\(\int \frac {e^{2 e^2} (-1+x)-25 x^2+25 x^3+e^x (5 x-5 x^2)+e^{e^2} (-5 x-e^x x+10 x^2)+(-e^{2 e^2} x+5 e^x x-25 x^2-25 x^3+e^{e^2} (e^x-5 x-10 x^2)+(e^{2 e^2}+10 e^{e^2} x+25 x^2) \log (x)) \log (\frac {-e^x+5 x+e^{e^2} x+5 x^2+(-e^{e^2}-5 x) \log (x)}{e^{e^2}+5 x})}{(-e^{2 e^2} x+5 e^x x-25 x^2-25 x^3+e^{e^2} (e^x-5 x-10 x^2)+(e^{2 e^2}+10 e^{e^2} x+25 x^2) \log (x)) \log ^2(\frac {-e^x+5 x+e^{e^2} x+5 x^2+(-e^{e^2}-5 x) \log (x)}{e^{e^2}+5 x})} \, dx\) [388]

Optimal result

Integrand size = 296, antiderivative size = 32 \[ \int \frac {e^{2 e^2} (-1+x)-25 x^2+25 x^3+e^x \left (5 x-5 x^2\right )+e^{e^2} \left (-5 x-e^x x+10 x^2\right )+\left (-e^{2 e^2} x+5 e^x x-25 x^2-25 x^3+e^{e^2} \left (e^x-5 x-10 x^2\right )+\left (e^{2 e^2}+10 e^{e^2} x+25 x^2\right ) \log (x)\right ) \log \left (\frac {-e^x+5 x+e^{e^2} x+5 x^2+\left (-e^{e^2}-5 x\right ) \log (x)}{e^{e^2}+5 x}\right )}{\left (-e^{2 e^2} x+5 e^x x-25 x^2-25 x^3+e^{e^2} \left (e^x-5 x-10 x^2\right )+\left (e^{2 e^2}+10 e^{e^2} x+25 x^2\right ) \log (x)\right ) \log ^2\left (\frac {-e^x+5 x+e^{e^2} x+5 x^2+\left (-e^{e^2}-5 x\right ) \log (x)}{e^{e^2}+5 x}\right )} \, dx=\frac {x}{\log \left (\frac {e^x-5 x}{-e^{e^2}-5 x}+x-\log (x)\right )} \] Output:

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

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.53 \[ \int \frac {e^{2 e^2} (-1+x)-25 x^2+25 x^3+e^x \left (5 x-5 x^2\right )+e^{e^2} \left (-5 x-e^x x+10 x^2\right )+\left (-e^{2 e^2} x+5 e^x x-25 x^2-25 x^3+e^{e^2} \left (e^x-5 x-10 x^2\right )+\left (e^{2 e^2}+10 e^{e^2} x+25 x^2\right ) \log (x)\right ) \log \left (\frac {-e^x+5 x+e^{e^2} x+5 x^2+\left (-e^{e^2}-5 x\right ) \log (x)}{e^{e^2}+5 x}\right )}{\left (-e^{2 e^2} x+5 e^x x-25 x^2-25 x^3+e^{e^2} \left (e^x-5 x-10 x^2\right )+\left (e^{2 e^2}+10 e^{e^2} x+25 x^2\right ) \log (x)\right ) \log ^2\left (\frac {-e^x+5 x+e^{e^2} x+5 x^2+\left (-e^{e^2}-5 x\right ) \log (x)}{e^{e^2}+5 x}\right )} \, dx=\frac {x}{\log \left (\frac {-e^x+e^{e^2} x+5 x (1+x)-\left (e^{e^2}+5 x\right ) \log (x)}{e^{e^2}+5 x}\right )} \] Input:

Integrate[(E^(2*E^2)*(-1 + x) - 25*x^2 + 25*x^3 + E^x*(5*x - 5*x^2) + E^E^ 
2*(-5*x - E^x*x + 10*x^2) + (-(E^(2*E^2)*x) + 5*E^x*x - 25*x^2 - 25*x^3 + 
E^E^2*(E^x - 5*x - 10*x^2) + (E^(2*E^2) + 10*E^E^2*x + 25*x^2)*Log[x])*Log 
[(-E^x + 5*x + E^E^2*x + 5*x^2 + (-E^E^2 - 5*x)*Log[x])/(E^E^2 + 5*x)])/(( 
-(E^(2*E^2)*x) + 5*E^x*x - 25*x^2 - 25*x^3 + E^E^2*(E^x - 5*x - 10*x^2) + 
(E^(2*E^2) + 10*E^E^2*x + 25*x^2)*Log[x])*Log[(-E^x + 5*x + E^E^2*x + 5*x^ 
2 + (-E^E^2 - 5*x)*Log[x])/(E^E^2 + 5*x)]^2),x]
 

Output:

x/Log[(-E^x + E^E^2*x + 5*x*(1 + x) - (E^E^2 + 5*x)*Log[x])/(E^E^2 + 5*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.

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

Int[(E^(2*E^2)*(-1 + x) - 25*x^2 + 25*x^3 + E^x*(5*x - 5*x^2) + E^E^2*(-5* 
x - E^x*x + 10*x^2) + (-(E^(2*E^2)*x) + 5*E^x*x - 25*x^2 - 25*x^3 + E^E^2* 
(E^x - 5*x - 10*x^2) + (E^(2*E^2) + 10*E^E^2*x + 25*x^2)*Log[x])*Log[(-E^x 
 + 5*x + E^E^2*x + 5*x^2 + (-E^E^2 - 5*x)*Log[x])/(E^E^2 + 5*x)])/((-(E^(2 
*E^2)*x) + 5*E^x*x - 25*x^2 - 25*x^3 + E^E^2*(E^x - 5*x - 10*x^2) + (E^(2* 
E^2) + 10*E^E^2*x + 25*x^2)*Log[x])*Log[(-E^x + 5*x + E^E^2*x + 5*x^2 + (- 
E^E^2 - 5*x)*Log[x])/(E^E^2 + 5*x)]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 60.46 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.44

method	result	size
parallelrisch	\(\frac {x}{\ln \left (\frac {\left (-{\mathrm e}^{{\mathrm e}^{2}}-5 x \right ) \ln \left (x \right )+x \,{\mathrm e}^{{\mathrm e}^{2}}-{\mathrm e}^{x}+5 x^{2}+5 x}{{\mathrm e}^{{\mathrm e}^{2}}+5 x}\right )}\)	\(46\)
risch	\(\frac {2 i x}{\pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{{\mathrm e}^{2}}+5 x}\right ) \operatorname {csgn}\left (i \left (-\left (x -\ln \left (x \right )\right ) {\mathrm e}^{{\mathrm e}^{2}}-5 x^{2}-\left (-5 \ln \left (x \right )+5\right ) x +{\mathrm e}^{x}\right )\right ) \operatorname {csgn}\left (\frac {i \left (-\left (x -\ln \left (x \right )\right ) {\mathrm e}^{{\mathrm e}^{2}}-5 x^{2}-\left (-5 \ln \left (x \right )+5\right ) x +{\mathrm e}^{x}\right )}{{\mathrm e}^{{\mathrm e}^{2}}+5 x}\right )-\pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{{\mathrm e}^{2}}+5 x}\right ) {\operatorname {csgn}\left (\frac {i \left (-\left (x -\ln \left (x \right )\right ) {\mathrm e}^{{\mathrm e}^{2}}-5 x^{2}-\left (-5 \ln \left (x \right )+5\right ) x +{\mathrm e}^{x}\right )}{{\mathrm e}^{{\mathrm e}^{2}}+5 x}\right )}^{2}+\pi \,\operatorname {csgn}\left (i \left (-\left (x -\ln \left (x \right )\right ) {\mathrm e}^{{\mathrm e}^{2}}-5 x^{2}-\left (-5 \ln \left (x \right )+5\right ) x +{\mathrm e}^{x}\right )\right ) {\operatorname {csgn}\left (\frac {i \left (-\left (x -\ln \left (x \right )\right ) {\mathrm e}^{{\mathrm e}^{2}}-5 x^{2}-\left (-5 \ln \left (x \right )+5\right ) x +{\mathrm e}^{x}\right )}{{\mathrm e}^{{\mathrm e}^{2}}+5 x}\right )}^{2}-\pi {\operatorname {csgn}\left (\frac {i \left (-\left (x -\ln \left (x \right )\right ) {\mathrm e}^{{\mathrm e}^{2}}-5 x^{2}-\left (-5 \ln \left (x \right )+5\right ) x +{\mathrm e}^{x}\right )}{{\mathrm e}^{{\mathrm e}^{2}}+5 x}\right )}^{3}-2 i \ln \left ({\mathrm e}^{{\mathrm e}^{2}}+5 x \right )+2 i \ln \left (\left (x -\ln \left (x \right )\right ) {\mathrm e}^{{\mathrm e}^{2}}+5 x^{2}+\left (-5 \ln \left (x \right )+5\right ) x -{\mathrm e}^{x}\right )}\)	\(321\)

Input:

int((((exp(exp(2))^2+10*x*exp(exp(2))+25*x^2)*ln(x)-x*exp(exp(2))^2+(exp(x 
)-10*x^2-5*x)*exp(exp(2))+5*exp(x)*x-25*x^3-25*x^2)*ln(((-exp(exp(2))-5*x) 
*ln(x)+x*exp(exp(2))-exp(x)+5*x^2+5*x)/(exp(exp(2))+5*x))+(-1+x)*exp(exp(2 
))^2+(-exp(x)*x+10*x^2-5*x)*exp(exp(2))+(-5*x^2+5*x)*exp(x)+25*x^3-25*x^2) 
/((exp(exp(2))^2+10*x*exp(exp(2))+25*x^2)*ln(x)-x*exp(exp(2))^2+(exp(x)-10 
*x^2-5*x)*exp(exp(2))+5*exp(x)*x-25*x^3-25*x^2)/ln(((-exp(exp(2))-5*x)*ln( 
x)+x*exp(exp(2))-exp(x)+5*x^2+5*x)/(exp(exp(2))+5*x))^2,x,method=_RETURNVE 
RBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.38 \[ \int \frac {e^{2 e^2} (-1+x)-25 x^2+25 x^3+e^x \left (5 x-5 x^2\right )+e^{e^2} \left (-5 x-e^x x+10 x^2\right )+\left (-e^{2 e^2} x+5 e^x x-25 x^2-25 x^3+e^{e^2} \left (e^x-5 x-10 x^2\right )+\left (e^{2 e^2}+10 e^{e^2} x+25 x^2\right ) \log (x)\right ) \log \left (\frac {-e^x+5 x+e^{e^2} x+5 x^2+\left (-e^{e^2}-5 x\right ) \log (x)}{e^{e^2}+5 x}\right )}{\left (-e^{2 e^2} x+5 e^x x-25 x^2-25 x^3+e^{e^2} \left (e^x-5 x-10 x^2\right )+\left (e^{2 e^2}+10 e^{e^2} x+25 x^2\right ) \log (x)\right ) \log ^2\left (\frac {-e^x+5 x+e^{e^2} x+5 x^2+\left (-e^{e^2}-5 x\right ) \log (x)}{e^{e^2}+5 x}\right )} \, dx=\frac {x}{\log \left (\frac {5 \, x^{2} + x e^{\left (e^{2}\right )} - {\left (5 \, x + e^{\left (e^{2}\right )}\right )} \log \left (x\right ) + 5 \, x - e^{x}}{5 \, x + e^{\left (e^{2}\right )}}\right )} \] Input:

integrate((((exp(exp(2))^2+10*x*exp(exp(2))+25*x^2)*log(x)-x*exp(exp(2))^2 
+(exp(x)-10*x^2-5*x)*exp(exp(2))+5*exp(x)*x-25*x^3-25*x^2)*log(((-exp(exp( 
2))-5*x)*log(x)+x*exp(exp(2))-exp(x)+5*x^2+5*x)/(exp(exp(2))+5*x))+(-1+x)* 
exp(exp(2))^2+(-exp(x)*x+10*x^2-5*x)*exp(exp(2))+(-5*x^2+5*x)*exp(x)+25*x^ 
3-25*x^2)/((exp(exp(2))^2+10*x*exp(exp(2))+25*x^2)*log(x)-x*exp(exp(2))^2+ 
(exp(x)-10*x^2-5*x)*exp(exp(2))+5*exp(x)*x-25*x^3-25*x^2)/log(((-exp(exp(2 
))-5*x)*log(x)+x*exp(exp(2))-exp(x)+5*x^2+5*x)/(exp(exp(2))+5*x))^2,x, alg 
orithm="fricas")
 

Output:

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

Sympy [A] (verification not implemented)

Time = 5.36 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.31 \[ \int \frac {e^{2 e^2} (-1+x)-25 x^2+25 x^3+e^x \left (5 x-5 x^2\right )+e^{e^2} \left (-5 x-e^x x+10 x^2\right )+\left (-e^{2 e^2} x+5 e^x x-25 x^2-25 x^3+e^{e^2} \left (e^x-5 x-10 x^2\right )+\left (e^{2 e^2}+10 e^{e^2} x+25 x^2\right ) \log (x)\right ) \log \left (\frac {-e^x+5 x+e^{e^2} x+5 x^2+\left (-e^{e^2}-5 x\right ) \log (x)}{e^{e^2}+5 x}\right )}{\left (-e^{2 e^2} x+5 e^x x-25 x^2-25 x^3+e^{e^2} \left (e^x-5 x-10 x^2\right )+\left (e^{2 e^2}+10 e^{e^2} x+25 x^2\right ) \log (x)\right ) \log ^2\left (\frac {-e^x+5 x+e^{e^2} x+5 x^2+\left (-e^{e^2}-5 x\right ) \log (x)}{e^{e^2}+5 x}\right )} \, dx=\frac {x}{\log {\left (\frac {5 x^{2} + 5 x + x e^{e^{2}} + \left (- 5 x - e^{e^{2}}\right ) \log {\left (x \right )} - e^{x}}{5 x + e^{e^{2}}} \right )}} \] Input:

integrate((((exp(exp(2))**2+10*x*exp(exp(2))+25*x**2)*ln(x)-x*exp(exp(2))* 
*2+(exp(x)-10*x**2-5*x)*exp(exp(2))+5*exp(x)*x-25*x**3-25*x**2)*ln(((-exp( 
exp(2))-5*x)*ln(x)+x*exp(exp(2))-exp(x)+5*x**2+5*x)/(exp(exp(2))+5*x))+(-1 
+x)*exp(exp(2))**2+(-exp(x)*x+10*x**2-5*x)*exp(exp(2))+(-5*x**2+5*x)*exp(x 
)+25*x**3-25*x**2)/((exp(exp(2))**2+10*x*exp(exp(2))+25*x**2)*ln(x)-x*exp( 
exp(2))**2+(exp(x)-10*x**2-5*x)*exp(exp(2))+5*exp(x)*x-25*x**3-25*x**2)/ln 
(((-exp(exp(2))-5*x)*ln(x)+x*exp(exp(2))-exp(x)+5*x**2+5*x)/(exp(exp(2))+5 
*x))**2,x)
 

Output:

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

Maxima [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.38 \[ \int \frac {e^{2 e^2} (-1+x)-25 x^2+25 x^3+e^x \left (5 x-5 x^2\right )+e^{e^2} \left (-5 x-e^x x+10 x^2\right )+\left (-e^{2 e^2} x+5 e^x x-25 x^2-25 x^3+e^{e^2} \left (e^x-5 x-10 x^2\right )+\left (e^{2 e^2}+10 e^{e^2} x+25 x^2\right ) \log (x)\right ) \log \left (\frac {-e^x+5 x+e^{e^2} x+5 x^2+\left (-e^{e^2}-5 x\right ) \log (x)}{e^{e^2}+5 x}\right )}{\left (-e^{2 e^2} x+5 e^x x-25 x^2-25 x^3+e^{e^2} \left (e^x-5 x-10 x^2\right )+\left (e^{2 e^2}+10 e^{e^2} x+25 x^2\right ) \log (x)\right ) \log ^2\left (\frac {-e^x+5 x+e^{e^2} x+5 x^2+\left (-e^{e^2}-5 x\right ) \log (x)}{e^{e^2}+5 x}\right )} \, dx=\frac {x}{\log \left (5 \, x^{2} + x {\left (e^{\left (e^{2}\right )} + 5\right )} - {\left (5 \, x + e^{\left (e^{2}\right )}\right )} \log \left (x\right ) - e^{x}\right ) - \log \left (5 \, x + e^{\left (e^{2}\right )}\right )} \] Input:

integrate((((exp(exp(2))^2+10*x*exp(exp(2))+25*x^2)*log(x)-x*exp(exp(2))^2 
+(exp(x)-10*x^2-5*x)*exp(exp(2))+5*exp(x)*x-25*x^3-25*x^2)*log(((-exp(exp( 
2))-5*x)*log(x)+x*exp(exp(2))-exp(x)+5*x^2+5*x)/(exp(exp(2))+5*x))+(-1+x)* 
exp(exp(2))^2+(-exp(x)*x+10*x^2-5*x)*exp(exp(2))+(-5*x^2+5*x)*exp(x)+25*x^ 
3-25*x^2)/((exp(exp(2))^2+10*x*exp(exp(2))+25*x^2)*log(x)-x*exp(exp(2))^2+ 
(exp(x)-10*x^2-5*x)*exp(exp(2))+5*exp(x)*x-25*x^3-25*x^2)/log(((-exp(exp(2 
))-5*x)*log(x)+x*exp(exp(2))-exp(x)+5*x^2+5*x)/(exp(exp(2))+5*x))^2,x, alg 
orithm="maxima")
 

Output:

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

Giac [A] (verification not implemented)

Time = 1.18 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.44 \[ \int \frac {e^{2 e^2} (-1+x)-25 x^2+25 x^3+e^x \left (5 x-5 x^2\right )+e^{e^2} \left (-5 x-e^x x+10 x^2\right )+\left (-e^{2 e^2} x+5 e^x x-25 x^2-25 x^3+e^{e^2} \left (e^x-5 x-10 x^2\right )+\left (e^{2 e^2}+10 e^{e^2} x+25 x^2\right ) \log (x)\right ) \log \left (\frac {-e^x+5 x+e^{e^2} x+5 x^2+\left (-e^{e^2}-5 x\right ) \log (x)}{e^{e^2}+5 x}\right )}{\left (-e^{2 e^2} x+5 e^x x-25 x^2-25 x^3+e^{e^2} \left (e^x-5 x-10 x^2\right )+\left (e^{2 e^2}+10 e^{e^2} x+25 x^2\right ) \log (x)\right ) \log ^2\left (\frac {-e^x+5 x+e^{e^2} x+5 x^2+\left (-e^{e^2}-5 x\right ) \log (x)}{e^{e^2}+5 x}\right )} \, dx=\frac {x}{\log \left (5 \, x^{2} + x e^{\left (e^{2}\right )} - 5 \, x \log \left (x\right ) - e^{\left (e^{2}\right )} \log \left (x\right ) + 5 \, x - e^{x}\right ) - \log \left (5 \, x + e^{\left (e^{2}\right )}\right )} \] Input:

integrate((((exp(exp(2))^2+10*x*exp(exp(2))+25*x^2)*log(x)-x*exp(exp(2))^2 
+(exp(x)-10*x^2-5*x)*exp(exp(2))+5*exp(x)*x-25*x^3-25*x^2)*log(((-exp(exp( 
2))-5*x)*log(x)+x*exp(exp(2))-exp(x)+5*x^2+5*x)/(exp(exp(2))+5*x))+(-1+x)* 
exp(exp(2))^2+(-exp(x)*x+10*x^2-5*x)*exp(exp(2))+(-5*x^2+5*x)*exp(x)+25*x^ 
3-25*x^2)/((exp(exp(2))^2+10*x*exp(exp(2))+25*x^2)*log(x)-x*exp(exp(2))^2+ 
(exp(x)-10*x^2-5*x)*exp(exp(2))+5*exp(x)*x-25*x^3-25*x^2)/log(((-exp(exp(2 
))-5*x)*log(x)+x*exp(exp(2))-exp(x)+5*x^2+5*x)/(exp(exp(2))+5*x))^2,x, alg 
orithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{2 e^2} (-1+x)-25 x^2+25 x^3+e^x \left (5 x-5 x^2\right )+e^{e^2} \left (-5 x-e^x x+10 x^2\right )+\left (-e^{2 e^2} x+5 e^x x-25 x^2-25 x^3+e^{e^2} \left (e^x-5 x-10 x^2\right )+\left (e^{2 e^2}+10 e^{e^2} x+25 x^2\right ) \log (x)\right ) \log \left (\frac {-e^x+5 x+e^{e^2} x+5 x^2+\left (-e^{e^2}-5 x\right ) \log (x)}{e^{e^2}+5 x}\right )}{\left (-e^{2 e^2} x+5 e^x x-25 x^2-25 x^3+e^{e^2} \left (e^x-5 x-10 x^2\right )+\left (e^{2 e^2}+10 e^{e^2} x+25 x^2\right ) \log (x)\right ) \log ^2\left (\frac {-e^x+5 x+e^{e^2} x+5 x^2+\left (-e^{e^2}-5 x\right ) \log (x)}{e^{e^2}+5 x}\right )} \, dx=-\int -\frac {\ln \left (\frac {5\,x-{\mathrm {e}}^x-\ln \left (x\right )\,\left (5\,x+{\mathrm {e}}^{{\mathrm {e}}^2}\right )+x\,{\mathrm {e}}^{{\mathrm {e}}^2}+5\,x^2}{5\,x+{\mathrm {e}}^{{\mathrm {e}}^2}}\right )\,\left (x\,{\mathrm {e}}^{2\,{\mathrm {e}}^2}-\ln \left (x\right )\,\left (25\,x^2+10\,{\mathrm {e}}^{{\mathrm {e}}^2}\,x+{\mathrm {e}}^{2\,{\mathrm {e}}^2}\right )-5\,x\,{\mathrm {e}}^x+{\mathrm {e}}^{{\mathrm {e}}^2}\,\left (5\,x-{\mathrm {e}}^x+10\,x^2\right )+25\,x^2+25\,x^3\right )+{\mathrm {e}}^{{\mathrm {e}}^2}\,\left (5\,x+x\,{\mathrm {e}}^x-10\,x^2\right )-{\mathrm {e}}^{2\,{\mathrm {e}}^2}\,\left (x-1\right )-{\mathrm {e}}^x\,\left (5\,x-5\,x^2\right )+25\,x^2-25\,x^3}{{\ln \left (\frac {5\,x-{\mathrm {e}}^x-\ln \left (x\right )\,\left (5\,x+{\mathrm {e}}^{{\mathrm {e}}^2}\right )+x\,{\mathrm {e}}^{{\mathrm {e}}^2}+5\,x^2}{5\,x+{\mathrm {e}}^{{\mathrm {e}}^2}}\right )}^2\,\left (x\,{\mathrm {e}}^{2\,{\mathrm {e}}^2}-\ln \left (x\right )\,\left (25\,x^2+10\,{\mathrm {e}}^{{\mathrm {e}}^2}\,x+{\mathrm {e}}^{2\,{\mathrm {e}}^2}\right )-5\,x\,{\mathrm {e}}^x+{\mathrm {e}}^{{\mathrm {e}}^2}\,\left (5\,x-{\mathrm {e}}^x+10\,x^2\right )+25\,x^2+25\,x^3\right )} \,d x \] Input:

int((log((5*x - exp(x) - log(x)*(5*x + exp(exp(2))) + x*exp(exp(2)) + 5*x^ 
2)/(5*x + exp(exp(2))))*(x*exp(2*exp(2)) - log(x)*(exp(2*exp(2)) + 10*x*ex 
p(exp(2)) + 25*x^2) - 5*x*exp(x) + exp(exp(2))*(5*x - exp(x) + 10*x^2) + 2 
5*x^2 + 25*x^3) + exp(exp(2))*(5*x + x*exp(x) - 10*x^2) - exp(2*exp(2))*(x 
 - 1) - exp(x)*(5*x - 5*x^2) + 25*x^2 - 25*x^3)/(log((5*x - exp(x) - log(x 
)*(5*x + exp(exp(2))) + x*exp(exp(2)) + 5*x^2)/(5*x + exp(exp(2))))^2*(x*e 
xp(2*exp(2)) - log(x)*(exp(2*exp(2)) + 10*x*exp(exp(2)) + 25*x^2) - 5*x*ex 
p(x) + exp(exp(2))*(5*x - exp(x) + 10*x^2) + 25*x^2 + 25*x^3)),x)
 

Output:

-int(-(log((5*x - exp(x) - log(x)*(5*x + exp(exp(2))) + x*exp(exp(2)) + 5* 
x^2)/(5*x + exp(exp(2))))*(x*exp(2*exp(2)) - log(x)*(exp(2*exp(2)) + 10*x* 
exp(exp(2)) + 25*x^2) - 5*x*exp(x) + exp(exp(2))*(5*x - exp(x) + 10*x^2) + 
 25*x^2 + 25*x^3) + exp(exp(2))*(5*x + x*exp(x) - 10*x^2) - exp(2*exp(2))* 
(x - 1) - exp(x)*(5*x - 5*x^2) + 25*x^2 - 25*x^3)/(log((5*x - exp(x) - log 
(x)*(5*x + exp(exp(2))) + x*exp(exp(2)) + 5*x^2)/(5*x + exp(exp(2))))^2*(x 
*exp(2*exp(2)) - log(x)*(exp(2*exp(2)) + 10*x*exp(exp(2)) + 25*x^2) - 5*x* 
exp(x) + exp(exp(2))*(5*x - exp(x) + 10*x^2) + 25*x^2 + 25*x^3)), x)
 

Reduce [B] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.62 \[ \int \frac {e^{2 e^2} (-1+x)-25 x^2+25 x^3+e^x \left (5 x-5 x^2\right )+e^{e^2} \left (-5 x-e^x x+10 x^2\right )+\left (-e^{2 e^2} x+5 e^x x-25 x^2-25 x^3+e^{e^2} \left (e^x-5 x-10 x^2\right )+\left (e^{2 e^2}+10 e^{e^2} x+25 x^2\right ) \log (x)\right ) \log \left (\frac {-e^x+5 x+e^{e^2} x+5 x^2+\left (-e^{e^2}-5 x\right ) \log (x)}{e^{e^2}+5 x}\right )}{\left (-e^{2 e^2} x+5 e^x x-25 x^2-25 x^3+e^{e^2} \left (e^x-5 x-10 x^2\right )+\left (e^{2 e^2}+10 e^{e^2} x+25 x^2\right ) \log (x)\right ) \log ^2\left (\frac {-e^x+5 x+e^{e^2} x+5 x^2+\left (-e^{e^2}-5 x\right ) \log (x)}{e^{e^2}+5 x}\right )} \, dx=\frac {x}{\mathrm {log}\left (\frac {-e^{e^{2}} \mathrm {log}\left (x \right )+e^{e^{2}} x -e^{x}-5 \,\mathrm {log}\left (x \right ) x +5 x^{2}+5 x}{e^{e^{2}}+5 x}\right )} \] Input:

int((((exp(exp(2))^2+10*x*exp(exp(2))+25*x^2)*log(x)-x*exp(exp(2))^2+(exp( 
x)-10*x^2-5*x)*exp(exp(2))+5*exp(x)*x-25*x^3-25*x^2)*log(((-exp(exp(2))-5* 
x)*log(x)+x*exp(exp(2))-exp(x)+5*x^2+5*x)/(exp(exp(2))+5*x))+(-1+x)*exp(ex 
p(2))^2+(-exp(x)*x+10*x^2-5*x)*exp(exp(2))+(-5*x^2+5*x)*exp(x)+25*x^3-25*x 
^2)/((exp(exp(2))^2+10*x*exp(exp(2))+25*x^2)*log(x)-x*exp(exp(2))^2+(exp(x 
)-10*x^2-5*x)*exp(exp(2))+5*exp(x)*x-25*x^3-25*x^2)/log(((-exp(exp(2))-5*x 
)*log(x)+x*exp(exp(2))-exp(x)+5*x^2+5*x)/(exp(exp(2))+5*x))^2,x)
 

Output:

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