\(\int \frac {e^{-\frac {-5 x+e^5 x+(-5+e^5) \log (x)+\log (\frac {x}{\log (x)})}{x+\log (x)}} (e^{\frac {-5 x+e^5 x+(-5+e^5) \log (x)+\log (\frac {x}{\log (x)})}{x+\log (x)}} (x^2 \log (x)+2 x \log ^2(x)+\log ^3(x))+e^{e^{-\frac {-5 x+e^5 x+(-5+e^5) \log (x)+\log (\frac {x}{\log (x)})}{x+\log (x)}} (-2 e^{\frac {-5 x+e^5 x+(-5+e^5) \log (x)+\log (\frac {x}{\log (x)})}{x+\log (x)}}+x)} (x+(1-x+x^2) \log (x)+(-1+2 x) \log ^2(x)+\log ^3(x)+(1+x) \log (x) \log (\frac {x}{\log (x)})))}{x^2 \log (x)+2 x \log ^2(x)+\log ^3(x)} \, dx\) [1148]

Optimal result

Integrand size = 223, antiderivative size = 32 \[ \int \frac {e^{-\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}} \left (e^{\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}} \left (x^2 \log (x)+2 x \log ^2(x)+\log ^3(x)\right )+e^{e^{-\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}} \left (-2 e^{\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}}+x\right )} \left (x+\left (1-x+x^2\right ) \log (x)+(-1+2 x) \log ^2(x)+\log ^3(x)+(1+x) \log (x) \log \left (\frac {x}{\log (x)}\right )\right )\right )}{x^2 \log (x)+2 x \log ^2(x)+\log ^3(x)} \, dx=e^{-2+e^{5-e^5-\frac {\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}} x}+x \] Output:

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

Mathematica [A] (verified)

Time = 1.41 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.75 \[ \int \frac {e^{-\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}} \left (e^{\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}} \left (x^2 \log (x)+2 x \log ^2(x)+\log ^3(x)\right )+e^{e^{-\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}} \left (-2 e^{\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}}+x\right )} \left (x+\left (1-x+x^2\right ) \log (x)+(-1+2 x) \log ^2(x)+\log ^3(x)+(1+x) \log (x) \log \left (\frac {x}{\log (x)}\right )\right )\right )}{x^2 \log (x)+2 x \log ^2(x)+\log ^3(x)} \, dx=e^{-2+e^{-\frac {\left (-5+e^5\right ) x}{x+\log (x)}+\frac {\left (5-e^5\right ) \log (x)}{x+\log (x)}} x \left (\frac {x}{\log (x)}\right )^{-\frac {1}{x+\log (x)}}}+x \] Input:

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

Output:

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

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7299

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

Input:

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

Output:

$Aborted
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 64.17 (sec) , antiderivative size = 179, normalized size of antiderivative = 5.59

method	result	size
risch	\(x +{\mathrm e}^{\left (x \,x^{\frac {4}{x +\ln \left (x \right )}} {\mathrm e}^{-\frac {-i \operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right )^{3} \pi +i \operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right )^{2} \operatorname {csgn}\left (i x \right ) \pi +i \operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right )^{2} \operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \pi -i \operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \pi +2 x \,{\mathrm e}^{5}-10 x}{2 \left (x +\ln \left (x \right )\right )}} \ln \left (x \right )^{\frac {1}{x +\ln \left (x \right )}}-2 x^{\frac {{\mathrm e}^{5}}{x +\ln \left (x \right )}}\right ) \ln \left (x \right )^{\frac {1}{x +\ln \left (x \right )}} \ln \left (x \right )^{-\frac {1}{x +\ln \left (x \right )}} x^{-\frac {{\mathrm e}^{5}}{x +\ln \left (x \right )}}}\)	\(179\)
parallelrisch	\(\frac {8 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )+4 x \ln \left (x \right )+8 x \ln \left (\ln \left (x \right )\right )-8 \ln \left (x \right )^{2}+12 x^{2}+8 \ln \left (x \right ) \ln \left (\frac {x}{\ln \left (x \right )}\right )+12 \,{\mathrm e}^{\left (-2 \,{\mathrm e}^{\frac {\ln \left (\frac {x}{\ln \left (x \right )}\right )+\left ({\mathrm e}^{5}-5\right ) \ln \left (x \right )+x \,{\mathrm e}^{5}-5 x}{x +\ln \left (x \right )}}+x \right ) {\mathrm e}^{-\frac {\ln \left (\frac {x}{\ln \left (x \right )}\right )+\left ({\mathrm e}^{5}-5\right ) \ln \left (x \right )+x \,{\mathrm e}^{5}-5 x}{x +\ln \left (x \right )}}} \ln \left (x \right )+12 \,{\mathrm e}^{\left (-2 \,{\mathrm e}^{\frac {\ln \left (\frac {x}{\ln \left (x \right )}\right )+\left ({\mathrm e}^{5}-5\right ) \ln \left (x \right )+x \,{\mathrm e}^{5}-5 x}{x +\ln \left (x \right )}}+x \right ) {\mathrm e}^{-\frac {\ln \left (\frac {x}{\ln \left (x \right )}\right )+\left ({\mathrm e}^{5}-5\right ) \ln \left (x \right )+x \,{\mathrm e}^{5}-5 x}{x +\ln \left (x \right )}}} x +8 x \ln \left (\frac {x}{\ln \left (x \right )}\right )}{12 x +12 \ln \left (x \right )}\)	\(203\)

Input:

int((((1+x)*ln(x)*ln(x/ln(x))+ln(x)^3+(-1+2*x)*ln(x)^2+(x^2-x+1)*ln(x)+x)* 
exp((-2*exp((ln(x/ln(x))+(exp(5)-5)*ln(x)+x*exp(5)-5*x)/(x+ln(x)))+x)/exp( 
(ln(x/ln(x))+(exp(5)-5)*ln(x)+x*exp(5)-5*x)/(x+ln(x))))+(ln(x)^3+2*x*ln(x) 
^2+x^2*ln(x))*exp((ln(x/ln(x))+(exp(5)-5)*ln(x)+x*exp(5)-5*x)/(x+ln(x))))/ 
(ln(x)^3+2*x*ln(x)^2+x^2*ln(x))/exp((ln(x/ln(x))+(exp(5)-5)*ln(x)+x*exp(5) 
-5*x)/(x+ln(x))),x,method=_RETURNVERBOSE)
 

Output:

x+exp((x*(x^(1/(x+ln(x))))^4*exp(-1/2*(-I*csgn(I*x/ln(x))^3*Pi+I*csgn(I*x/ 
ln(x))^2*csgn(I*x)*Pi+I*csgn(I*x/ln(x))^2*csgn(I/ln(x))*Pi-I*csgn(I*x/ln(x 
))*csgn(I*x)*csgn(I/ln(x))*Pi+2*x*exp(5)-10*x)/(x+ln(x)))*ln(x)^(1/(x+ln(x 
)))-2*x^(1/(x+ln(x))*exp(5)))/(ln(x)^(-1/(x+ln(x))))/(ln(x)^(1/(x+ln(x)))) 
/(x^(1/(x+ln(x))*exp(5))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (29) = 58\).

Time = 0.11 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.16 \[ \int \frac {e^{-\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}} \left (e^{\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}} \left (x^2 \log (x)+2 x \log ^2(x)+\log ^3(x)\right )+e^{e^{-\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}} \left (-2 e^{\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}}+x\right )} \left (x+\left (1-x+x^2\right ) \log (x)+(-1+2 x) \log ^2(x)+\log ^3(x)+(1+x) \log (x) \log \left (\frac {x}{\log (x)}\right )\right )\right )}{x^2 \log (x)+2 x \log ^2(x)+\log ^3(x)} \, dx=x + e^{\left ({\left (x - 2 \, e^{\left (\frac {x e^{5} + {\left (e^{5} - 5\right )} \log \left (x\right ) - 5 \, x + \log \left (\frac {x}{\log \left (x\right )}\right )}{x + \log \left (x\right )}\right )}\right )} e^{\left (-\frac {x e^{5} + {\left (e^{5} - 5\right )} \log \left (x\right ) - 5 \, x + \log \left (\frac {x}{\log \left (x\right )}\right )}{x + \log \left (x\right )}\right )}\right )} \] Input:

integrate((((1+x)*log(x)*log(x/log(x))+log(x)^3+(-1+2*x)*log(x)^2+(x^2-x+1 
)*log(x)+x)*exp((-2*exp((log(x/log(x))+(exp(5)-5)*log(x)+x*exp(5)-5*x)/(x+ 
log(x)))+x)/exp((log(x/log(x))+(exp(5)-5)*log(x)+x*exp(5)-5*x)/(x+log(x))) 
)+(log(x)^3+2*x*log(x)^2+x^2*log(x))*exp((log(x/log(x))+(exp(5)-5)*log(x)+ 
x*exp(5)-5*x)/(x+log(x))))/(log(x)^3+2*x*log(x)^2+x^2*log(x))/exp((log(x/l 
og(x))+(exp(5)-5)*log(x)+x*exp(5)-5*x)/(x+log(x))),x, algorithm="fricas")
 

Output:

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

Sympy [F(-2)]

Exception generated. \[ \int \frac {e^{-\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}} \left (e^{\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}} \left (x^2 \log (x)+2 x \log ^2(x)+\log ^3(x)\right )+e^{e^{-\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}} \left (-2 e^{\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}}+x\right )} \left (x+\left (1-x+x^2\right ) \log (x)+(-1+2 x) \log ^2(x)+\log ^3(x)+(1+x) \log (x) \log \left (\frac {x}{\log (x)}\right )\right )\right )}{x^2 \log (x)+2 x \log ^2(x)+\log ^3(x)} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> '>' not supported between instances of 'Pol 
y' and 'int'
 

Maxima [A] (verification not implemented)

Time = 0.92 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.19 \[ \int \frac {e^{-\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}} \left (e^{\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}} \left (x^2 \log (x)+2 x \log ^2(x)+\log ^3(x)\right )+e^{e^{-\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}} \left (-2 e^{\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}}+x\right )} \left (x+\left (1-x+x^2\right ) \log (x)+(-1+2 x) \log ^2(x)+\log ^3(x)+(1+x) \log (x) \log \left (\frac {x}{\log (x)}\right )\right )\right )}{x^2 \log (x)+2 x \log ^2(x)+\log ^3(x)} \, dx={\left (x e^{2} + e^{\left (x e^{\left (-\frac {\log \left (x\right )}{x + \log \left (x\right )} + \frac {\log \left (\log \left (x\right )\right )}{x + \log \left (x\right )} - e^{5} + 5\right )}\right )}\right )} e^{\left (-2\right )} \] Input:

integrate((((1+x)*log(x)*log(x/log(x))+log(x)^3+(-1+2*x)*log(x)^2+(x^2-x+1 
)*log(x)+x)*exp((-2*exp((log(x/log(x))+(exp(5)-5)*log(x)+x*exp(5)-5*x)/(x+ 
log(x)))+x)/exp((log(x/log(x))+(exp(5)-5)*log(x)+x*exp(5)-5*x)/(x+log(x))) 
)+(log(x)^3+2*x*log(x)^2+x^2*log(x))*exp((log(x/log(x))+(exp(5)-5)*log(x)+ 
x*exp(5)-5*x)/(x+log(x))))/(log(x)^3+2*x*log(x)^2+x^2*log(x))/exp((log(x/l 
og(x))+(exp(5)-5)*log(x)+x*exp(5)-5*x)/(x+log(x))),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((((1+x)*log(x)*log(x/log(x))+log(x)^3+(-1+2*x)*log(x)^2+(x^2-x+1 
)*log(x)+x)*exp((-2*exp((log(x/log(x))+(exp(5)-5)*log(x)+x*exp(5)-5*x)/(x+ 
log(x)))+x)/exp((log(x/log(x))+(exp(5)-5)*log(x)+x*exp(5)-5*x)/(x+log(x))) 
)+(log(x)^3+2*x*log(x)^2+x^2*log(x))*exp((log(x/log(x))+(exp(5)-5)*log(x)+ 
x*exp(5)-5*x)/(x+log(x))))/(log(x)^3+2*x*log(x)^2+x^2*log(x))/exp((log(x/l 
og(x))+(exp(5)-5)*log(x)+x*exp(5)-5*x)/(x+log(x))),x, algorithm="giac")
 

Output:

undef
 

Mupad [B] (verification not implemented)

Time = 9.78 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.12 \[ \int \frac {e^{-\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}} \left (e^{\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}} \left (x^2 \log (x)+2 x \log ^2(x)+\log ^3(x)\right )+e^{e^{-\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}} \left (-2 e^{\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}}+x\right )} \left (x+\left (1-x+x^2\right ) \log (x)+(-1+2 x) \log ^2(x)+\log ^3(x)+(1+x) \log (x) \log \left (\frac {x}{\log (x)}\right )\right )\right )}{x^2 \log (x)+2 x \log ^2(x)+\log ^3(x)} \, dx=x+{\mathrm {e}}^{\frac {x\,x^{\frac {5}{x+\ln \left (x\right )}}\,{\mathrm {e}}^{\frac {5\,x}{x+\ln \left (x\right )}}\,{\mathrm {e}}^{-\frac {x\,{\mathrm {e}}^5}{x+\ln \left (x\right )}}}{x^{\frac {{\mathrm {e}}^5}{x+\ln \left (x\right )}}\,{\left (\frac {x}{\ln \left (x\right )}\right )}^{\frac {1}{x+\ln \left (x\right )}}}}\,{\mathrm {e}}^{-2} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.47 \[ \int \frac {e^{-\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}} \left (e^{\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}} \left (x^2 \log (x)+2 x \log ^2(x)+\log ^3(x)\right )+e^{e^{-\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}} \left (-2 e^{\frac {-5 x+e^5 x+\left (-5+e^5\right ) \log (x)+\log \left (\frac {x}{\log (x)}\right )}{x+\log (x)}}+x\right )} \left (x+\left (1-x+x^2\right ) \log (x)+(-1+2 x) \log ^2(x)+\log ^3(x)+(1+x) \log (x) \log \left (\frac {x}{\log (x)}\right )\right )\right )}{x^2 \log (x)+2 x \log ^2(x)+\log ^3(x)} \, dx=\frac {e^{\frac {e^{5} x}{e^{\frac {\mathrm {log}\left (\frac {x}{\mathrm {log}\left (x \right )}\right )+\mathrm {log}\left (x \right ) e^{5}+e^{5} x}{\mathrm {log}\left (x \right )+x}}}}+e^{2} x}{e^{2}} \] Input:

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

Output:

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