\(\int \frac {8-2 x+((8-4 x) \log (x) \log (\frac {1}{\log (x)}) \log (\log (\frac {1}{\log (x)}))+(4-2 x) \log (3) \log (x) \log (\frac {1}{\log (x)}) \log ^2(\log (\frac {1}{\log (x)}))) \log (\frac {2+\log (3) \log (\log (\frac {1}{\log (x)}))}{\log (\log (\frac {1}{\log (x)}))})}{2 \log (x) \log (\frac {1}{\log (x)}) \log (\log (\frac {1}{\log (x)}))+\log (3) \log (x) \log (\frac {1}{\log (x)}) \log ^2(\log (\frac {1}{\log (x)}))} \, dx\) [1227]

Optimal result

Integrand size = 107, antiderivative size = 21 \[ \int \frac {8-2 x+\left ((8-4 x) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+(4-2 x) \log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )\right ) \log \left (\frac {2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right )}{2 \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+\log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )} \, dx=(4-x) x \log \left (\log (3)+\frac {2}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right ) \] Output:

x*ln(2/ln(ln(1/ln(x)))+ln(3))*(4-x)
 

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {8-2 x+\left ((8-4 x) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+(4-2 x) \log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )\right ) \log \left (\frac {2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right )}{2 \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+\log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )} \, dx=-\left ((-4+x) x \log \left (\log (3)+\frac {2}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right )\right ) \] Input:

Integrate[(8 - 2*x + ((8 - 4*x)*Log[x]*Log[Log[x]^(-1)]*Log[Log[Log[x]^(-1 
)]] + (4 - 2*x)*Log[3]*Log[x]*Log[Log[x]^(-1)]*Log[Log[Log[x]^(-1)]]^2)*Lo 
g[(2 + Log[3]*Log[Log[Log[x]^(-1)]])/Log[Log[Log[x]^(-1)]]])/(2*Log[x]*Log 
[Log[x]^(-1)]*Log[Log[Log[x]^(-1)]] + Log[3]*Log[x]*Log[Log[x]^(-1)]*Log[L 
og[Log[x]^(-1)]]^2),x]
 

Output:

-((-4 + x)*x*Log[Log[3] + 2/Log[Log[Log[x]^(-1)]]])
 

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.

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

Int[(8 - 2*x + ((8 - 4*x)*Log[x]*Log[Log[x]^(-1)]*Log[Log[Log[x]^(-1)]] + 
(4 - 2*x)*Log[3]*Log[x]*Log[Log[x]^(-1)]*Log[Log[Log[x]^(-1)]]^2)*Log[(2 + 
 Log[3]*Log[Log[Log[x]^(-1)]])/Log[Log[Log[x]^(-1)]]])/(2*Log[x]*Log[Log[x 
]^(-1)]*Log[Log[Log[x]^(-1)]] + Log[3]*Log[x]*Log[Log[x]^(-1)]*Log[Log[Log 
[x]^(-1)]]^2),x]
 

Output:

$Aborted
 

Maple [C] (warning: unable to verify)

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

Time = 0.24 (sec) , antiderivative size = 397, normalized size of antiderivative = 18.90

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

Input:

int((((4-2*x)*ln(3)*ln(x)*ln(1/ln(x))*ln(ln(1/ln(x)))^2+(-4*x+8)*ln(x)*ln( 
1/ln(x))*ln(ln(1/ln(x))))*ln((ln(3)*ln(ln(1/ln(x)))+2)/ln(ln(1/ln(x))))-2* 
x+8)/(ln(3)*ln(x)*ln(1/ln(x))*ln(ln(1/ln(x)))^2+2*ln(x)*ln(1/ln(x))*ln(ln( 
1/ln(x)))),x)
 

Output:

(-x^2+4*x)*ln(ln(3)*ln(-ln(ln(x)))+2)+x^2*ln(ln(-ln(ln(x))))-4*x*ln(ln(-ln 
(ln(x))))+1/2*I*Pi*x^2*csgn(I*(ln(3)*ln(-ln(ln(x)))+2))*csgn(I/ln(-ln(ln(x 
))))*csgn(I/ln(-ln(ln(x)))*(ln(3)*ln(-ln(ln(x)))+2))-1/2*I*Pi*x^2*csgn(I*( 
ln(3)*ln(-ln(ln(x)))+2))*csgn(I/ln(-ln(ln(x)))*(ln(3)*ln(-ln(ln(x)))+2))^2 
-1/2*I*Pi*x^2*csgn(I/ln(-ln(ln(x))))*csgn(I/ln(-ln(ln(x)))*(ln(3)*ln(-ln(l 
n(x)))+2))^2+1/2*I*Pi*x^2*csgn(I/ln(-ln(ln(x)))*(ln(3)*ln(-ln(ln(x)))+2))^ 
3-2*I*Pi*x*csgn(I*(ln(3)*ln(-ln(ln(x)))+2))*csgn(I/ln(-ln(ln(x))))*csgn(I/ 
ln(-ln(ln(x)))*(ln(3)*ln(-ln(ln(x)))+2))+2*I*Pi*x*csgn(I*(ln(3)*ln(-ln(ln( 
x)))+2))*csgn(I/ln(-ln(ln(x)))*(ln(3)*ln(-ln(ln(x)))+2))^2+2*I*Pi*x*csgn(I 
/ln(-ln(ln(x))))*csgn(I/ln(-ln(ln(x)))*(ln(3)*ln(-ln(ln(x)))+2))^2-2*I*Pi* 
x*csgn(I/ln(-ln(ln(x)))*(ln(3)*ln(-ln(ln(x)))+2))^3
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.43 \[ \int \frac {8-2 x+\left ((8-4 x) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+(4-2 x) \log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )\right ) \log \left (\frac {2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right )}{2 \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+\log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )} \, dx=-{\left (x^{2} - 4 \, x\right )} \log \left (\frac {\log \left (3\right ) \log \left (\log \left (\frac {1}{\log \left (x\right )}\right )\right ) + 2}{\log \left (\log \left (\frac {1}{\log \left (x\right )}\right )\right )}\right ) \] Input:

integrate((((4-2*x)*log(3)*log(x)*log(1/log(x))*log(log(1/log(x)))^2+(-4*x 
+8)*log(x)*log(1/log(x))*log(log(1/log(x))))*log((log(3)*log(log(1/log(x)) 
)+2)/log(log(1/log(x))))-2*x+8)/(log(3)*log(x)*log(1/log(x))*log(log(1/log 
(x)))^2+2*log(x)*log(1/log(x))*log(log(1/log(x)))),x, algorithm="fricas")
 

Output:

-(x^2 - 4*x)*log((log(3)*log(log(1/log(x))) + 2)/log(log(1/log(x))))
 

Sympy [A] (verification not implemented)

Time = 1.09 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.38 \[ \int \frac {8-2 x+\left ((8-4 x) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+(4-2 x) \log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )\right ) \log \left (\frac {2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right )}{2 \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+\log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )} \, dx=\left (- x^{2} + 4 x\right ) \log {\left (\frac {\log {\left (3 \right )} \log {\left (\log {\left (\frac {1}{\log {\left (x \right )}} \right )} \right )} + 2}{\log {\left (\log {\left (\frac {1}{\log {\left (x \right )}} \right )} \right )}} \right )} \] Input:

integrate((((4-2*x)*ln(3)*ln(x)*ln(1/ln(x))*ln(ln(1/ln(x)))**2+(-4*x+8)*ln 
(x)*ln(1/ln(x))*ln(ln(1/ln(x))))*ln((ln(3)*ln(ln(1/ln(x)))+2)/ln(ln(1/ln(x 
))))-2*x+8)/(ln(3)*ln(x)*ln(1/ln(x))*ln(ln(1/ln(x)))**2+2*ln(x)*ln(1/ln(x) 
)*ln(ln(1/ln(x)))),x)
 

Output:

(-x**2 + 4*x)*log((log(3)*log(log(1/log(x))) + 2)/log(log(1/log(x))))
 

Maxima [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.76 \[ \int \frac {8-2 x+\left ((8-4 x) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+(4-2 x) \log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )\right ) \log \left (\frac {2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right )}{2 \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+\log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )} \, dx=-{\left (x^{2} - 4 \, x\right )} \log \left (\log \left (3\right ) \log \left (-\log \left (\log \left (x\right )\right )\right ) + 2\right ) + {\left (x^{2} - 4 \, x\right )} \log \left (\log \left (-\log \left (\log \left (x\right )\right )\right )\right ) \] Input:

integrate((((4-2*x)*log(3)*log(x)*log(1/log(x))*log(log(1/log(x)))^2+(-4*x 
+8)*log(x)*log(1/log(x))*log(log(1/log(x))))*log((log(3)*log(log(1/log(x)) 
)+2)/log(log(1/log(x))))-2*x+8)/(log(3)*log(x)*log(1/log(x))*log(log(1/log 
(x)))^2+2*log(x)*log(1/log(x))*log(log(1/log(x)))),x, algorithm="maxima")
 

Output:

-(x^2 - 4*x)*log(log(3)*log(-log(log(x))) + 2) + (x^2 - 4*x)*log(log(-log( 
log(x))))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (20) = 40\).

Time = 1.10 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.57 \[ \int \frac {8-2 x+\left ((8-4 x) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+(4-2 x) \log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )\right ) \log \left (\frac {2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right )}{2 \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+\log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )} \, dx=-x^{2} \log \left (\log \left (3\right ) \log \left (-\log \left (\log \left (x\right )\right )\right ) + 2\right ) + x^{2} \log \left (\log \left (-\log \left (\log \left (x\right )\right )\right )\right ) + 4 \, x \log \left (\log \left (3\right ) \log \left (-\log \left (\log \left (x\right )\right )\right ) + 2\right ) - 4 \, x \log \left (\log \left (-\log \left (\log \left (x\right )\right )\right )\right ) \] Input:

integrate((((4-2*x)*log(3)*log(x)*log(1/log(x))*log(log(1/log(x)))^2+(-4*x 
+8)*log(x)*log(1/log(x))*log(log(1/log(x))))*log((log(3)*log(log(1/log(x)) 
)+2)/log(log(1/log(x))))-2*x+8)/(log(3)*log(x)*log(1/log(x))*log(log(1/log 
(x)))^2+2*log(x)*log(1/log(x))*log(log(1/log(x)))),x, algorithm="giac")
 

Output:

-x^2*log(log(3)*log(-log(log(x))) + 2) + x^2*log(log(-log(log(x)))) + 4*x* 
log(log(3)*log(-log(log(x))) + 2) - 4*x*log(log(-log(log(x))))
 

Mupad [B] (verification not implemented)

Time = 3.96 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.29 \[ \int \frac {8-2 x+\left ((8-4 x) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+(4-2 x) \log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )\right ) \log \left (\frac {2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right )}{2 \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+\log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )} \, dx=-x\,\ln \left (\frac {\ln \left (\ln \left (\frac {1}{\ln \left (x\right )}\right )\right )\,\ln \left (3\right )+2}{\ln \left (\ln \left (\frac {1}{\ln \left (x\right )}\right )\right )}\right )\,\left (x-4\right ) \] Input:

int(-(2*x + log((log(log(1/log(x)))*log(3) + 2)/log(log(1/log(x))))*(log(l 
og(1/log(x)))*log(1/log(x))*log(x)*(4*x - 8) + log(log(1/log(x)))^2*log(3) 
*log(1/log(x))*log(x)*(2*x - 4)) - 8)/(2*log(log(1/log(x)))*log(1/log(x))* 
log(x) + log(log(1/log(x)))^2*log(3)*log(1/log(x))*log(x)),x)
 

Output:

-x*log((log(log(1/log(x)))*log(3) + 2)/log(log(1/log(x))))*(x - 4)
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.33 \[ \int \frac {8-2 x+\left ((8-4 x) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+(4-2 x) \log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )\right ) \log \left (\frac {2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right )}{2 \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+\log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )} \, dx=\mathrm {log}\left (\frac {\mathrm {log}\left (-\mathrm {log}\left (\mathrm {log}\left (x \right )\right )\right ) \mathrm {log}\left (3\right )+2}{\mathrm {log}\left (-\mathrm {log}\left (\mathrm {log}\left (x \right )\right )\right )}\right ) x \left (-x +4\right ) \] Input:

int((((4-2*x)*log(3)*log(x)*log(1/log(x))*log(log(1/log(x)))^2+(-4*x+8)*lo 
g(x)*log(1/log(x))*log(log(1/log(x))))*log((log(3)*log(log(1/log(x)))+2)/l 
og(log(1/log(x))))-2*x+8)/(log(3)*log(x)*log(1/log(x))*log(log(1/log(x)))^ 
2+2*log(x)*log(1/log(x))*log(log(1/log(x)))),x)
 

Output:

log((log( - log(log(x)))*log(3) + 2)/log( - log(log(x))))*x*( - x + 4)