3.10.22 \(\int \frac {(-2 x+x^2+(2-2 x) \log (x)) \log (3 x)-x^2 \log (x) \log (3 x) \log (\log (x))+((2 x-x^2) \log (x)+(-2 x+x^2) \log (x) \log (3 x)+((2 x^2-x^3) \log (x)+(-2 x^2+x^3) \log (x) \log (3 x)) \log (\log (x))) \log (\frac {-1-x \log (\log (x))}{-2 x+x^2}) \log (\log (\frac {-1-x \log (\log (x))}{-2 x+x^2}))+((2-x) \log (x)+(2 x-x^2) \log (x) \log (\log (x))) \log (\frac {-1-x \log (\log (x))}{-2 x+x^2}) \log (\log (\frac {-1-x \log (\log (x))}{-2 x+x^2})) \log (\log (\log (\frac {-1-x \log (\log (x))}{-2 x+x^2})))}{((-2 x+x^2) \log (x) \log ^2(3 x)+(-2 x^2+x^3) \log (x) \log ^2(3 x) \log (\log (x))) \log (\frac {-1-x \log (\log (x))}{-2 x+x^2}) \log (\log (\frac {-1-x \log (\log (x))}{-2 x+x^2}))} \, dx\) [922]

3.10.22.1 Optimal result

Integrand size = 306, antiderivative size = 27 \begin {dmath*} \int \frac {\left (-2 x+x^2+(2-2 x) \log (x)\right ) \log (3 x)-x^2 \log (x) \log (3 x) \log (\log (x))+\left (\left (2 x-x^2\right ) \log (x)+\left (-2 x+x^2\right ) \log (x) \log (3 x)+\left (\left (2 x^2-x^3\right ) \log (x)+\left (-2 x^2+x^3\right ) \log (x) \log (3 x)\right ) \log (\log (x))\right ) \log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right ) \log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right )+\left ((2-x) \log (x)+\left (2 x-x^2\right ) \log (x) \log (\log (x))\right ) \log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right ) \log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right ) \log \left (\log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right )\right )}{\left (\left (-2 x+x^2\right ) \log (x) \log ^2(3 x)+\left (-2 x^2+x^3\right ) \log (x) \log ^2(3 x) \log (\log (x))\right ) \log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right ) \log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right )} \, dx=\frac {x+\log \left (\log \left (\log \left (\frac {\frac {1}{x}+\log (\log (x))}{2-x}\right )\right )\right )}{\log (3 x)} \end {dmath*}

output

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

3.10.22.2 Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33 \begin {dmath*} \int \frac {\left (-2 x+x^2+(2-2 x) \log (x)\right ) \log (3 x)-x^2 \log (x) \log (3 x) \log (\log (x))+\left (\left (2 x-x^2\right ) \log (x)+\left (-2 x+x^2\right ) \log (x) \log (3 x)+\left (\left (2 x^2-x^3\right ) \log (x)+\left (-2 x^2+x^3\right ) \log (x) \log (3 x)\right ) \log (\log (x))\right ) \log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right ) \log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right )+\left ((2-x) \log (x)+\left (2 x-x^2\right ) \log (x) \log (\log (x))\right ) \log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right ) \log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right ) \log \left (\log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right )\right )}{\left (\left (-2 x+x^2\right ) \log (x) \log ^2(3 x)+\left (-2 x^2+x^3\right ) \log (x) \log ^2(3 x) \log (\log (x))\right ) \log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right ) \log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right )} \, dx=\frac {x}{\log (3 x)}+\frac {\log \left (\log \left (\log \left (-\frac {1+x \log (\log (x))}{(-2+x) x}\right )\right )\right )}{\log (3 x)} \end {dmath*}

input

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

output

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

3.10.22.3 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.

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

input

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

output

$Aborted
 

3.10.22.3.1 Defintions of rubi rules used

rule 7239

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl 
erIntegrandQ[v, u, x]]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.10.22.4 Maple [C] (warning: unable to verify)

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

Time = 11.71 (sec) , antiderivative size = 274, normalized size of antiderivative = 10.15

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

input

int((((-x^2+2*x)*ln(x)*ln(ln(x))+(2-x)*ln(x))*ln((-x*ln(ln(x))-1)/(x^2-2*x 
))*ln(ln((-x*ln(ln(x))-1)/(x^2-2*x)))*ln(ln(ln((-x*ln(ln(x))-1)/(x^2-2*x)) 
))+(((x^3-2*x^2)*ln(x)*ln(3*x)+(-x^3+2*x^2)*ln(x))*ln(ln(x))+(x^2-2*x)*ln( 
x)*ln(3*x)+(-x^2+2*x)*ln(x))*ln((-x*ln(ln(x))-1)/(x^2-2*x))*ln(ln((-x*ln(l 
n(x))-1)/(x^2-2*x)))-x^2*ln(x)*ln(3*x)*ln(ln(x))+((2-2*x)*ln(x)+x^2-2*x)*l 
n(3*x))/((x^3-2*x^2)*ln(x)*ln(3*x)^2*ln(ln(x))+(x^2-2*x)*ln(x)*ln(3*x)^2)/ 
ln((-x*ln(ln(x))-1)/(x^2-2*x))/ln(ln((-x*ln(ln(x))-1)/(x^2-2*x))),x)
 

output

2*I/(2*I*ln(3)+2*I*ln(x))*ln(ln(I*Pi-ln(x)-ln(-2+x)+ln(x*ln(ln(x))+1)-1/2* 
I*Pi*csgn(I/(-2+x)*(x*ln(ln(x))+1))*(-csgn(I/(-2+x)*(x*ln(ln(x))+1))+csgn( 
I/(-2+x)))*(-csgn(I/(-2+x)*(x*ln(ln(x))+1))+csgn(I*(x*ln(ln(x))+1)))-1/2*I 
*Pi*csgn(I/x/(-2+x)*(x*ln(ln(x))+1))*(-csgn(I/x/(-2+x)*(x*ln(ln(x))+1))+cs 
gn(I/x))*(-csgn(I/x/(-2+x)*(x*ln(ln(x))+1))+csgn(I/(-2+x)*(x*ln(ln(x))+1)) 
)+I*Pi*csgn(I/x/(-2+x)*(x*ln(ln(x))+1))^2*(csgn(I/x/(-2+x)*(x*ln(ln(x))+1) 
)-1)))+2*I*x/(2*I*ln(3)+2*I*ln(x))
 

3.10.22.5 Fricas [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \begin {dmath*} \int \frac {\left (-2 x+x^2+(2-2 x) \log (x)\right ) \log (3 x)-x^2 \log (x) \log (3 x) \log (\log (x))+\left (\left (2 x-x^2\right ) \log (x)+\left (-2 x+x^2\right ) \log (x) \log (3 x)+\left (\left (2 x^2-x^3\right ) \log (x)+\left (-2 x^2+x^3\right ) \log (x) \log (3 x)\right ) \log (\log (x))\right ) \log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right ) \log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right )+\left ((2-x) \log (x)+\left (2 x-x^2\right ) \log (x) \log (\log (x))\right ) \log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right ) \log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right ) \log \left (\log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right )\right )}{\left (\left (-2 x+x^2\right ) \log (x) \log ^2(3 x)+\left (-2 x^2+x^3\right ) \log (x) \log ^2(3 x) \log (\log (x))\right ) \log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right ) \log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right )} \, dx=\frac {x + \log \left (\log \left (\log \left (-\frac {x \log \left (\log \left (x\right )\right ) + 1}{x^{2} - 2 \, x}\right )\right )\right )}{\log \left (3\right ) + \log \left (x\right )} \end {dmath*}

input

integrate((((-x^2+2*x)*log(x)*log(log(x))+(2-x)*log(x))*log((-x*log(log(x) 
)-1)/(x^2-2*x))*log(log((-x*log(log(x))-1)/(x^2-2*x)))*log(log(log((-x*log 
(log(x))-1)/(x^2-2*x))))+(((x^3-2*x^2)*log(x)*log(3*x)+(-x^3+2*x^2)*log(x) 
)*log(log(x))+(x^2-2*x)*log(x)*log(3*x)+(-x^2+2*x)*log(x))*log((-x*log(log 
(x))-1)/(x^2-2*x))*log(log((-x*log(log(x))-1)/(x^2-2*x)))-x^2*log(x)*log(3 
*x)*log(log(x))+((2-2*x)*log(x)+x^2-2*x)*log(3*x))/((x^3-2*x^2)*log(x)*log 
(3*x)^2*log(log(x))+(x^2-2*x)*log(x)*log(3*x)^2)/log((-x*log(log(x))-1)/(x 
^2-2*x))/log(log((-x*log(log(x))-1)/(x^2-2*x))),x, algorithm=\
 

output

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

3.10.22.6 Sympy [F(-1)]

Timed out. \begin {dmath*} \int \frac {\left (-2 x+x^2+(2-2 x) \log (x)\right ) \log (3 x)-x^2 \log (x) \log (3 x) \log (\log (x))+\left (\left (2 x-x^2\right ) \log (x)+\left (-2 x+x^2\right ) \log (x) \log (3 x)+\left (\left (2 x^2-x^3\right ) \log (x)+\left (-2 x^2+x^3\right ) \log (x) \log (3 x)\right ) \log (\log (x))\right ) \log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right ) \log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right )+\left ((2-x) \log (x)+\left (2 x-x^2\right ) \log (x) \log (\log (x))\right ) \log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right ) \log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right ) \log \left (\log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right )\right )}{\left (\left (-2 x+x^2\right ) \log (x) \log ^2(3 x)+\left (-2 x^2+x^3\right ) \log (x) \log ^2(3 x) \log (\log (x))\right ) \log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right ) \log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right )} \, dx=\text {Timed out} \end {dmath*}

input

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

output

Timed out
 

3.10.22.7 Maxima [A] (verification not implemented)

Time = 0.49 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22 \begin {dmath*} \int \frac {\left (-2 x+x^2+(2-2 x) \log (x)\right ) \log (3 x)-x^2 \log (x) \log (3 x) \log (\log (x))+\left (\left (2 x-x^2\right ) \log (x)+\left (-2 x+x^2\right ) \log (x) \log (3 x)+\left (\left (2 x^2-x^3\right ) \log (x)+\left (-2 x^2+x^3\right ) \log (x) \log (3 x)\right ) \log (\log (x))\right ) \log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right ) \log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right )+\left ((2-x) \log (x)+\left (2 x-x^2\right ) \log (x) \log (\log (x))\right ) \log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right ) \log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right ) \log \left (\log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right )\right )}{\left (\left (-2 x+x^2\right ) \log (x) \log ^2(3 x)+\left (-2 x^2+x^3\right ) \log (x) \log ^2(3 x) \log (\log (x))\right ) \log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right ) \log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right )} \, dx=\frac {x + \log \left (\log \left (\log \left (x \log \left (\log \left (x\right )\right ) + 1\right ) - \log \left (x\right ) - \log \left (-x + 2\right )\right )\right )}{\log \left (3\right ) + \log \left (x\right )} \end {dmath*}

input

integrate((((-x^2+2*x)*log(x)*log(log(x))+(2-x)*log(x))*log((-x*log(log(x) 
)-1)/(x^2-2*x))*log(log((-x*log(log(x))-1)/(x^2-2*x)))*log(log(log((-x*log 
(log(x))-1)/(x^2-2*x))))+(((x^3-2*x^2)*log(x)*log(3*x)+(-x^3+2*x^2)*log(x) 
)*log(log(x))+(x^2-2*x)*log(x)*log(3*x)+(-x^2+2*x)*log(x))*log((-x*log(log 
(x))-1)/(x^2-2*x))*log(log((-x*log(log(x))-1)/(x^2-2*x)))-x^2*log(x)*log(3 
*x)*log(log(x))+((2-2*x)*log(x)+x^2-2*x)*log(3*x))/((x^3-2*x^2)*log(x)*log 
(3*x)^2*log(log(x))+(x^2-2*x)*log(x)*log(3*x)^2)/log((-x*log(log(x))-1)/(x 
^2-2*x))/log(log((-x*log(log(x))-1)/(x^2-2*x))),x, algorithm=\
 

output

(x + log(log(log(x*log(log(x)) + 1) - log(x) - log(-x + 2))))/(log(3) + lo 
g(x))
 

3.10.22.8 Giac [A] (verification not implemented)

Time = 1.84 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.48 \begin {dmath*} \int \frac {\left (-2 x+x^2+(2-2 x) \log (x)\right ) \log (3 x)-x^2 \log (x) \log (3 x) \log (\log (x))+\left (\left (2 x-x^2\right ) \log (x)+\left (-2 x+x^2\right ) \log (x) \log (3 x)+\left (\left (2 x^2-x^3\right ) \log (x)+\left (-2 x^2+x^3\right ) \log (x) \log (3 x)\right ) \log (\log (x))\right ) \log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right ) \log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right )+\left ((2-x) \log (x)+\left (2 x-x^2\right ) \log (x) \log (\log (x))\right ) \log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right ) \log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right ) \log \left (\log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right )\right )}{\left (\left (-2 x+x^2\right ) \log (x) \log ^2(3 x)+\left (-2 x^2+x^3\right ) \log (x) \log ^2(3 x) \log (\log (x))\right ) \log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right ) \log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right )} \, dx=\frac {x}{\log \left (3\right ) + \log \left (x\right )} + \frac {\log \left (\log \left (\log \left (-x \log \left (\log \left (x\right )\right ) - 1\right ) - \log \left (x - 2\right ) - \log \left (x\right )\right )\right )}{\log \left (3\right ) + \log \left (x\right )} \end {dmath*}

input

integrate((((-x^2+2*x)*log(x)*log(log(x))+(2-x)*log(x))*log((-x*log(log(x) 
)-1)/(x^2-2*x))*log(log((-x*log(log(x))-1)/(x^2-2*x)))*log(log(log((-x*log 
(log(x))-1)/(x^2-2*x))))+(((x^3-2*x^2)*log(x)*log(3*x)+(-x^3+2*x^2)*log(x) 
)*log(log(x))+(x^2-2*x)*log(x)*log(3*x)+(-x^2+2*x)*log(x))*log((-x*log(log 
(x))-1)/(x^2-2*x))*log(log((-x*log(log(x))-1)/(x^2-2*x)))-x^2*log(x)*log(3 
*x)*log(log(x))+((2-2*x)*log(x)+x^2-2*x)*log(3*x))/((x^3-2*x^2)*log(x)*log 
(3*x)^2*log(log(x))+(x^2-2*x)*log(x)*log(3*x)^2)/log((-x*log(log(x))-1)/(x 
^2-2*x))/log(log((-x*log(log(x))-1)/(x^2-2*x))),x, algorithm=\
 

output

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

3.10.22.9 Mupad [B] (verification not implemented)

Time = 20.56 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \begin {dmath*} \int \frac {\left (-2 x+x^2+(2-2 x) \log (x)\right ) \log (3 x)-x^2 \log (x) \log (3 x) \log (\log (x))+\left (\left (2 x-x^2\right ) \log (x)+\left (-2 x+x^2\right ) \log (x) \log (3 x)+\left (\left (2 x^2-x^3\right ) \log (x)+\left (-2 x^2+x^3\right ) \log (x) \log (3 x)\right ) \log (\log (x))\right ) \log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right ) \log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right )+\left ((2-x) \log (x)+\left (2 x-x^2\right ) \log (x) \log (\log (x))\right ) \log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right ) \log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right ) \log \left (\log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right )\right )}{\left (\left (-2 x+x^2\right ) \log (x) \log ^2(3 x)+\left (-2 x^2+x^3\right ) \log (x) \log ^2(3 x) \log (\log (x))\right ) \log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right ) \log \left (\log \left (\frac {-1-x \log (\log (x))}{-2 x+x^2}\right )\right )} \, dx=\frac {x+\ln \left (\ln \left (\ln \left (\frac {x\,\ln \left (\ln \left (x\right )\right )+1}{2\,x-x^2}\right )\right )\right )}{\ln \left (3\,x\right )} \end {dmath*}

input

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

output

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