\(\int \frac {e^{e^{-2+x}} (e^8 (25-10 x+x^2)+e^4 (5 x^2-x^3))+e^{e^{-2+x}} (e^8 (-25+10 x-x^2)+e^4 (-15 x^2+2 x^3)+e^{-2+x} (e^8 (25 x-10 x^2+x^3)+e^4 (5 x^3-x^4))) \log (x) \log (\log (x))}{(x^6+e^8 (25 x^2-10 x^3+x^4)+e^4 (10 x^4-2 x^5)) \log (x)} \, dx\) [1839]

Optimal result

Integrand size = 162, antiderivative size = 32 \[ \int \frac {e^{e^{-2+x}} \left (e^8 \left (25-10 x+x^2\right )+e^4 \left (5 x^2-x^3\right )\right )+e^{e^{-2+x}} \left (e^8 \left (-25+10 x-x^2\right )+e^4 \left (-15 x^2+2 x^3\right )+e^{-2+x} \left (e^8 \left (25 x-10 x^2+x^3\right )+e^4 \left (5 x^3-x^4\right )\right )\right ) \log (x) \log (\log (x))}{\left (x^6+e^8 \left (25 x^2-10 x^3+x^4\right )+e^4 \left (10 x^4-2 x^5\right )\right ) \log (x)} \, dx=\frac {e^{e^{-2+x}} \log (\log (x))}{x \left (1+\frac {x^2}{e^4 (5-x)}\right )} \] Output:

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

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \frac {e^{e^{-2+x}} \left (e^8 \left (25-10 x+x^2\right )+e^4 \left (5 x^2-x^3\right )\right )+e^{e^{-2+x}} \left (e^8 \left (-25+10 x-x^2\right )+e^4 \left (-15 x^2+2 x^3\right )+e^{-2+x} \left (e^8 \left (25 x-10 x^2+x^3\right )+e^4 \left (5 x^3-x^4\right )\right )\right ) \log (x) \log (\log (x))}{\left (x^6+e^8 \left (25 x^2-10 x^3+x^4\right )+e^4 \left (10 x^4-2 x^5\right )\right ) \log (x)} \, dx=\frac {e^{4+e^{-2+x}} (-5+x) \log (\log (x))}{e^4 (-5+x) x-x^3} \] Input:

Integrate[(E^E^(-2 + x)*(E^8*(25 - 10*x + x^2) + E^4*(5*x^2 - x^3)) + E^E^ 
(-2 + x)*(E^8*(-25 + 10*x - x^2) + E^4*(-15*x^2 + 2*x^3) + E^(-2 + x)*(E^8 
*(25*x - 10*x^2 + x^3) + E^4*(5*x^3 - x^4)))*Log[x]*Log[Log[x]])/((x^6 + E 
^8*(25*x^2 - 10*x^3 + x^4) + E^4*(10*x^4 - 2*x^5))*Log[x]),x]
 

Output:

(E^(4 + E^(-2 + x))*(-5 + x)*Log[Log[x]])/(E^4*(-5 + x)*x - x^3)
 

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.

\(\displaystyle \int \frac {e^{e^{x-2}} \left (e^8 \left (x^2-10 x+25\right )+e^4 \left (5 x^2-x^3\right )\right )+e^{e^{x-2}} \left (e^8 \left (-x^2+10 x-25\right )+e^4 \left (2 x^3-15 x^2\right )+e^{x-2} \left (e^4 \left (5 x^3-x^4\right )+e^8 \left (x^3-10 x^2+25 x\right )\right )\right ) \log (x) \log (\log (x))}{\left (x^6+e^4 \left (10 x^4-2 x^5\right )+e^8 \left (x^4-10 x^3+25 x^2\right )\right ) \log (x)} \, dx\)

\(\Big \downarrow \) 2026

\(\displaystyle \int \frac {e^{e^{x-2}} \left (e^8 \left (x^2-10 x+25\right )+e^4 \left (5 x^2-x^3\right )\right )+e^{e^{x-2}} \left (e^8 \left (-x^2+10 x-25\right )+e^4 \left (2 x^3-15 x^2\right )+e^{x-2} \left (e^4 \left (5 x^3-x^4\right )+e^8 \left (x^3-10 x^2+25 x\right )\right )\right ) \log (x) \log (\log (x))}{x^2 \left (x^4-2 e^4 x^3+e^4 \left (10+e^4\right ) x^2-10 e^8 x+25 e^8\right ) \log (x)}dx\)

\(\Big \downarrow \) 2463

\(\displaystyle \int \left (\frac {4 \left (e^{e^{x-2}} \left (e^8 \left (x^2-10 x+25\right )+e^4 \left (5 x^2-x^3\right )\right )+e^{e^{x-2}} \left (e^8 \left (-x^2+10 x-25\right )+e^4 \left (2 x^3-15 x^2\right )+e^{x-2} \left (e^4 \left (5 x^3-x^4\right )+e^8 \left (x^3-10 x^2+25 x\right )\right )\right ) \log (x) \log (\log (x))\right )}{e^6 \left (e^4-20\right )^{3/2} x^2 \left (2 x+e^2 \sqrt {e^4-20}-e^4\right ) \log (x)}+\frac {4 \left (e^{e^{x-2}} \left (e^8 \left (x^2-10 x+25\right )+e^4 \left (5 x^2-x^3\right )\right )+e^{e^{x-2}} \left (e^8 \left (-x^2+10 x-25\right )+e^4 \left (2 x^3-15 x^2\right )+e^{x-2} \left (e^4 \left (5 x^3-x^4\right )+e^8 \left (x^3-10 x^2+25 x\right )\right )\right ) \log (x) \log (\log (x))\right )}{e^6 \left (e^4-20\right )^{3/2} \left (-2 x+e^2 \sqrt {e^4-20}+e^4\right ) x^2 \log (x)}+\frac {4 \left (e^{e^{x-2}} \left (e^8 \left (x^2-10 x+25\right )+e^4 \left (5 x^2-x^3\right )\right )+e^{e^{x-2}} \left (e^8 \left (-x^2+10 x-25\right )+e^4 \left (2 x^3-15 x^2\right )+e^{x-2} \left (e^4 \left (5 x^3-x^4\right )+e^8 \left (x^3-10 x^2+25 x\right )\right )\right ) \log (x) \log (\log (x))\right )}{e^4 \left (e^4-20\right ) \left (-2 x+e^2 \sqrt {e^4-20}+e^4\right )^2 x^2 \log (x)}+\frac {4 \left (e^{e^{x-2}} \left (e^8 \left (x^2-10 x+25\right )+e^4 \left (5 x^2-x^3\right )\right )+e^{e^{x-2}} \left (e^8 \left (-x^2+10 x-25\right )+e^4 \left (2 x^3-15 x^2\right )+e^{x-2} \left (e^4 \left (5 x^3-x^4\right )+e^8 \left (x^3-10 x^2+25 x\right )\right )\right ) \log (x) \log (\log (x))\right )}{e^4 \left (e^4-20\right ) x^2 \left (2 x+e^2 \sqrt {e^4-20}-e^4\right )^2 \log (x)}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {e^{e^{x-2}+2} \left (-e^2 (x-5) \left (x^2-e^4 (x-5)\right )-\left (e^x (x-5) x^3+e^2 (15-2 x) x^2-e^{x+4} (x-5)^2 x+e^6 (x-5)^2\right ) \log (x) \log (\log (x))\right )}{x^2 \left (x^2-e^4 x+5 e^4\right )^2 \log (x)}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^{e^{x-2}+4} \left (-x^3+2 x^3 \log (x) \log (\log (x))+5 \left (1+\frac {e^4}{5}\right ) x^2-15 \left (1+\frac {e^4}{15}\right ) x^2 \log (x) \log (\log (x))-10 e^4 x+10 e^4 x \log (x) \log (\log (x))-25 e^4 \log (x) \log (\log (x))+25 e^4\right )}{x^2 \left (x^2-e^4 x+5 e^4\right )^2 \log (x)}-\frac {e^{x+e^{x-2}+2} (x-5) \log (\log (x))}{x \left (x^2-e^4 x+5 e^4\right )}\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^{x+e^{x-2}+2} (x-5) \log (\log (x))}{x \left (-x^2+e^4 x-5 e^4\right )}+\frac {e^{e^{x-2}+4} \left (-x^3+2 x^3 \log (x) \log (\log (x))+5 \left (1+\frac {e^4}{5}\right ) x^2-15 \left (1+\frac {e^4}{15}\right ) x^2 \log (x) \log (\log (x))-10 e^4 x+10 e^4 x \log (x) \log (\log (x))-25 e^4 \log (x) \log (\log (x))+25 e^4\right )}{x^2 \left (x^2-e^4 x+5 e^4\right )^2 \log (x)}\right )dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \left (\frac {e^{x+e^{x-2}+2} (x-5) \log (\log (x))}{x \left (-x^2+e^4 x-5 e^4\right )}+\frac {e^{e^{x-2}+4} \left (-x^3+2 x^3 \log (x) \log (\log (x))+5 \left (1+\frac {e^4}{5}\right ) x^2-15 \left (1+\frac {e^4}{15}\right ) x^2 \log (x) \log (\log (x))-10 e^4 x+10 e^4 x \log (x) \log (\log (x))-25 e^4 \log (x) \log (\log (x))+25 e^4\right )}{x^2 \left (x^2-e^4 x+5 e^4\right )^2 \log (x)}\right )dx\)

Input:

Int[(E^E^(-2 + x)*(E^8*(25 - 10*x + x^2) + E^4*(5*x^2 - x^3)) + E^E^(-2 + 
x)*(E^8*(-25 + 10*x - x^2) + E^4*(-15*x^2 + 2*x^3) + E^(-2 + x)*(E^8*(25*x 
 - 10*x^2 + x^3) + E^4*(5*x^3 - x^4)))*Log[x]*Log[Log[x]])/((x^6 + E^8*(25 
*x^2 - 10*x^3 + x^4) + E^4*(10*x^4 - 2*x^5))*Log[x]),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06

\[\frac {\ln \left (\ln \left (x \right )\right ) \left (-5+x \right ) {\mathrm e}^{{\mathrm e}^{-2+x}+4}}{\left (x \,{\mathrm e}^{4}-x^{2}-5 \,{\mathrm e}^{4}\right ) x}\]

Input:

int(((((x^3-10*x^2+25*x)*exp(4)^2+(-x^4+5*x^3)*exp(4))*exp(-2+x)+(-x^2+10* 
x-25)*exp(4)^2+(2*x^3-15*x^2)*exp(4))*ln(x)*exp(exp(-2+x))*ln(ln(x))+((x^2 
-10*x+25)*exp(4)^2+(-x^3+5*x^2)*exp(4))*exp(exp(-2+x)))/((x^4-10*x^3+25*x^ 
2)*exp(4)^2+(-2*x^5+10*x^4)*exp(4)+x^6)/ln(x),x)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \frac {e^{e^{-2+x}} \left (e^8 \left (25-10 x+x^2\right )+e^4 \left (5 x^2-x^3\right )\right )+e^{e^{-2+x}} \left (e^8 \left (-25+10 x-x^2\right )+e^4 \left (-15 x^2+2 x^3\right )+e^{-2+x} \left (e^8 \left (25 x-10 x^2+x^3\right )+e^4 \left (5 x^3-x^4\right )\right )\right ) \log (x) \log (\log (x))}{\left (x^6+e^8 \left (25 x^2-10 x^3+x^4\right )+e^4 \left (10 x^4-2 x^5\right )\right ) \log (x)} \, dx=-\frac {{\left (x - 5\right )} e^{\left (e^{\left (x - 2\right )} + 4\right )} \log \left (\log \left (x\right )\right )}{x^{3} - {\left (x^{2} - 5 \, x\right )} e^{4}} \] Input:

integrate(((((x^3-10*x^2+25*x)*exp(4)^2+(-x^4+5*x^3)*exp(4))*exp(-2+x)+(-x 
^2+10*x-25)*exp(4)^2+(2*x^3-15*x^2)*exp(4))*log(x)*exp(exp(-2+x))*log(log( 
x))+((x^2-10*x+25)*exp(4)^2+(-x^3+5*x^2)*exp(4))*exp(exp(-2+x)))/((x^4-10* 
x^3+25*x^2)*exp(4)^2+(-2*x^5+10*x^4)*exp(4)+x^6)/log(x),x, algorithm="fric 
as")
 

Output:

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

Sympy [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.31 \[ \int \frac {e^{e^{-2+x}} \left (e^8 \left (25-10 x+x^2\right )+e^4 \left (5 x^2-x^3\right )\right )+e^{e^{-2+x}} \left (e^8 \left (-25+10 x-x^2\right )+e^4 \left (-15 x^2+2 x^3\right )+e^{-2+x} \left (e^8 \left (25 x-10 x^2+x^3\right )+e^4 \left (5 x^3-x^4\right )\right )\right ) \log (x) \log (\log (x))}{\left (x^6+e^8 \left (25 x^2-10 x^3+x^4\right )+e^4 \left (10 x^4-2 x^5\right )\right ) \log (x)} \, dx=\frac {\left (- x e^{4} \log {\left (\log {\left (x \right )} \right )} + 5 e^{4} \log {\left (\log {\left (x \right )} \right )}\right ) e^{e^{x - 2}}}{x^{3} - x^{2} e^{4} + 5 x e^{4}} \] Input:

integrate(((((x**3-10*x**2+25*x)*exp(4)**2+(-x**4+5*x**3)*exp(4))*exp(-2+x 
)+(-x**2+10*x-25)*exp(4)**2+(2*x**3-15*x**2)*exp(4))*ln(x)*exp(exp(-2+x))* 
ln(ln(x))+((x**2-10*x+25)*exp(4)**2+(-x**3+5*x**2)*exp(4))*exp(exp(-2+x))) 
/((x**4-10*x**3+25*x**2)*exp(4)**2+(-2*x**5+10*x**4)*exp(4)+x**6)/ln(x),x)
 

Output:

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

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16 \[ \int \frac {e^{e^{-2+x}} \left (e^8 \left (25-10 x+x^2\right )+e^4 \left (5 x^2-x^3\right )\right )+e^{e^{-2+x}} \left (e^8 \left (-25+10 x-x^2\right )+e^4 \left (-15 x^2+2 x^3\right )+e^{-2+x} \left (e^8 \left (25 x-10 x^2+x^3\right )+e^4 \left (5 x^3-x^4\right )\right )\right ) \log (x) \log (\log (x))}{\left (x^6+e^8 \left (25 x^2-10 x^3+x^4\right )+e^4 \left (10 x^4-2 x^5\right )\right ) \log (x)} \, dx=-\frac {{\left (x e^{4} - 5 \, e^{4}\right )} e^{\left (e^{\left (x - 2\right )}\right )} \log \left (\log \left (x\right )\right )}{x^{3} - x^{2} e^{4} + 5 \, x e^{4}} \] Input:

integrate(((((x^3-10*x^2+25*x)*exp(4)^2+(-x^4+5*x^3)*exp(4))*exp(-2+x)+(-x 
^2+10*x-25)*exp(4)^2+(2*x^3-15*x^2)*exp(4))*log(x)*exp(exp(-2+x))*log(log( 
x))+((x^2-10*x+25)*exp(4)^2+(-x^3+5*x^2)*exp(4))*exp(exp(-2+x)))/((x^4-10* 
x^3+25*x^2)*exp(4)^2+(-2*x^5+10*x^4)*exp(4)+x^6)/log(x),x, algorithm="maxi 
ma")
 

Output:

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

Giac [F]

\[ \int \frac {e^{e^{-2+x}} \left (e^8 \left (25-10 x+x^2\right )+e^4 \left (5 x^2-x^3\right )\right )+e^{e^{-2+x}} \left (e^8 \left (-25+10 x-x^2\right )+e^4 \left (-15 x^2+2 x^3\right )+e^{-2+x} \left (e^8 \left (25 x-10 x^2+x^3\right )+e^4 \left (5 x^3-x^4\right )\right )\right ) \log (x) \log (\log (x))}{\left (x^6+e^8 \left (25 x^2-10 x^3+x^4\right )+e^4 \left (10 x^4-2 x^5\right )\right ) \log (x)} \, dx=\int { -\frac {{\left ({\left (x^{2} - 10 \, x + 25\right )} e^{8} - {\left (2 \, x^{3} - 15 \, x^{2}\right )} e^{4} - {\left ({\left (x^{3} - 10 \, x^{2} + 25 \, x\right )} e^{8} - {\left (x^{4} - 5 \, x^{3}\right )} e^{4}\right )} e^{\left (x - 2\right )}\right )} e^{\left (e^{\left (x - 2\right )}\right )} \log \left (x\right ) \log \left (\log \left (x\right )\right ) - {\left ({\left (x^{2} - 10 \, x + 25\right )} e^{8} - {\left (x^{3} - 5 \, x^{2}\right )} e^{4}\right )} e^{\left (e^{\left (x - 2\right )}\right )}}{{\left (x^{6} + {\left (x^{4} - 10 \, x^{3} + 25 \, x^{2}\right )} e^{8} - 2 \, {\left (x^{5} - 5 \, x^{4}\right )} e^{4}\right )} \log \left (x\right )} \,d x } \] Input:

integrate(((((x^3-10*x^2+25*x)*exp(4)^2+(-x^4+5*x^3)*exp(4))*exp(-2+x)+(-x 
^2+10*x-25)*exp(4)^2+(2*x^3-15*x^2)*exp(4))*log(x)*exp(exp(-2+x))*log(log( 
x))+((x^2-10*x+25)*exp(4)^2+(-x^3+5*x^2)*exp(4))*exp(exp(-2+x)))/((x^4-10* 
x^3+25*x^2)*exp(4)^2+(-2*x^5+10*x^4)*exp(4)+x^6)/log(x),x, algorithm="giac 
")
 

Output:

integrate(-(((x^2 - 10*x + 25)*e^8 - (2*x^3 - 15*x^2)*e^4 - ((x^3 - 10*x^2 
 + 25*x)*e^8 - (x^4 - 5*x^3)*e^4)*e^(x - 2))*e^(e^(x - 2))*log(x)*log(log( 
x)) - ((x^2 - 10*x + 25)*e^8 - (x^3 - 5*x^2)*e^4)*e^(e^(x - 2)))/((x^6 + ( 
x^4 - 10*x^3 + 25*x^2)*e^8 - 2*(x^5 - 5*x^4)*e^4)*log(x)), x)
 

Mupad [B] (verification not implemented)

Time = 2.46 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {e^{e^{-2+x}} \left (e^8 \left (25-10 x+x^2\right )+e^4 \left (5 x^2-x^3\right )\right )+e^{e^{-2+x}} \left (e^8 \left (-25+10 x-x^2\right )+e^4 \left (-15 x^2+2 x^3\right )+e^{-2+x} \left (e^8 \left (25 x-10 x^2+x^3\right )+e^4 \left (5 x^3-x^4\right )\right )\right ) \log (x) \log (\log (x))}{\left (x^6+e^8 \left (25 x^2-10 x^3+x^4\right )+e^4 \left (10 x^4-2 x^5\right )\right ) \log (x)} \, dx=-\frac {\ln \left (\ln \left (x\right )\right )\,{\mathrm {e}}^{{\mathrm {e}}^{-2}\,{\mathrm {e}}^x+4}\,\left (x-5\right )}{x\,\left (x^2-{\mathrm {e}}^4\,x+5\,{\mathrm {e}}^4\right )} \] Input:

int((exp(exp(x - 2))*(exp(4)*(5*x^2 - x^3) + exp(8)*(x^2 - 10*x + 25)) - l 
og(log(x))*exp(exp(x - 2))*log(x)*(exp(4)*(15*x^2 - 2*x^3) - exp(x - 2)*(e 
xp(8)*(25*x - 10*x^2 + x^3) + exp(4)*(5*x^3 - x^4)) + exp(8)*(x^2 - 10*x + 
 25)))/(log(x)*(exp(8)*(25*x^2 - 10*x^3 + x^4) + exp(4)*(10*x^4 - 2*x^5) + 
 x^6)),x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.25 \[ \int \frac {e^{e^{-2+x}} \left (e^8 \left (25-10 x+x^2\right )+e^4 \left (5 x^2-x^3\right )\right )+e^{e^{-2+x}} \left (e^8 \left (-25+10 x-x^2\right )+e^4 \left (-15 x^2+2 x^3\right )+e^{-2+x} \left (e^8 \left (25 x-10 x^2+x^3\right )+e^4 \left (5 x^3-x^4\right )\right )\right ) \log (x) \log (\log (x))}{\left (x^6+e^8 \left (25 x^2-10 x^3+x^4\right )+e^4 \left (10 x^4-2 x^5\right )\right ) \log (x)} \, dx=\frac {e^{\frac {e^{x}}{e^{2}}} \mathrm {log}\left (\mathrm {log}\left (x \right )\right ) e^{4} \left (-5+x \right )}{x \left (e^{4} x -5 e^{4}-x^{2}\right )} \] Input:

int(((((x^3-10*x^2+25*x)*exp(4)^2+(-x^4+5*x^3)*exp(4))*exp(-2+x)+(-x^2+10* 
x-25)*exp(4)^2+(2*x^3-15*x^2)*exp(4))*log(x)*exp(exp(-2+x))*log(log(x))+(( 
x^2-10*x+25)*exp(4)^2+(-x^3+5*x^2)*exp(4))*exp(exp(-2+x)))/((x^4-10*x^3+25 
*x^2)*exp(4)^2+(-2*x^5+10*x^4)*exp(4)+x^6)/log(x),x)
 

Output:

(e**(e**x/e**2)*log(log(x))*e**4*(x - 5))/(x*(e**4*x - 5*e**4 - x**2))