\(\int \frac {-5 e^4+5 x^2-5 x^3-5 x^4+(10 e^4-5 x^3-10 x^4) \log (x) \log (\log (x))+(e^4 (18-3 x)-8 x^3-18 x^4+x^5) \log (x) \log ^2(\log (x))}{25 x^3 \log (x)+(90 x^3-10 x^4) \log (x) \log (\log (x))+(81 x^3-18 x^4+x^5) \log (x) \log ^2(\log (x))} \, dx\) [754]

Optimal result

Integrand size = 124, antiderivative size = 33 \[ \int \frac {-5 e^4+5 x^2-5 x^3-5 x^4+\left (10 e^4-5 x^3-10 x^4\right ) \log (x) \log (\log (x))+\left (e^4 (18-3 x)-8 x^3-18 x^4+x^5\right ) \log (x) \log ^2(\log (x))}{25 x^3 \log (x)+\left (90 x^3-10 x^4\right ) \log (x) \log (\log (x))+\left (81 x^3-18 x^4+x^5\right ) \log (x) \log ^2(\log (x))} \, dx=\frac {1-\frac {e^4}{x^2}-x-x^2}{9-x+\frac {5}{\log (\log (x))}} \] Output:

(1-x^2-x-exp(4)/x^2)/(5/ln(ln(x))-x+9)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.24 \[ \int \frac {-5 e^4+5 x^2-5 x^3-5 x^4+\left (10 e^4-5 x^3-10 x^4\right ) \log (x) \log (\log (x))+\left (e^4 (18-3 x)-8 x^3-18 x^4+x^5\right ) \log (x) \log ^2(\log (x))}{25 x^3 \log (x)+\left (90 x^3-10 x^4\right ) \log (x) \log (\log (x))+\left (81 x^3-18 x^4+x^5\right ) \log (x) \log ^2(\log (x))} \, dx=\frac {50 x^2+\left (e^4+x^2 \left (89-9 x+x^2\right )\right ) \log (\log (x))}{x^2 (-5+(-9+x) \log (\log (x)))} \] Input:

Integrate[(-5*E^4 + 5*x^2 - 5*x^3 - 5*x^4 + (10*E^4 - 5*x^3 - 10*x^4)*Log[ 
x]*Log[Log[x]] + (E^4*(18 - 3*x) - 8*x^3 - 18*x^4 + x^5)*Log[x]*Log[Log[x] 
]^2)/(25*x^3*Log[x] + (90*x^3 - 10*x^4)*Log[x]*Log[Log[x]] + (81*x^3 - 18* 
x^4 + x^5)*Log[x]*Log[Log[x]]^2),x]
 

Output:

(50*x^2 + (E^4 + x^2*(89 - 9*x + x^2))*Log[Log[x]])/(x^2*(-5 + (-9 + x)*Lo 
g[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.

\(\displaystyle \int \frac {-5 x^4-5 x^3+5 x^2+\left (-10 x^4-5 x^3+10 e^4\right ) \log (x) \log (\log (x))+\left (x^5-18 x^4-8 x^3+e^4 (18-3 x)\right ) \log (x) \log ^2(\log (x))-5 e^4}{25 x^3 \log (x)+\left (90 x^3-10 x^4\right ) \log (x) \log (\log (x))+\left (x^5-18 x^4+81 x^3\right ) \log (x) \log ^2(\log (x))} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {-5 x^4-5 x^3+5 x^2+\left (-10 x^4-5 x^3+10 e^4\right ) \log (x) \log (\log (x))+\left (x^5-18 x^4-8 x^3+e^4 (18-3 x)\right ) \log (x) \log ^2(\log (x))-5 e^4}{x^3 \log (x) (-x \log (\log (x))+9 \log (\log (x))+5)^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {5 \left (19 x^4+7 x^3+4 e^4 x-18 e^4\right )}{(x-9)^2 x^3 (x \log (\log (x))-9 \log (\log (x))-5)}+\frac {x^5-18 x^4-8 x^3-3 e^4 x+18 e^4}{(x-9)^2 x^3}-\frac {5 \left (x^4+x^3-x^2+e^4\right ) \left (x^2-18 x+5 x \log (x)+81\right )}{(x-9)^2 x^3 \log (x) (x \log (\log (x))-9 \log (\log (x))-5)^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -5 e^4 \int \frac {1}{x^3 \log (x) (x \log (\log (x))-9 \log (\log (x))-5)^2}dx+\frac {10}{9} e^4 \int \frac {1}{x^3 (x \log (\log (x))-9 \log (\log (x))-5)}dx-\frac {25}{81} e^4 \int \frac {1}{x^2 (x \log (\log (x))-9 \log (\log (x))-5)^2}dx-\frac {25 \left (2214-e^4\right ) \int \frac {1}{(x \log (\log (x))-9 \log (\log (x))-5)^2}dx}{2187}+\frac {25 \left (27-e^4\right ) \int \frac {1}{(x \log (\log (x))-9 \log (\log (x))-5)^2}dx}{2187}-\frac {25}{81} \left (7209+e^4\right ) \int \frac {1}{(x-9)^2 (x \log (\log (x))-9 \log (\log (x))-5)^2}dx-\frac {25}{729} \left (7209+e^4\right ) \int \frac {1}{(x-9) (x \log (\log (x))-9 \log (\log (x))-5)^2}dx-\frac {25}{243} \left (2214-e^4\right ) \int \frac {1}{(x-9) (x \log (\log (x))-9 \log (\log (x))-5)^2}dx-\frac {50}{729} e^4 \int \frac {1}{x (x \log (\log (x))-9 \log (\log (x))-5)^2}dx-\frac {5}{729} \left (7209+e^4\right ) \int \frac {1}{\log (x) (x \log (\log (x))-9 \log (\log (x))-5)^2}dx+\frac {5}{243} \left (2214-e^4\right ) \int \frac {1}{\log (x) (x \log (\log (x))-9 \log (\log (x))-5)^2}dx-\frac {10}{243} \left (27-e^4\right ) \int \frac {1}{\log (x) (x \log (\log (x))-9 \log (\log (x))-5)^2}dx-\frac {10}{729} e^4 \int \frac {1}{\log (x) (x \log (\log (x))-9 \log (\log (x))-5)^2}dx+\frac {5}{27} \left (27-e^4\right ) \int \frac {1}{x \log (x) (x \log (\log (x))-9 \log (\log (x))-5)^2}dx+\frac {5}{27} e^4 \int \frac {1}{x \log (x) (x \log (\log (x))-9 \log (\log (x))-5)^2}dx-\frac {5 \left (2214-e^4\right ) \int \frac {x}{\log (x) (x \log (\log (x))-9 \log (\log (x))-5)^2}dx}{2187}+\frac {5 \left (27-e^4\right ) \int \frac {x}{\log (x) (x \log (\log (x))-9 \log (\log (x))-5)^2}dx}{2187}-\frac {10}{81} \left (7209+e^4\right ) \int \frac {1}{(x-9)^2 (x \log (\log (x))-9 \log (\log (x))-5)}dx-\frac {5}{729} \left (13851-2 e^4\right ) \int \frac {1}{(x-9) (x \log (\log (x))-9 \log (\log (x))-5)}dx-\frac {10}{729} e^4 \int \frac {1}{x (x \log (\log (x))-9 \log (\log (x))-5)}dx-\frac {e^4}{9 x^2}+x-\frac {7209+e^4}{81 (9-x)}-\frac {e^4}{81 x}\)

Input:

Int[(-5*E^4 + 5*x^2 - 5*x^3 - 5*x^4 + (10*E^4 - 5*x^3 - 10*x^4)*Log[x]*Log 
[Log[x]] + (E^4*(18 - 3*x) - 8*x^3 - 18*x^4 + x^5)*Log[x]*Log[Log[x]]^2)/( 
25*x^3*Log[x] + (90*x^3 - 10*x^4)*Log[x]*Log[Log[x]] + (81*x^3 - 18*x^4 + 
x^5)*Log[x]*Log[Log[x]]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 5.94 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.39

method	result	size
parallelrisch	\(\frac {\ln \left (\ln \left (x \right )\right ) x^{4}+8 x^{2} \ln \left (\ln \left (x \right )\right )+{\mathrm e}^{4} \ln \left (\ln \left (x \right )\right )+5 x^{2}}{x^{2} \left (x \ln \left (\ln \left (x \right )\right )-9 \ln \left (\ln \left (x \right )\right )-5\right )}\)	\(46\)
risch	\(\frac {x^{4}-9 x^{3}+89 x^{2}+{\mathrm e}^{4}}{x^{2} \left (x -9\right )}+\frac {5 x^{4}+5 x^{3}-5 x^{2}+5 \,{\mathrm e}^{4}}{x^{2} \left (x -9\right ) \left (x \ln \left (\ln \left (x \right )\right )-9 \ln \left (\ln \left (x \right )\right )-5\right )}\)	\(65\)

Input:

int((((-3*x+18)*exp(4)+x^5-18*x^4-8*x^3)*ln(x)*ln(ln(x))^2+(10*exp(4)-10*x 
^4-5*x^3)*ln(x)*ln(ln(x))-5*exp(4)-5*x^4-5*x^3+5*x^2)/((x^5-18*x^4+81*x^3) 
*ln(x)*ln(ln(x))^2+(-10*x^4+90*x^3)*ln(x)*ln(ln(x))+25*x^3*ln(x)),x,method 
=_RETURNVERBOSE)
 

Output:

1/x^2*(ln(ln(x))*x^4+8*x^2*ln(ln(x))+exp(4)*ln(ln(x))+5*x^2)/(x*ln(ln(x))- 
9*ln(ln(x))-5)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.52 \[ \int \frac {-5 e^4+5 x^2-5 x^3-5 x^4+\left (10 e^4-5 x^3-10 x^4\right ) \log (x) \log (\log (x))+\left (e^4 (18-3 x)-8 x^3-18 x^4+x^5\right ) \log (x) \log ^2(\log (x))}{25 x^3 \log (x)+\left (90 x^3-10 x^4\right ) \log (x) \log (\log (x))+\left (81 x^3-18 x^4+x^5\right ) \log (x) \log ^2(\log (x))} \, dx=-\frac {50 \, x^{2} + {\left (x^{4} - 9 \, x^{3} + 89 \, x^{2} + e^{4}\right )} \log \left (\log \left (x\right )\right )}{5 \, x^{2} - {\left (x^{3} - 9 \, x^{2}\right )} \log \left (\log \left (x\right )\right )} \] Input:

integrate((((-3*x+18)*exp(4)+x^5-18*x^4-8*x^3)*log(x)*log(log(x))^2+(10*ex 
p(4)-10*x^4-5*x^3)*log(x)*log(log(x))-5*exp(4)-5*x^4-5*x^3+5*x^2)/((x^5-18 
*x^4+81*x^3)*log(x)*log(log(x))^2+(-10*x^4+90*x^3)*log(x)*log(log(x))+25*x 
^3*log(x)),x, algorithm="fricas")
 

Output:

-(50*x^2 + (x^4 - 9*x^3 + 89*x^2 + e^4)*log(log(x)))/(5*x^2 - (x^3 - 9*x^2 
)*log(log(x)))
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (22) = 44\).

Time = 0.33 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.00 \[ \int \frac {-5 e^4+5 x^2-5 x^3-5 x^4+\left (10 e^4-5 x^3-10 x^4\right ) \log (x) \log (\log (x))+\left (e^4 (18-3 x)-8 x^3-18 x^4+x^5\right ) \log (x) \log ^2(\log (x))}{25 x^3 \log (x)+\left (90 x^3-10 x^4\right ) \log (x) \log (\log (x))+\left (81 x^3-18 x^4+x^5\right ) \log (x) \log ^2(\log (x))} \, dx=x + \frac {89 x^{2} + e^{4}}{x^{3} - 9 x^{2}} + \frac {5 x^{4} + 5 x^{3} - 5 x^{2} + 5 e^{4}}{- 5 x^{3} + 45 x^{2} + \left (x^{4} - 18 x^{3} + 81 x^{2}\right ) \log {\left (\log {\left (x \right )} \right )}} \] Input:

integrate((((-3*x+18)*exp(4)+x**5-18*x**4-8*x**3)*ln(x)*ln(ln(x))**2+(10*e 
xp(4)-10*x**4-5*x**3)*ln(x)*ln(ln(x))-5*exp(4)-5*x**4-5*x**3+5*x**2)/((x** 
5-18*x**4+81*x**3)*ln(x)*ln(ln(x))**2+(-10*x**4+90*x**3)*ln(x)*ln(ln(x))+2 
5*x**3*ln(x)),x)
 

Output:

x + (89*x**2 + exp(4))/(x**3 - 9*x**2) + (5*x**4 + 5*x**3 - 5*x**2 + 5*exp 
(4))/(-5*x**3 + 45*x**2 + (x**4 - 18*x**3 + 81*x**2)*log(log(x)))
 

Maxima [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.52 \[ \int \frac {-5 e^4+5 x^2-5 x^3-5 x^4+\left (10 e^4-5 x^3-10 x^4\right ) \log (x) \log (\log (x))+\left (e^4 (18-3 x)-8 x^3-18 x^4+x^5\right ) \log (x) \log ^2(\log (x))}{25 x^3 \log (x)+\left (90 x^3-10 x^4\right ) \log (x) \log (\log (x))+\left (81 x^3-18 x^4+x^5\right ) \log (x) \log ^2(\log (x))} \, dx=-\frac {50 \, x^{2} + {\left (x^{4} - 9 \, x^{3} + 89 \, x^{2} + e^{4}\right )} \log \left (\log \left (x\right )\right )}{5 \, x^{2} - {\left (x^{3} - 9 \, x^{2}\right )} \log \left (\log \left (x\right )\right )} \] Input:

integrate((((-3*x+18)*exp(4)+x^5-18*x^4-8*x^3)*log(x)*log(log(x))^2+(10*ex 
p(4)-10*x^4-5*x^3)*log(x)*log(log(x))-5*exp(4)-5*x^4-5*x^3+5*x^2)/((x^5-18 
*x^4+81*x^3)*log(x)*log(log(x))^2+(-10*x^4+90*x^3)*log(x)*log(log(x))+25*x 
^3*log(x)),x, algorithm="maxima")
 

Output:

-(50*x^2 + (x^4 - 9*x^3 + 89*x^2 + e^4)*log(log(x)))/(5*x^2 - (x^3 - 9*x^2 
)*log(log(x)))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (25) = 50\).

Time = 0.16 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.79 \[ \int \frac {-5 e^4+5 x^2-5 x^3-5 x^4+\left (10 e^4-5 x^3-10 x^4\right ) \log (x) \log (\log (x))+\left (e^4 (18-3 x)-8 x^3-18 x^4+x^5\right ) \log (x) \log ^2(\log (x))}{25 x^3 \log (x)+\left (90 x^3-10 x^4\right ) \log (x) \log (\log (x))+\left (81 x^3-18 x^4+x^5\right ) \log (x) \log ^2(\log (x))} \, dx=\frac {x^{4} \log \left (\log \left (x\right )\right ) - 9 \, x^{3} \log \left (\log \left (x\right )\right ) + 89 \, x^{2} \log \left (\log \left (x\right )\right ) + 50 \, x^{2} + e^{4} \log \left (\log \left (x\right )\right )}{x^{3} \log \left (\log \left (x\right )\right ) - 9 \, x^{2} \log \left (\log \left (x\right )\right ) - 5 \, x^{2}} \] Input:

integrate((((-3*x+18)*exp(4)+x^5-18*x^4-8*x^3)*log(x)*log(log(x))^2+(10*ex 
p(4)-10*x^4-5*x^3)*log(x)*log(log(x))-5*exp(4)-5*x^4-5*x^3+5*x^2)/((x^5-18 
*x^4+81*x^3)*log(x)*log(log(x))^2+(-10*x^4+90*x^3)*log(x)*log(log(x))+25*x 
^3*log(x)),x, algorithm="giac")
 

Output:

(x^4*log(log(x)) - 9*x^3*log(log(x)) + 89*x^2*log(log(x)) + 50*x^2 + e^4*l 
og(log(x)))/(x^3*log(log(x)) - 9*x^2*log(log(x)) - 5*x^2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {-5 e^4+5 x^2-5 x^3-5 x^4+\left (10 e^4-5 x^3-10 x^4\right ) \log (x) \log (\log (x))+\left (e^4 (18-3 x)-8 x^3-18 x^4+x^5\right ) \log (x) \log ^2(\log (x))}{25 x^3 \log (x)+\left (90 x^3-10 x^4\right ) \log (x) \log (\log (x))+\left (81 x^3-18 x^4+x^5\right ) \log (x) \log ^2(\log (x))} \, dx=\int -\frac {5\,{\mathrm {e}}^4-5\,x^2+5\,x^3+5\,x^4+\ln \left (\ln \left (x\right )\right )\,\ln \left (x\right )\,\left (10\,x^4+5\,x^3-10\,{\mathrm {e}}^4\right )+{\ln \left (\ln \left (x\right )\right )}^2\,\ln \left (x\right )\,\left (8\,x^3+18\,x^4-x^5+{\mathrm {e}}^4\,\left (3\,x-18\right )\right )}{25\,x^3\,\ln \left (x\right )+\ln \left (\ln \left (x\right )\right )\,\ln \left (x\right )\,\left (90\,x^3-10\,x^4\right )+{\ln \left (\ln \left (x\right )\right )}^2\,\ln \left (x\right )\,\left (x^5-18\,x^4+81\,x^3\right )} \,d x \] Input:

int(-(5*exp(4) - 5*x^2 + 5*x^3 + 5*x^4 + log(log(x))*log(x)*(5*x^3 - 10*ex 
p(4) + 10*x^4) + log(log(x))^2*log(x)*(8*x^3 + 18*x^4 - x^5 + exp(4)*(3*x 
- 18)))/(25*x^3*log(x) + log(log(x))*log(x)*(90*x^3 - 10*x^4) + log(log(x) 
)^2*log(x)*(81*x^3 - 18*x^4 + x^5)),x)
 

Output:

int(-(5*exp(4) - 5*x^2 + 5*x^3 + 5*x^4 + log(log(x))*log(x)*(5*x^3 - 10*ex 
p(4) + 10*x^4) + log(log(x))^2*log(x)*(8*x^3 + 18*x^4 - x^5 + exp(4)*(3*x 
- 18)))/(25*x^3*log(x) + log(log(x))*log(x)*(90*x^3 - 10*x^4) + log(log(x) 
)^2*log(x)*(81*x^3 - 18*x^4 + x^5)), x)
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09 \[ \int \frac {-5 e^4+5 x^2-5 x^3-5 x^4+\left (10 e^4-5 x^3-10 x^4\right ) \log (x) \log (\log (x))+\left (e^4 (18-3 x)-8 x^3-18 x^4+x^5\right ) \log (x) \log ^2(\log (x))}{25 x^3 \log (x)+\left (90 x^3-10 x^4\right ) \log (x) \log (\log (x))+\left (81 x^3-18 x^4+x^5\right ) \log (x) \log ^2(\log (x))} \, dx=\frac {\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) \left (e^{4}+x^{4}+x^{3}-x^{2}\right )}{x^{2} \left (\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) x -9 \,\mathrm {log}\left (\mathrm {log}\left (x \right )\right )-5\right )} \] Input:

int((((-3*x+18)*exp(4)+x^5-18*x^4-8*x^3)*log(x)*log(log(x))^2+(10*exp(4)-1 
0*x^4-5*x^3)*log(x)*log(log(x))-5*exp(4)-5*x^4-5*x^3+5*x^2)/((x^5-18*x^4+8 
1*x^3)*log(x)*log(log(x))^2+(-10*x^4+90*x^3)*log(x)*log(log(x))+25*x^3*log 
(x)),x)
 

Output:

(log(log(x))*(e**4 + x**4 + x**3 - x**2))/(x**2*(log(log(x))*x - 9*log(log 
(x)) - 5))