\(\int \frac {4+(-4 x+2 e^5 x^3-6 e^8 x^5) \log (x)+(-2 e^5 x^2+6 e^8 x^4) \log (x) \log (\log (x))+(2 x \log (x)-2 \log (x) \log (\log (x))) \log (x^2-2 x \log (\log (x))+\log ^2(\log (x)))}{(-e^{10} x^5+2 e^{13} x^7-e^{16} x^9) \log (x)+(e^{10} x^4-2 e^{13} x^6+e^{16} x^8) \log (x) \log (\log (x))+((2 e^5 x^3-2 e^8 x^5) \log (x)+(-2 e^5 x^2+2 e^8 x^4) \log (x) \log (\log (x))) \log (x^2-2 x \log (\log (x))+\log ^2(\log (x)))+(-x \log (x)+\log (x) \log (\log (x))) \log ^2(x^2-2 x \log (\log (x))+\log ^2(\log (x)))} \, dx\) [1123]

Optimal result

Integrand size = 231, antiderivative size = 34 \[ \int \frac {4+\left (-4 x+2 e^5 x^3-6 e^8 x^5\right ) \log (x)+\left (-2 e^5 x^2+6 e^8 x^4\right ) \log (x) \log (\log (x))+(2 x \log (x)-2 \log (x) \log (\log (x))) \log \left (x^2-2 x \log (\log (x))+\log ^2(\log (x))\right )}{\left (-e^{10} x^5+2 e^{13} x^7-e^{16} x^9\right ) \log (x)+\left (e^{10} x^4-2 e^{13} x^6+e^{16} x^8\right ) \log (x) \log (\log (x))+\left (\left (2 e^5 x^3-2 e^8 x^5\right ) \log (x)+\left (-2 e^5 x^2+2 e^8 x^4\right ) \log (x) \log (\log (x))\right ) \log \left (x^2-2 x \log (\log (x))+\log ^2(\log (x))\right )+(-x \log (x)+\log (x) \log (\log (x))) \log ^2\left (x^2-2 x \log (\log (x))+\log ^2(\log (x))\right )} \, dx=\frac {2 x}{x^2 \left (e^5-e^8 x^2\right )-\log \left ((x-\log (\log (x)))^2\right )} \] Output:

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.91 \[ \int \frac {4+\left (-4 x+2 e^5 x^3-6 e^8 x^5\right ) \log (x)+\left (-2 e^5 x^2+6 e^8 x^4\right ) \log (x) \log (\log (x))+(2 x \log (x)-2 \log (x) \log (\log (x))) \log \left (x^2-2 x \log (\log (x))+\log ^2(\log (x))\right )}{\left (-e^{10} x^5+2 e^{13} x^7-e^{16} x^9\right ) \log (x)+\left (e^{10} x^4-2 e^{13} x^6+e^{16} x^8\right ) \log (x) \log (\log (x))+\left (\left (2 e^5 x^3-2 e^8 x^5\right ) \log (x)+\left (-2 e^5 x^2+2 e^8 x^4\right ) \log (x) \log (\log (x))\right ) \log \left (x^2-2 x \log (\log (x))+\log ^2(\log (x))\right )+(-x \log (x)+\log (x) \log (\log (x))) \log ^2\left (x^2-2 x \log (\log (x))+\log ^2(\log (x))\right )} \, dx=-\frac {2 x}{-e^5 x^2+e^8 x^4+\log \left ((x-\log (\log (x)))^2\right )} \] Input:

Integrate[(4 + (-4*x + 2*E^5*x^3 - 6*E^8*x^5)*Log[x] + (-2*E^5*x^2 + 6*E^8 
*x^4)*Log[x]*Log[Log[x]] + (2*x*Log[x] - 2*Log[x]*Log[Log[x]])*Log[x^2 - 2 
*x*Log[Log[x]] + Log[Log[x]]^2])/((-(E^10*x^5) + 2*E^13*x^7 - E^16*x^9)*Lo 
g[x] + (E^10*x^4 - 2*E^13*x^6 + E^16*x^8)*Log[x]*Log[Log[x]] + ((2*E^5*x^3 
 - 2*E^8*x^5)*Log[x] + (-2*E^5*x^2 + 2*E^8*x^4)*Log[x]*Log[Log[x]])*Log[x^ 
2 - 2*x*Log[Log[x]] + Log[Log[x]]^2] + (-(x*Log[x]) + Log[x]*Log[Log[x]])* 
Log[x^2 - 2*x*Log[Log[x]] + Log[Log[x]]^2]^2),x]
 

Output:

(-2*x)/(-(E^5*x^2) + E^8*x^4 + Log[(x - Log[Log[x]])^2])
 

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.

\(\displaystyle \int \frac {(2 x \log (x)-2 \log (x) \log (\log (x))) \log \left (x^2+\log ^2(\log (x))-2 x \log (\log (x))\right )+\left (-6 e^8 x^5+2 e^5 x^3-4 x\right ) \log (x)+\left (6 e^8 x^4-2 e^5 x^2\right ) \log (\log (x)) \log (x)+4}{(\log (x) \log (\log (x))-x \log (x)) \log ^2\left (x^2+\log ^2(\log (x))-2 x \log (\log (x))\right )+\left (-e^{16} x^9+2 e^{13} x^7-e^{10} x^5\right ) \log (x)+\left (e^{16} x^8-2 e^{13} x^6+e^{10} x^4\right ) \log (x) \log (\log (x))+\left (\left (2 e^5 x^3-2 e^8 x^5\right ) \log (x)+\left (2 e^8 x^4-2 e^5 x^2\right ) \log (\log (x)) \log (x)\right ) \log \left (x^2+\log ^2(\log (x))-2 x \log (\log (x))\right )} \, dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {4 \left (2 e^8 x^5 \log (x)-2 e^8 x^4 \log (x) \log (\log (x))-e^5 x^3 \log (x)+e^5 x^2 \log (x) \log (\log (x))+x \log (x)-1\right )}{\log (x) (x-\log (\log (x))) \left (e^8 x^4-e^5 x^2+\log \left ((x-\log (\log (x)))^2\right )\right )^2}-\frac {2}{e^8 x^4-e^5 x^2+\log \left ((x-\log (\log (x)))^2\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 4 \int \frac {x}{(x-\log (\log (x))) \left (e^8 x^4-e^5 x^2+\log \left ((x-\log (\log (x)))^2\right )\right )^2}dx-4 \int \frac {1}{\log (x) (x-\log (\log (x))) \left (e^8 x^4-e^5 x^2+\log \left ((x-\log (\log (x)))^2\right )\right )^2}dx+4 e^5 \int \frac {x^2 \log (\log (x))}{(x-\log (\log (x))) \left (e^8 x^4-e^5 x^2+\log \left ((x-\log (\log (x)))^2\right )\right )^2}dx-8 e^8 \int \frac {x^4 \log (\log (x))}{(x-\log (\log (x))) \left (e^8 x^4-e^5 x^2+\log \left ((x-\log (\log (x)))^2\right )\right )^2}dx-2 \int \frac {1}{e^8 x^4-e^5 x^2+\log \left ((x-\log (\log (x)))^2\right )}dx+8 e^8 \int \frac {x^5}{(x-\log (\log (x))) \left (e^8 x^4-e^5 x^2+\log \left ((x-\log (\log (x)))^2\right )\right )^2}dx-4 e^5 \int \frac {x^3}{(x-\log (\log (x))) \left (e^8 x^4-e^5 x^2+\log \left ((x-\log (\log (x)))^2\right )\right )^2}dx\)

Input:

Int[(4 + (-4*x + 2*E^5*x^3 - 6*E^8*x^5)*Log[x] + (-2*E^5*x^2 + 6*E^8*x^4)* 
Log[x]*Log[Log[x]] + (2*x*Log[x] - 2*Log[x]*Log[Log[x]])*Log[x^2 - 2*x*Log 
[Log[x]] + Log[Log[x]]^2])/((-(E^10*x^5) + 2*E^13*x^7 - E^16*x^9)*Log[x] + 
 (E^10*x^4 - 2*E^13*x^6 + E^16*x^8)*Log[x]*Log[Log[x]] + ((2*E^5*x^3 - 2*E 
^8*x^5)*Log[x] + (-2*E^5*x^2 + 2*E^8*x^4)*Log[x]*Log[Log[x]])*Log[x^2 - 2* 
x*Log[Log[x]] + Log[Log[x]]^2] + (-(x*Log[x]) + Log[x]*Log[Log[x]])*Log[x^ 
2 - 2*x*Log[Log[x]] + Log[Log[x]]^2]^2),x]
 

Output:

$Aborted
 

Maple [C] (warning: unable to verify)

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

Time = 1.27 (sec) , antiderivative size = 387, normalized size of antiderivative = 11.38

\[\frac {4 x}{\left (2 \,{\mathrm e}^{8} x^{5} \ln \left (x \right )-2 \,{\mathrm e}^{8} x^{4} \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )-x^{3} {\mathrm e}^{5} \ln \left (x \right )+{\mathrm e}^{5} \ln \left (x \right ) \ln \left (\ln \left (x \right )\right ) x^{2}+x \ln \left (x \right )-1\right ) \left (2 x^{4} {\mathrm e}^{8}-2 x^{2} {\mathrm e}^{5}-i \pi \operatorname {csgn}\left (i \left (x -\ln \left (\ln \left (x \right )\right )\right )\right )^{2} \operatorname {csgn}\left (i \left (x -\ln \left (\ln \left (x \right )\right )\right )^{2}\right )+2 i \pi \,\operatorname {csgn}\left (i \left (x -\ln \left (\ln \left (x \right )\right )\right )\right ) \operatorname {csgn}\left (i \left (x -\ln \left (\ln \left (x \right )\right )\right )^{2}\right )^{2}-i \pi \operatorname {csgn}\left (i \left (x -\ln \left (\ln \left (x \right )\right )\right )^{2}\right )^{3}+4 \ln \left (x -\ln \left (\ln \left (x \right )\right )\right )\right )}-\frac {4 \ln \left (x \right ) x^{2} \left (2 x^{4} {\mathrm e}^{8}-2 x^{3} \ln \left (\ln \left (x \right )\right ) {\mathrm e}^{8}-x^{2} {\mathrm e}^{5}+x \,{\mathrm e}^{5} \ln \left (\ln \left (x \right )\right )+1\right )}{\left (2 \,{\mathrm e}^{8} x^{5} \ln \left (x \right )-2 \,{\mathrm e}^{8} x^{4} \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )-x^{3} {\mathrm e}^{5} \ln \left (x \right )+{\mathrm e}^{5} \ln \left (x \right ) \ln \left (\ln \left (x \right )\right ) x^{2}+x \ln \left (x \right )-1\right ) \left (2 x^{4} {\mathrm e}^{8}-2 x^{2} {\mathrm e}^{5}-i \pi \operatorname {csgn}\left (i \left (x -\ln \left (\ln \left (x \right )\right )\right )\right )^{2} \operatorname {csgn}\left (i \left (x -\ln \left (\ln \left (x \right )\right )\right )^{2}\right )+2 i \pi \,\operatorname {csgn}\left (i \left (x -\ln \left (\ln \left (x \right )\right )\right )\right ) \operatorname {csgn}\left (i \left (x -\ln \left (\ln \left (x \right )\right )\right )^{2}\right )^{2}-i \pi \operatorname {csgn}\left (i \left (x -\ln \left (\ln \left (x \right )\right )\right )^{2}\right )^{3}+4 \ln \left (x -\ln \left (\ln \left (x \right )\right )\right )\right )}\]

Input:

int(((-2*ln(x)*ln(ln(x))+2*x*ln(x))*ln(ln(ln(x))^2-2*x*ln(ln(x))+x^2)+(-2* 
x^2*exp(5)+6*x^4*exp(4)^2)*ln(x)*ln(ln(x))+(2*x^3*exp(5)-6*x^5*exp(4)^2-4* 
x)*ln(x)+4)/((ln(x)*ln(ln(x))-x*ln(x))*ln(ln(ln(x))^2-2*x*ln(ln(x))+x^2)^2 
+((-2*x^2*exp(5)+2*x^4*exp(4)^2)*ln(x)*ln(ln(x))+(2*x^3*exp(5)-2*x^5*exp(4 
)^2)*ln(x))*ln(ln(ln(x))^2-2*x*ln(ln(x))+x^2)+(x^4*exp(5)^2-2*x^6*exp(4)^2 
*exp(5)+x^8*exp(4)^4)*ln(x)*ln(ln(x))+(-x^5*exp(5)^2+2*x^7*exp(4)^2*exp(5) 
-x^9*exp(4)^4)*ln(x)),x)
 

Output:

4/(2*exp(1)^8*x^5*ln(x)-2*exp(1)^8*x^4*ln(x)*ln(ln(x))-exp(1)^5*x^3*ln(x)+ 
exp(1)^5*x^2*ln(x)*ln(ln(x))+x*ln(x)-1)*x/(2*exp(1)^8*x^4-2*exp(1)^5*x^2-I 
*Pi*csgn(I*(x-ln(ln(x))))^2*csgn(I*(x-ln(ln(x)))^2)+2*I*Pi*csgn(I*(x-ln(ln 
(x))))*csgn(I*(x-ln(ln(x)))^2)^2-I*Pi*csgn(I*(x-ln(ln(x)))^2)^3+4*ln(x-ln( 
ln(x))))-4/(2*exp(1)^8*x^5*ln(x)-2*exp(1)^8*x^4*ln(x)*ln(ln(x))-exp(1)^5*x 
^3*ln(x)+exp(1)^5*x^2*ln(x)*ln(ln(x))+x*ln(x)-1)*ln(x)*x^2*(2*exp(1)^8*x^4 
-2*x^3*ln(ln(x))*exp(1)^8-exp(1)^5*x^2+x*exp(1)^5*ln(ln(x))+1)/(2*exp(1)^8 
*x^4-2*exp(1)^5*x^2-I*Pi*csgn(I*(x-ln(ln(x))))^2*csgn(I*(x-ln(ln(x)))^2)+2 
*I*Pi*csgn(I*(x-ln(ln(x))))*csgn(I*(x-ln(ln(x)))^2)^2-I*Pi*csgn(I*(x-ln(ln 
(x)))^2)^3+4*ln(x-ln(ln(x))))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.03 \[ \int \frac {4+\left (-4 x+2 e^5 x^3-6 e^8 x^5\right ) \log (x)+\left (-2 e^5 x^2+6 e^8 x^4\right ) \log (x) \log (\log (x))+(2 x \log (x)-2 \log (x) \log (\log (x))) \log \left (x^2-2 x \log (\log (x))+\log ^2(\log (x))\right )}{\left (-e^{10} x^5+2 e^{13} x^7-e^{16} x^9\right ) \log (x)+\left (e^{10} x^4-2 e^{13} x^6+e^{16} x^8\right ) \log (x) \log (\log (x))+\left (\left (2 e^5 x^3-2 e^8 x^5\right ) \log (x)+\left (-2 e^5 x^2+2 e^8 x^4\right ) \log (x) \log (\log (x))\right ) \log \left (x^2-2 x \log (\log (x))+\log ^2(\log (x))\right )+(-x \log (x)+\log (x) \log (\log (x))) \log ^2\left (x^2-2 x \log (\log (x))+\log ^2(\log (x))\right )} \, dx=-\frac {2 \, x}{x^{4} e^{8} - x^{2} e^{5} + \log \left (x^{2} - 2 \, x \log \left (\log \left (x\right )\right ) + \log \left (\log \left (x\right )\right )^{2}\right )} \] Input:

integrate(((-2*log(x)*log(log(x))+2*x*log(x))*log(log(log(x))^2-2*x*log(lo 
g(x))+x^2)+(-2*x^2*exp(5)+6*x^4*exp(4)^2)*log(x)*log(log(x))+(2*x^3*exp(5) 
-6*x^5*exp(4)^2-4*x)*log(x)+4)/((log(x)*log(log(x))-x*log(x))*log(log(log( 
x))^2-2*x*log(log(x))+x^2)^2+((-2*x^2*exp(5)+2*x^4*exp(4)^2)*log(x)*log(lo 
g(x))+(2*x^3*exp(5)-2*x^5*exp(4)^2)*log(x))*log(log(log(x))^2-2*x*log(log( 
x))+x^2)+(x^4*exp(5)^2-2*x^6*exp(4)^2*exp(5)+x^8*exp(4)^4)*log(x)*log(log( 
x))+(-x^5*exp(5)^2+2*x^7*exp(4)^2*exp(5)-x^9*exp(4)^4)*log(x)),x, algorith 
m="fricas")
 

Output:

-2*x/(x^4*e^8 - x^2*e^5 + log(x^2 - 2*x*log(log(x)) + log(log(x))^2))
 

Sympy [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.09 \[ \int \frac {4+\left (-4 x+2 e^5 x^3-6 e^8 x^5\right ) \log (x)+\left (-2 e^5 x^2+6 e^8 x^4\right ) \log (x) \log (\log (x))+(2 x \log (x)-2 \log (x) \log (\log (x))) \log \left (x^2-2 x \log (\log (x))+\log ^2(\log (x))\right )}{\left (-e^{10} x^5+2 e^{13} x^7-e^{16} x^9\right ) \log (x)+\left (e^{10} x^4-2 e^{13} x^6+e^{16} x^8\right ) \log (x) \log (\log (x))+\left (\left (2 e^5 x^3-2 e^8 x^5\right ) \log (x)+\left (-2 e^5 x^2+2 e^8 x^4\right ) \log (x) \log (\log (x))\right ) \log \left (x^2-2 x \log (\log (x))+\log ^2(\log (x))\right )+(-x \log (x)+\log (x) \log (\log (x))) \log ^2\left (x^2-2 x \log (\log (x))+\log ^2(\log (x))\right )} \, dx=- \frac {2 x}{x^{4} e^{8} - x^{2} e^{5} + \log {\left (x^{2} - 2 x \log {\left (\log {\left (x \right )} \right )} + \log {\left (\log {\left (x \right )} \right )}^{2} \right )}} \] Input:

integrate(((-2*ln(x)*ln(ln(x))+2*x*ln(x))*ln(ln(ln(x))**2-2*x*ln(ln(x))+x* 
*2)+(-2*x**2*exp(5)+6*x**4*exp(4)**2)*ln(x)*ln(ln(x))+(2*x**3*exp(5)-6*x** 
5*exp(4)**2-4*x)*ln(x)+4)/((ln(x)*ln(ln(x))-x*ln(x))*ln(ln(ln(x))**2-2*x*l 
n(ln(x))+x**2)**2+((-2*x**2*exp(5)+2*x**4*exp(4)**2)*ln(x)*ln(ln(x))+(2*x* 
*3*exp(5)-2*x**5*exp(4)**2)*ln(x))*ln(ln(ln(x))**2-2*x*ln(ln(x))+x**2)+(x* 
*4*exp(5)**2-2*x**6*exp(4)**2*exp(5)+x**8*exp(4)**4)*ln(x)*ln(ln(x))+(-x** 
5*exp(5)**2+2*x**7*exp(4)**2*exp(5)-x**9*exp(4)**4)*ln(x)),x)
 

Output:

-2*x/(x**4*exp(8) - x**2*exp(5) + log(x**2 - 2*x*log(log(x)) + log(log(x)) 
**2))
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.85 \[ \int \frac {4+\left (-4 x+2 e^5 x^3-6 e^8 x^5\right ) \log (x)+\left (-2 e^5 x^2+6 e^8 x^4\right ) \log (x) \log (\log (x))+(2 x \log (x)-2 \log (x) \log (\log (x))) \log \left (x^2-2 x \log (\log (x))+\log ^2(\log (x))\right )}{\left (-e^{10} x^5+2 e^{13} x^7-e^{16} x^9\right ) \log (x)+\left (e^{10} x^4-2 e^{13} x^6+e^{16} x^8\right ) \log (x) \log (\log (x))+\left (\left (2 e^5 x^3-2 e^8 x^5\right ) \log (x)+\left (-2 e^5 x^2+2 e^8 x^4\right ) \log (x) \log (\log (x))\right ) \log \left (x^2-2 x \log (\log (x))+\log ^2(\log (x))\right )+(-x \log (x)+\log (x) \log (\log (x))) \log ^2\left (x^2-2 x \log (\log (x))+\log ^2(\log (x))\right )} \, dx=-\frac {2 \, x}{x^{4} e^{8} - x^{2} e^{5} + 2 \, \log \left (-x + \log \left (\log \left (x\right )\right )\right )} \] Input:

integrate(((-2*log(x)*log(log(x))+2*x*log(x))*log(log(log(x))^2-2*x*log(lo 
g(x))+x^2)+(-2*x^2*exp(5)+6*x^4*exp(4)^2)*log(x)*log(log(x))+(2*x^3*exp(5) 
-6*x^5*exp(4)^2-4*x)*log(x)+4)/((log(x)*log(log(x))-x*log(x))*log(log(log( 
x))^2-2*x*log(log(x))+x^2)^2+((-2*x^2*exp(5)+2*x^4*exp(4)^2)*log(x)*log(lo 
g(x))+(2*x^3*exp(5)-2*x^5*exp(4)^2)*log(x))*log(log(log(x))^2-2*x*log(log( 
x))+x^2)+(x^4*exp(5)^2-2*x^6*exp(4)^2*exp(5)+x^8*exp(4)^4)*log(x)*log(log( 
x))+(-x^5*exp(5)^2+2*x^7*exp(4)^2*exp(5)-x^9*exp(4)^4)*log(x)),x, algorith 
m="maxima")
 

Output:

-2*x/(x^4*e^8 - x^2*e^5 + 2*log(-x + log(log(x))))
 

Giac [A] (verification not implemented)

Time = 81.86 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.03 \[ \int \frac {4+\left (-4 x+2 e^5 x^3-6 e^8 x^5\right ) \log (x)+\left (-2 e^5 x^2+6 e^8 x^4\right ) \log (x) \log (\log (x))+(2 x \log (x)-2 \log (x) \log (\log (x))) \log \left (x^2-2 x \log (\log (x))+\log ^2(\log (x))\right )}{\left (-e^{10} x^5+2 e^{13} x^7-e^{16} x^9\right ) \log (x)+\left (e^{10} x^4-2 e^{13} x^6+e^{16} x^8\right ) \log (x) \log (\log (x))+\left (\left (2 e^5 x^3-2 e^8 x^5\right ) \log (x)+\left (-2 e^5 x^2+2 e^8 x^4\right ) \log (x) \log (\log (x))\right ) \log \left (x^2-2 x \log (\log (x))+\log ^2(\log (x))\right )+(-x \log (x)+\log (x) \log (\log (x))) \log ^2\left (x^2-2 x \log (\log (x))+\log ^2(\log (x))\right )} \, dx=-\frac {2 \, x}{x^{4} e^{8} - x^{2} e^{5} + \log \left (x^{2} - 2 \, x \log \left (\log \left (x\right )\right ) + \log \left (\log \left (x\right )\right )^{2}\right )} \] Input:

integrate(((-2*log(x)*log(log(x))+2*x*log(x))*log(log(log(x))^2-2*x*log(lo 
g(x))+x^2)+(-2*x^2*exp(5)+6*x^4*exp(4)^2)*log(x)*log(log(x))+(2*x^3*exp(5) 
-6*x^5*exp(4)^2-4*x)*log(x)+4)/((log(x)*log(log(x))-x*log(x))*log(log(log( 
x))^2-2*x*log(log(x))+x^2)^2+((-2*x^2*exp(5)+2*x^4*exp(4)^2)*log(x)*log(lo 
g(x))+(2*x^3*exp(5)-2*x^5*exp(4)^2)*log(x))*log(log(log(x))^2-2*x*log(log( 
x))+x^2)+(x^4*exp(5)^2-2*x^6*exp(4)^2*exp(5)+x^8*exp(4)^4)*log(x)*log(log( 
x))+(-x^5*exp(5)^2+2*x^7*exp(4)^2*exp(5)-x^9*exp(4)^4)*log(x)),x, algorith 
m="giac")
 

Output:

-2*x/(x^4*e^8 - x^2*e^5 + log(x^2 - 2*x*log(log(x)) + log(log(x))^2))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {4+\left (-4 x+2 e^5 x^3-6 e^8 x^5\right ) \log (x)+\left (-2 e^5 x^2+6 e^8 x^4\right ) \log (x) \log (\log (x))+(2 x \log (x)-2 \log (x) \log (\log (x))) \log \left (x^2-2 x \log (\log (x))+\log ^2(\log (x))\right )}{\left (-e^{10} x^5+2 e^{13} x^7-e^{16} x^9\right ) \log (x)+\left (e^{10} x^4-2 e^{13} x^6+e^{16} x^8\right ) \log (x) \log (\log (x))+\left (\left (2 e^5 x^3-2 e^8 x^5\right ) \log (x)+\left (-2 e^5 x^2+2 e^8 x^4\right ) \log (x) \log (\log (x))\right ) \log \left (x^2-2 x \log (\log (x))+\log ^2(\log (x))\right )+(-x \log (x)+\log (x) \log (\log (x))) \log ^2\left (x^2-2 x \log (\log (x))+\log ^2(\log (x))\right )} \, dx=\int -\frac {\ln \left (x^2-2\,x\,\ln \left (\ln \left (x\right )\right )+{\ln \left (\ln \left (x\right )\right )}^2\right )\,\left (2\,\ln \left (\ln \left (x\right )\right )\,\ln \left (x\right )-2\,x\,\ln \left (x\right )\right )+\ln \left (x\right )\,\left (6\,{\mathrm {e}}^8\,x^5-2\,{\mathrm {e}}^5\,x^3+4\,x\right )+\ln \left (\ln \left (x\right )\right )\,\ln \left (x\right )\,\left (2\,x^2\,{\mathrm {e}}^5-6\,x^4\,{\mathrm {e}}^8\right )-4}{\left (\ln \left (\ln \left (x\right )\right )\,\ln \left (x\right )-x\,\ln \left (x\right )\right )\,{\ln \left (x^2-2\,x\,\ln \left (\ln \left (x\right )\right )+{\ln \left (\ln \left (x\right )\right )}^2\right )}^2+\left (\ln \left (x\right )\,\left (2\,x^3\,{\mathrm {e}}^5-2\,x^5\,{\mathrm {e}}^8\right )-\ln \left (\ln \left (x\right )\right )\,\ln \left (x\right )\,\left (2\,x^2\,{\mathrm {e}}^5-2\,x^4\,{\mathrm {e}}^8\right )\right )\,\ln \left (x^2-2\,x\,\ln \left (\ln \left (x\right )\right )+{\ln \left (\ln \left (x\right )\right )}^2\right )-\ln \left (x\right )\,\left ({\mathrm {e}}^{16}\,x^9-2\,{\mathrm {e}}^{13}\,x^7+{\mathrm {e}}^{10}\,x^5\right )+\ln \left (\ln \left (x\right )\right )\,\ln \left (x\right )\,\left ({\mathrm {e}}^{16}\,x^8-2\,{\mathrm {e}}^{13}\,x^6+{\mathrm {e}}^{10}\,x^4\right )} \,d x \] Input:

int(-(log(log(log(x))^2 - 2*x*log(log(x)) + x^2)*(2*log(log(x))*log(x) - 2 
*x*log(x)) + log(x)*(4*x - 2*x^3*exp(5) + 6*x^5*exp(8)) + log(log(x))*log( 
x)*(2*x^2*exp(5) - 6*x^4*exp(8)) - 4)/(log(log(log(x))^2 - 2*x*log(log(x)) 
 + x^2)*(log(x)*(2*x^3*exp(5) - 2*x^5*exp(8)) - log(log(x))*log(x)*(2*x^2* 
exp(5) - 2*x^4*exp(8))) - log(x)*(x^5*exp(10) - 2*x^7*exp(13) + x^9*exp(16 
)) + log(log(log(x))^2 - 2*x*log(log(x)) + x^2)^2*(log(log(x))*log(x) - x* 
log(x)) + log(log(x))*log(x)*(x^4*exp(10) - 2*x^6*exp(13) + x^8*exp(16))), 
x)
 

Output:

int(-(log(log(log(x))^2 - 2*x*log(log(x)) + x^2)*(2*log(log(x))*log(x) - 2 
*x*log(x)) + log(x)*(4*x - 2*x^3*exp(5) + 6*x^5*exp(8)) + log(log(x))*log( 
x)*(2*x^2*exp(5) - 6*x^4*exp(8)) - 4)/(log(log(log(x))^2 - 2*x*log(log(x)) 
 + x^2)*(log(x)*(2*x^3*exp(5) - 2*x^5*exp(8)) - log(log(x))*log(x)*(2*x^2* 
exp(5) - 2*x^4*exp(8))) - log(x)*(x^5*exp(10) - 2*x^7*exp(13) + x^9*exp(16 
)) + log(log(log(x))^2 - 2*x*log(log(x)) + x^2)^2*(log(log(x))*log(x) - x* 
log(x)) + log(log(x))*log(x)*(x^4*exp(10) - 2*x^6*exp(13) + x^8*exp(16))), 
 x)
 

Reduce [F]

\[ \int \frac {4+\left (-4 x+2 e^5 x^3-6 e^8 x^5\right ) \log (x)+\left (-2 e^5 x^2+6 e^8 x^4\right ) \log (x) \log (\log (x))+(2 x \log (x)-2 \log (x) \log (\log (x))) \log \left (x^2-2 x \log (\log (x))+\log ^2(\log (x))\right )}{\left (-e^{10} x^5+2 e^{13} x^7-e^{16} x^9\right ) \log (x)+\left (e^{10} x^4-2 e^{13} x^6+e^{16} x^8\right ) \log (x) \log (\log (x))+\left (\left (2 e^5 x^3-2 e^8 x^5\right ) \log (x)+\left (-2 e^5 x^2+2 e^8 x^4\right ) \log (x) \log (\log (x))\right ) \log \left (x^2-2 x \log (\log (x))+\log ^2(\log (x))\right )+(-x \log (x)+\log (x) \log (\log (x))) \log ^2\left (x^2-2 x \log (\log (x))+\log ^2(\log (x))\right )} \, dx=\int \frac {\left (-2 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (\mathrm {log}\left (x \right )\right )+2 \,\mathrm {log}\left (x \right ) x \right ) \mathrm {log}\left (\mathrm {log}\left (\mathrm {log}\left (x \right )\right )^{2}-2 \,\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) x +x^{2}\right )+\left (-2 x^{2} {\mathrm e}^{5}+6 x^{4} \left ({\mathrm e}^{4}\right )^{2}\right ) \mathrm {log}\left (x \right ) \mathrm {log}\left (\mathrm {log}\left (x \right )\right )+\left (2 x^{3} {\mathrm e}^{5}-6 x^{5} \left ({\mathrm e}^{4}\right )^{2}-4 x \right ) \mathrm {log}\left (x \right )+4}{\left (\mathrm {log}\left (x \right ) \mathrm {log}\left (\mathrm {log}\left (x \right )\right )-\mathrm {log}\left (x \right ) x \right ) \mathrm {log}\left (\mathrm {log}\left (\mathrm {log}\left (x \right )\right )^{2}-2 \,\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) x +x^{2}\right )^{2}+\left (\left (-2 x^{2} {\mathrm e}^{5}+2 x^{4} \left ({\mathrm e}^{4}\right )^{2}\right ) \mathrm {log}\left (x \right ) \mathrm {log}\left (\mathrm {log}\left (x \right )\right )+\left (2 x^{3} {\mathrm e}^{5}-2 x^{5} \left ({\mathrm e}^{4}\right )^{2}\right ) \mathrm {log}\left (x \right )\right ) \mathrm {log}\left (\mathrm {log}\left (\mathrm {log}\left (x \right )\right )^{2}-2 \,\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) x +x^{2}\right )+\left (x^{4} \left ({\mathrm e}^{5}\right )^{2}-2 x^{6} \left ({\mathrm e}^{4}\right )^{2} {\mathrm e}^{5}+x^{8} \left ({\mathrm e}^{4}\right )^{4}\right ) \mathrm {log}\left (x \right ) \mathrm {log}\left (\mathrm {log}\left (x \right )\right )+\left (-x^{5} \left ({\mathrm e}^{5}\right )^{2}+2 x^{7} \left ({\mathrm e}^{4}\right )^{2} {\mathrm e}^{5}-x^{9} \left ({\mathrm e}^{4}\right )^{4}\right ) \mathrm {log}\left (x \right )}d x \] Input:

int(((-2*log(x)*log(log(x))+2*x*log(x))*log(log(log(x))^2-2*x*log(log(x))+ 
x^2)+(-2*x^2*exp(5)+6*x^4*exp(4)^2)*log(x)*log(log(x))+(2*x^3*exp(5)-6*x^5 
*exp(4)^2-4*x)*log(x)+4)/((log(x)*log(log(x))-x*log(x))*log(log(log(x))^2- 
2*x*log(log(x))+x^2)^2+((-2*x^2*exp(5)+2*x^4*exp(4)^2)*log(x)*log(log(x))+ 
(2*x^3*exp(5)-2*x^5*exp(4)^2)*log(x))*log(log(log(x))^2-2*x*log(log(x))+x^ 
2)+(x^4*exp(5)^2-2*x^6*exp(4)^2*exp(5)+x^8*exp(4)^4)*log(x)*log(log(x))+(- 
x^5*exp(5)^2+2*x^7*exp(4)^2*exp(5)-x^9*exp(4)^4)*log(x)),x)
 

Output:

int(((-2*log(x)*log(log(x))+2*x*log(x))*log(log(log(x))^2-2*x*log(log(x))+ 
x^2)+(-2*x^2*exp(5)+6*x^4*exp(4)^2)*log(x)*log(log(x))+(2*x^3*exp(5)-6*x^5 
*exp(4)^2-4*x)*log(x)+4)/((log(x)*log(log(x))-x*log(x))*log(log(log(x))^2- 
2*x*log(log(x))+x^2)^2+((-2*x^2*exp(5)+2*x^4*exp(4)^2)*log(x)*log(log(x))+ 
(2*x^3*exp(5)-2*x^5*exp(4)^2)*log(x))*log(log(log(x))^2-2*x*log(log(x))+x^ 
2)+(x^4*exp(5)^2-2*x^6*exp(4)^2*exp(5)+x^8*exp(4)^4)*log(x)*log(log(x))+(- 
x^5*exp(5)^2+2*x^7*exp(4)^2*exp(5)-x^9*exp(4)^4)*log(x)),x)