \left (\int \frac {e^{\frac {4 e^{4 e^{e^{x}+x} e^{25}+e^{x}+x} e^{25}+e^{\frac {4 e^{e^{x}+x} e^{25}}{x}} x^{2}+e^{4 e^{e^{x}+x} e^{25}+x} x +e^{4 e^{e^{x}+x} e^{25}} x^{2}}{e^{4 e^{e^{x}+x} e^{25}} x}}}{e^{4 e^{e^{x}+x} e^{25}} x}d x \right ) e^{25}+4 \left (\int \frac {e^{\frac {4 e^{4 e^{e^{x}+x} e^{25}+e^{x}+x} e^{25}+e^{\frac {4 e^{e^{x}+x} e^{25}}{x}} x^{2}+e^{4 e^{e^{x}+x} e^{25}+x} x +2 e^{4 e^{e^{x}+x} e^{25}} x^{2}}{e^{4 e^{e^{x}+x} e^{25}} x}}}{e^{4 e^{e^{x}+x} e^{25}}}d x \right ) e^{25}+\int \frac {e^{\frac {4 e^{4 e^{e^{x}+x} e^{25}+e^{x}+x} e^{25}+e^{\frac {4 e^{e^{x}+x} e^{25}}{x}} x^{2}}{e^{4 e^{e^{x}+x} e^{25}} x}}}{e^{4 e^{e^{x}+x} e^{25}}}d x -4 \left (\int \frac {e^{\frac {4 e^{4 e^{e^{x}+x} e^{25}+e^{x}+x} e^{25}+e^{\frac {4 e^{e^{x}+x} e^{25}}{x}} x^{2}+e^{4 e^{e^{x}+x} e^{25}+x} x +e^{4 e^{e^{x}+x} e^{25}} x^{2}}{e^{4 e^{e^{x}+x} e^{25}} x}} x}{e^{4 e^{e^{x}+x} e^{25}}}d x \right ) e^{25}-4 \left (\int \frac {e^{\frac {4 e^{4 e^{e^{x}+x} e^{25}+e^{x}+x} e^{25}+e^{\frac {4 e^{e^{x}+x} e^{25}}{x}} x^{2}+e^{4 e^{e^{x}+x} e^{25}+x} x +2 e^{4 e^{e^{x}+x} e^{25}} x^{2}}{e^{4 e^{e^{x}+x} e^{25}} x}} x}{e^{4 e^{e^{x}+x} e^{25}}}d x \right ) e^{25} \] Input:

int((((-4*x^2+4*x)*exp(x)-4*x^2+4*x-4)*exp(log(exp(25)/x)+exp(x)+x)+1)*exp 
(x/exp((-4+4*x)*exp(log(exp(25)/x)+exp(x)+x)))/exp((-4+4*x)*exp(log(exp(25 
)/x)+exp(x)+x)),x)
 

Output:

4*int(e**((4*e**(4*e**(e**x + x)*e**25 + e**x + x)*e**25 + e**((4*e**(e**x 
 + x)*e**25)/x)*x**2 + e**(4*e**(e**x + x)*e**25 + x)*x + e**(4*e**(e**x + 
 x)*e**25)*x**2)/(e**(4*e**(e**x + x)*e**25)*x))/e**(4*e**(e**x + x)*e**25 
),x)*e**25 - 4*int(e**((4*e**(4*e**(e**x + x)*e**25 + e**x + x)*e**25 + e* 
*((4*e**(e**x + x)*e**25)/x)*x**2 + e**(4*e**(e**x + x)*e**25 + x)*x + e** 
(4*e**(e**x + x)*e**25)*x**2)/(e**(4*e**(e**x + x)*e**25)*x))/(e**(4*e**(e 
**x + x)*e**25)*x),x)*e**25 + 4*int(e**((4*e**(4*e**(e**x + x)*e**25 + e** 
x + x)*e**25 + e**((4*e**(e**x + x)*e**25)/x)*x**2 + e**(4*e**(e**x + x)*e 
**25 + x)*x + 2*e**(4*e**(e**x + x)*e**25)*x**2)/(e**(4*e**(e**x + x)*e**2 
5)*x))/e**(4*e**(e**x + x)*e**25),x)*e**25 + int(e**((4*e**(4*e**(e**x + x 
)*e**25 + e**x + x)*e**25 + e**((4*e**(e**x + x)*e**25)/x)*x**2)/(e**(4*e* 
*(e**x + x)*e**25)*x))/e**(4*e**(e**x + x)*e**25),x) - 4*int((e**((4*e**(4 
*e**(e**x + x)*e**25 + e**x + x)*e**25 + e**((4*e**(e**x + x)*e**25)/x)*x* 
*2 + e**(4*e**(e**x + x)*e**25 + x)*x + e**(4*e**(e**x + x)*e**25)*x**2)/( 
e**(4*e**(e**x + x)*e**25)*x))*x)/e**(4*e**(e**x + x)*e**25),x)*e**25 - 4* 
int((e**((4*e**(4*e**(e**x + x)*e**25 + e**x + x)*e**25 + e**((4*e**(e**x 
+ x)*e**25)/x)*x**2 + e**(4*e**(e**x + x)*e**25 + x)*x + 2*e**(4*e**(e**x 
+ x)*e**25)*x**2)/(e**(4*e**(e**x + x)*e**25)*x))*x)/e**(4*e**(e**x + x)*e 
**25),x)*e**25