\(\int \frac {625 e^x+e^x (-625-625 x) \log (x) \log (\log (x))+(625-1250 x) \log (x) \log ^2(\log (x))}{625 e^{2 x} x^2 \log (x)+e^x (-50 x-1250 x^2+1250 x^3) \log (x) \log (\log (x))+(1+50 x+575 x^2-1250 x^3+625 x^4) \log (x) \log ^2(\log (x))} \, dx\) [168]

Optimal result

Integrand size = 100, antiderivative size = 25 \[ \int \frac {625 e^x+e^x (-625-625 x) \log (x) \log (\log (x))+(625-1250 x) \log (x) \log ^2(\log (x))}{625 e^{2 x} x^2 \log (x)+e^x \left (-50 x-1250 x^2+1250 x^3\right ) \log (x) \log (\log (x))+\left (1+50 x+575 x^2-1250 x^3+625 x^4\right ) \log (x) \log ^2(\log (x))} \, dx=\frac {1}{x \left (-1-\frac {1}{25 x}+x+\frac {e^x}{\log (\log (x))}\right )} \] Output:

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

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {625 e^x+e^x (-625-625 x) \log (x) \log (\log (x))+(625-1250 x) \log (x) \log ^2(\log (x))}{625 e^{2 x} x^2 \log (x)+e^x \left (-50 x-1250 x^2+1250 x^3\right ) \log (x) \log (\log (x))+\left (1+50 x+575 x^2-1250 x^3+625 x^4\right ) \log (x) \log ^2(\log (x))} \, dx=\frac {25 \log (\log (x))}{25 e^x x+\left (-1-25 x+25 x^2\right ) \log (\log (x))} \] Input:

Integrate[(625*E^x + E^x*(-625 - 625*x)*Log[x]*Log[Log[x]] + (625 - 1250*x 
)*Log[x]*Log[Log[x]]^2)/(625*E^(2*x)*x^2*Log[x] + E^x*(-50*x - 1250*x^2 + 
1250*x^3)*Log[x]*Log[Log[x]] + (1 + 50*x + 575*x^2 - 1250*x^3 + 625*x^4)*L 
og[x]*Log[Log[x]]^2),x]
 

Output:

(25*Log[Log[x]])/(25*E^x*x + (-1 - 25*x + 25*x^2)*Log[Log[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.

\(\displaystyle \int \frac {625 e^x+(625-1250 x) \log (x) \log ^2(\log (x))+e^x (-625 x-625) \log (x) \log (\log (x))}{625 e^{2 x} x^2 \log (x)+e^x \left (1250 x^3-1250 x^2-50 x\right ) \log (x) \log (\log (x))+\left (625 x^4-1250 x^3+575 x^2+50 x+1\right ) \log (x) \log ^2(\log (x))} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {625 \left (e^x-\log (x) \log (\log (x)) \left (e^x (x+1)+(2 x-1) \log (\log (x))\right )\right )}{\log (x) \left (\left (25 x^2-25 x-1\right ) \log (\log (x))+25 e^x x\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 625 \int \frac {e^x-\log (x) \log (\log (x)) \left (e^x (x+1)-(1-2 x) \log (\log (x))\right )}{\log (x) \left (25 e^x x-\left (-25 x^2+25 x+1\right ) \log (\log (x))\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 625 \int \left (\frac {\log (\log (x)) \left (25 \log (x) \log (\log (x)) x^3-50 \log (x) \log (\log (x)) x^2-25 x^2-\log (x) \log (\log (x)) x+25 x-\log (x) \log (\log (x))+1\right )}{25 x \log (x) \left (25 \log (\log (x)) x^2+25 e^x x-25 \log (\log (x)) x-\log (\log (x))\right )^2}-\frac {x \log (x) \log (\log (x))+\log (x) \log (\log (x))-1}{25 x \log (x) \left (25 \log (\log (x)) x^2+25 e^x x-25 \log (\log (x)) x-\log (\log (x))\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 625 \left (-\frac {1}{25} \int \frac {\log ^2(\log (x))}{\left (25 \log (\log (x)) x^2+25 e^x x-25 \log (\log (x)) x-\log (\log (x))\right )^2}dx-\frac {1}{25} \int \frac {\log ^2(\log (x))}{x \left (25 \log (\log (x)) x^2+25 e^x x-25 \log (\log (x)) x-\log (\log (x))\right )^2}dx-2 \int \frac {x \log ^2(\log (x))}{\left (25 \log (\log (x)) x^2+25 e^x x-25 \log (\log (x)) x-\log (\log (x))\right )^2}dx+\int \frac {x^2 \log ^2(\log (x))}{\left (25 \log (\log (x)) x^2+25 e^x x-25 \log (\log (x)) x-\log (\log (x))\right )^2}dx+\int \frac {\log (\log (x))}{\log (x) \left (25 \log (\log (x)) x^2+25 e^x x-25 \log (\log (x)) x-\log (\log (x))\right )^2}dx+\frac {1}{25} \int \frac {\log (\log (x))}{x \log (x) \left (25 \log (\log (x)) x^2+25 e^x x-25 \log (\log (x)) x-\log (\log (x))\right )^2}dx-\int \frac {x \log (\log (x))}{\log (x) \left (25 \log (\log (x)) x^2+25 e^x x-25 \log (\log (x)) x-\log (\log (x))\right )^2}dx+\frac {1}{25} \int \frac {1}{x \log (x) \left (25 \log (\log (x)) x^2+25 e^x x-25 \log (\log (x)) x-\log (\log (x))\right )}dx-\frac {1}{25} \int \frac {\log (\log (x))}{25 \log (\log (x)) x^2+25 e^x x-25 \log (\log (x)) x-\log (\log (x))}dx-\frac {1}{25} \int \frac {\log (\log (x))}{x \left (25 \log (\log (x)) x^2+25 e^x x-25 \log (\log (x)) x-\log (\log (x))\right )}dx\right )\)

Input:

Int[(625*E^x + E^x*(-625 - 625*x)*Log[x]*Log[Log[x]] + (625 - 1250*x)*Log[ 
x]*Log[Log[x]]^2)/(625*E^(2*x)*x^2*Log[x] + E^x*(-50*x - 1250*x^2 + 1250*x 
^3)*Log[x]*Log[Log[x]] + (1 + 50*x + 575*x^2 - 1250*x^3 + 625*x^4)*Log[x]* 
Log[Log[x]]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 6.31 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32

method	result	size
parallelrisch	\(\frac {25 \ln \left (\ln \left (x \right )\right )}{25 x^{2} \ln \left (\ln \left (x \right )\right )+25 \,{\mathrm e}^{x} x -25 x \ln \left (\ln \left (x \right )\right )-\ln \left (\ln \left (x \right )\right )}\)	\(33\)
risch	\(\frac {25}{25 x^{2}-25 x -1}-\frac {625 x \,{\mathrm e}^{x}}{\left (25 x^{2}-25 x -1\right ) \left (25 x^{2} \ln \left (\ln \left (x \right )\right )+25 \,{\mathrm e}^{x} x -25 x \ln \left (\ln \left (x \right )\right )-\ln \left (\ln \left (x \right )\right )\right )}\)	\(60\)

Input:

int(((-1250*x+625)*ln(x)*ln(ln(x))^2+(-625*x-625)*exp(x)*ln(x)*ln(ln(x))+6 
25*exp(x))/((625*x^4-1250*x^3+575*x^2+50*x+1)*ln(x)*ln(ln(x))^2+(1250*x^3- 
1250*x^2-50*x)*exp(x)*ln(x)*ln(ln(x))+625*x^2*exp(x)^2*ln(x)),x,method=_RE 
TURNVERBOSE)
 

Output:

25*ln(ln(x))/(25*x^2*ln(ln(x))+25*exp(x)*x-25*x*ln(ln(x))-ln(ln(x)))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {625 e^x+e^x (-625-625 x) \log (x) \log (\log (x))+(625-1250 x) \log (x) \log ^2(\log (x))}{625 e^{2 x} x^2 \log (x)+e^x \left (-50 x-1250 x^2+1250 x^3\right ) \log (x) \log (\log (x))+\left (1+50 x+575 x^2-1250 x^3+625 x^4\right ) \log (x) \log ^2(\log (x))} \, dx=\frac {25 \, \log \left (\log \left (x\right )\right )}{25 \, x e^{x} + {\left (25 \, x^{2} - 25 \, x - 1\right )} \log \left (\log \left (x\right )\right )} \] Input:

integrate(((-1250*x+625)*log(x)*log(log(x))^2+(-625*x-625)*exp(x)*log(x)*l 
og(log(x))+625*exp(x))/((625*x^4-1250*x^3+575*x^2+50*x+1)*log(x)*log(log(x 
))^2+(1250*x^3-1250*x^2-50*x)*exp(x)*log(x)*log(log(x))+625*x^2*exp(x)^2*l 
og(x)),x, algorithm="fricas")
 

Output:

25*log(log(x))/(25*x*e^x + (25*x^2 - 25*x - 1)*log(log(x)))
 

Sympy [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44 \[ \int \frac {625 e^x+e^x (-625-625 x) \log (x) \log (\log (x))+(625-1250 x) \log (x) \log ^2(\log (x))}{625 e^{2 x} x^2 \log (x)+e^x \left (-50 x-1250 x^2+1250 x^3\right ) \log (x) \log (\log (x))+\left (1+50 x+575 x^2-1250 x^3+625 x^4\right ) \log (x) \log ^2(\log (x))} \, dx=\frac {25 \log {\left (\log {\left (x \right )} \right )}}{25 x^{2} \log {\left (\log {\left (x \right )} \right )} + 25 x e^{x} - 25 x \log {\left (\log {\left (x \right )} \right )} - \log {\left (\log {\left (x \right )} \right )}} \] Input:

integrate(((-1250*x+625)*ln(x)*ln(ln(x))**2+(-625*x-625)*exp(x)*ln(x)*ln(l 
n(x))+625*exp(x))/((625*x**4-1250*x**3+575*x**2+50*x+1)*ln(x)*ln(ln(x))**2 
+(1250*x**3-1250*x**2-50*x)*exp(x)*ln(x)*ln(ln(x))+625*x**2*exp(x)**2*ln(x 
)),x)
 

Output:

25*log(log(x))/(25*x**2*log(log(x)) + 25*x*exp(x) - 25*x*log(log(x)) - log 
(log(x)))
 

Maxima [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {625 e^x+e^x (-625-625 x) \log (x) \log (\log (x))+(625-1250 x) \log (x) \log ^2(\log (x))}{625 e^{2 x} x^2 \log (x)+e^x \left (-50 x-1250 x^2+1250 x^3\right ) \log (x) \log (\log (x))+\left (1+50 x+575 x^2-1250 x^3+625 x^4\right ) \log (x) \log ^2(\log (x))} \, dx=\frac {25 \, \log \left (\log \left (x\right )\right )}{25 \, x e^{x} + {\left (25 \, x^{2} - 25 \, x - 1\right )} \log \left (\log \left (x\right )\right )} \] Input:

integrate(((-1250*x+625)*log(x)*log(log(x))^2+(-625*x-625)*exp(x)*log(x)*l 
og(log(x))+625*exp(x))/((625*x^4-1250*x^3+575*x^2+50*x+1)*log(x)*log(log(x 
))^2+(1250*x^3-1250*x^2-50*x)*exp(x)*log(x)*log(log(x))+625*x^2*exp(x)^2*l 
og(x)),x, algorithm="maxima")
 

Output:

25*log(log(x))/(25*x*e^x + (25*x^2 - 25*x - 1)*log(log(x)))
 

Giac [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28 \[ \int \frac {625 e^x+e^x (-625-625 x) \log (x) \log (\log (x))+(625-1250 x) \log (x) \log ^2(\log (x))}{625 e^{2 x} x^2 \log (x)+e^x \left (-50 x-1250 x^2+1250 x^3\right ) \log (x) \log (\log (x))+\left (1+50 x+575 x^2-1250 x^3+625 x^4\right ) \log (x) \log ^2(\log (x))} \, dx=\frac {25 \, \log \left (\log \left (x\right )\right )}{25 \, x^{2} \log \left (\log \left (x\right )\right ) + 25 \, x e^{x} - 25 \, x \log \left (\log \left (x\right )\right ) - \log \left (\log \left (x\right )\right )} \] Input:

integrate(((-1250*x+625)*log(x)*log(log(x))^2+(-625*x-625)*exp(x)*log(x)*l 
og(log(x))+625*exp(x))/((625*x^4-1250*x^3+575*x^2+50*x+1)*log(x)*log(log(x 
))^2+(1250*x^3-1250*x^2-50*x)*exp(x)*log(x)*log(log(x))+625*x^2*exp(x)^2*l 
og(x)),x, algorithm="giac")
 

Output:

25*log(log(x))/(25*x^2*log(log(x)) + 25*x*e^x - 25*x*log(log(x)) - log(log 
(x)))
 

Mupad [B] (verification not implemented)

Time = 3.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.20 \[ \int \frac {625 e^x+e^x (-625-625 x) \log (x) \log (\log (x))+(625-1250 x) \log (x) \log ^2(\log (x))}{625 e^{2 x} x^2 \log (x)+e^x \left (-50 x-1250 x^2+1250 x^3\right ) \log (x) \log (\log (x))+\left (1+50 x+575 x^2-1250 x^3+625 x^4\right ) \log (x) \log ^2(\log (x))} \, dx=-\frac {25\,\ln \left (\ln \left (x\right )\right )}{\ln \left (\ln \left (x\right )\right )+25\,x\,\ln \left (\ln \left (x\right )\right )-25\,x^2\,\ln \left (\ln \left (x\right )\right )-25\,x\,{\mathrm {e}}^x} \] Input:

int(-(log(log(x))^2*log(x)*(1250*x - 625) - 625*exp(x) + log(log(x))*exp(x 
)*log(x)*(625*x + 625))/(625*x^2*exp(2*x)*log(x) + log(log(x))^2*log(x)*(5 
0*x + 575*x^2 - 1250*x^3 + 625*x^4 + 1) - log(log(x))*exp(x)*log(x)*(50*x 
+ 1250*x^2 - 1250*x^3)),x)
 

Output:

-(25*log(log(x)))/(log(log(x)) + 25*x*log(log(x)) - 25*x^2*log(log(x)) - 2 
5*x*exp(x))
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32 \[ \int \frac {625 e^x+e^x (-625-625 x) \log (x) \log (\log (x))+(625-1250 x) \log (x) \log ^2(\log (x))}{625 e^{2 x} x^2 \log (x)+e^x \left (-50 x-1250 x^2+1250 x^3\right ) \log (x) \log (\log (x))+\left (1+50 x+575 x^2-1250 x^3+625 x^4\right ) \log (x) \log ^2(\log (x))} \, dx=\frac {25 \,\mathrm {log}\left (\mathrm {log}\left (x \right )\right )}{25 e^{x} x +25 \,\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) x^{2}-25 \,\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) x -\mathrm {log}\left (\mathrm {log}\left (x \right )\right )} \] Input:

int(((-1250*x+625)*log(x)*log(log(x))^2+(-625*x-625)*exp(x)*log(x)*log(log 
(x))+625*exp(x))/((625*x^4-1250*x^3+575*x^2+50*x+1)*log(x)*log(log(x))^2+( 
1250*x^3-1250*x^2-50*x)*exp(x)*log(x)*log(log(x))+625*x^2*exp(x)^2*log(x)) 
,x)
 

Output:

(25*log(log(x)))/(25*e**x*x + 25*log(log(x))*x**2 - 25*log(log(x))*x - log 
(log(x)))