\(\int \frac {-36+54 x+(-12+63 x-27 x^2) \log (x)+(30-36 x+27 x^2) \log ^2(x)}{((6 x-9 x^2) \log (x)+(10 x-21 x^2+9 x^3) \log ^2(x)) \log (\frac {(-6+9 x) \log ^2(x)}{-3 x+(-5 x+3 x^2) \log (x)}) \log ^2(\log (\frac {(-6+9 x) \log ^2(x)}{-3 x+(-5 x+3 x^2) \log (x)}))} \, dx\) [1265]

Optimal result

Integrand size = 131, antiderivative size = 32 \[ \int \frac {-36+54 x+\left (-12+63 x-27 x^2\right ) \log (x)+\left (30-36 x+27 x^2\right ) \log ^2(x)}{\left (\left (6 x-9 x^2\right ) \log (x)+\left (10 x-21 x^2+9 x^3\right ) \log ^2(x)\right ) \log \left (\frac {(-6+9 x) \log ^2(x)}{-3 x+\left (-5 x+3 x^2\right ) \log (x)}\right ) \log ^2\left (\log \left (\frac {(-6+9 x) \log ^2(x)}{-3 x+\left (-5 x+3 x^2\right ) \log (x)}\right )\right )} \, dx=\frac {3}{\log \left (\log \left (\frac {3 \log (x)}{x+\frac {x+\frac {x}{\log (x)}}{\frac {2}{3}-x}}\right )\right )} \] Output:

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \frac {-36+54 x+\left (-12+63 x-27 x^2\right ) \log (x)+\left (30-36 x+27 x^2\right ) \log ^2(x)}{\left (\left (6 x-9 x^2\right ) \log (x)+\left (10 x-21 x^2+9 x^3\right ) \log ^2(x)\right ) \log \left (\frac {(-6+9 x) \log ^2(x)}{-3 x+\left (-5 x+3 x^2\right ) \log (x)}\right ) \log ^2\left (\log \left (\frac {(-6+9 x) \log ^2(x)}{-3 x+\left (-5 x+3 x^2\right ) \log (x)}\right )\right )} \, dx=\frac {3}{\log \left (\log \left (\frac {3 (-2+3 x) \log ^2(x)}{x (-3+(-5+3 x) \log (x))}\right )\right )} \] Input:

Integrate[(-36 + 54*x + (-12 + 63*x - 27*x^2)*Log[x] + (30 - 36*x + 27*x^2 
)*Log[x]^2)/(((6*x - 9*x^2)*Log[x] + (10*x - 21*x^2 + 9*x^3)*Log[x]^2)*Log 
[((-6 + 9*x)*Log[x]^2)/(-3*x + (-5*x + 3*x^2)*Log[x])]*Log[Log[((-6 + 9*x) 
*Log[x]^2)/(-3*x + (-5*x + 3*x^2)*Log[x])]]^2),x]
 

Output:

3/Log[Log[(3*(-2 + 3*x)*Log[x]^2)/(x*(-3 + (-5 + 3*x)*Log[x]))]]
 

Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (27 x^2-36 x+30\right ) \log ^2(x)+\left (-27 x^2+63 x-12\right ) \log (x)+54 x-36}{\left (\left (6 x-9 x^2\right ) \log (x)+\left (9 x^3-21 x^2+10 x\right ) \log ^2(x)\right ) \log \left (\frac {(9 x-6) \log ^2(x)}{\left (3 x^2-5 x\right ) \log (x)-3 x}\right ) \log ^2\left (\log \left (\frac {(9 x-6) \log ^2(x)}{\left (3 x^2-5 x\right ) \log (x)-3 x}\right )\right )} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {\left (27 x^2-36 x+30\right ) \log ^2(x)+\left (-27 x^2+63 x-12\right ) \log (x)+54 x-36}{(2-3 x) x \log (x) (-3 x \log (x)+5 \log (x)+3) \log \left (\frac {(9 x-6) \log ^2(x)}{\left (3 x^2-5 x\right ) \log (x)-3 x}\right ) \log ^2\left (\log \left (\frac {(9 x-6) \log ^2(x)}{\left (3 x^2-5 x\right ) \log (x)-3 x}\right )\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {9 \left (9 x^2 \log ^2(x)-9 x^2 \log (x)+18 x-12 x \log ^2(x)+10 \log ^2(x)+21 x \log (x)-4 \log (x)-12\right )}{2 (3 x-2) \log (x) (3 x \log (x)-5 \log (x)-3) \log \left (\frac {3 (3 x-2) \log ^2(x)}{x ((3 x-5) \log (x)-3)}\right ) \log ^2\left (\log \left (\frac {3 (3 x-2) \log ^2(x)}{x ((3 x-5) \log (x)-3)}\right )\right )}-\frac {3 \left (9 x^2 \log ^2(x)-9 x^2 \log (x)+18 x-12 x \log ^2(x)+10 \log ^2(x)+21 x \log (x)-4 \log (x)-12\right )}{2 x \log (x) (3 x \log (x)-5 \log (x)-3) \log \left (\frac {3 (3 x-2) \log ^2(x)}{x ((3 x-5) \log (x)-3)}\right ) \log ^2\left (\log \left (\frac {3 (3 x-2) \log ^2(x)}{x ((3 x-5) \log (x)-3)}\right )\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {3 \left (\left (9 x^2-12 x+10\right ) \log ^2(x)+\left (-9 x^2+21 x-4\right ) \log (x)+6 (3 x-2)\right )}{(2-3 x) x \log (x) (3-(3 x-5) \log (x)) \log \left (\frac {3 (3 x-2) \log ^2(x)}{x ((3 x-5) \log (x)-3)}\right ) \log ^2\left (\log \left (\frac {3 (3 x-2) \log ^2(x)}{x ((3 x-5) \log (x)-3)}\right )\right )}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 3 \int -\frac {-\left (\left (9 x^2-12 x+10\right ) \log ^2(x)\right )+\left (9 x^2-21 x+4\right ) \log (x)+6 (2-3 x)}{(2-3 x) x \log (x) ((5-3 x) \log (x)+3) \log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right ) \log ^2\left (\log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right )\right )}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -3 \int \frac {-\left (\left (9 x^2-12 x+10\right ) \log ^2(x)\right )+\left (9 x^2-21 x+4\right ) \log (x)+6 (2-3 x)}{(2-3 x) x \log (x) ((5-3 x) \log (x)+3) \log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right ) \log ^2\left (\log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right )\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -3 \int \left (\frac {9 \log ^2(x) x^2-9 \log (x) x^2-12 \log ^2(x) x+21 \log (x) x+18 x+10 \log ^2(x)-4 \log (x)-12}{2 x \log (x) (3 x \log (x)-5 \log (x)-3) \log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right ) \log ^2\left (\log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right )\right )}-\frac {3 \left (9 \log ^2(x) x^2-9 \log (x) x^2-12 \log ^2(x) x+21 \log (x) x+18 x+10 \log ^2(x)-4 \log (x)-12\right )}{2 (3 x-2) \log (x) (3 x \log (x)-5 \log (x)-3) \log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right ) \log ^2\left (\log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right )\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -3 \int \frac {\left (-9 x^2+12 x-10\right ) \log ^2(x)+\left (9 x^2-21 x+4\right ) \log (x)-18 x+12}{(2-3 x) x \log (x) (3-(3 x-5) \log (x)) \log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right ) \log ^2\left (\log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right )\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -3 \int \left (\frac {9 \log ^2(x) x^2-9 \log (x) x^2-12 \log ^2(x) x+21 \log (x) x+18 x+10 \log ^2(x)-4 \log (x)-12}{2 x \log (x) (3 x \log (x)-5 \log (x)-3) \log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right ) \log ^2\left (\log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right )\right )}-\frac {3 \left (9 \log ^2(x) x^2-9 \log (x) x^2-12 \log ^2(x) x+21 \log (x) x+18 x+10 \log ^2(x)-4 \log (x)-12\right )}{2 (3 x-2) \log (x) (3 x \log (x)-5 \log (x)-3) \log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right ) \log ^2\left (\log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right )\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -3 \int \frac {\left (-9 x^2+12 x-10\right ) \log ^2(x)+\left (9 x^2-21 x+4\right ) \log (x)-18 x+12}{(2-3 x) x \log (x) (3-(3 x-5) \log (x)) \log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right ) \log ^2\left (\log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right )\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -3 \int \left (\frac {9 \log ^2(x) x^2-9 \log (x) x^2-12 \log ^2(x) x+21 \log (x) x+18 x+10 \log ^2(x)-4 \log (x)-12}{2 x \log (x) (3 x \log (x)-5 \log (x)-3) \log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right ) \log ^2\left (\log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right )\right )}-\frac {3 \left (9 \log ^2(x) x^2-9 \log (x) x^2-12 \log ^2(x) x+21 \log (x) x+18 x+10 \log ^2(x)-4 \log (x)-12\right )}{2 (3 x-2) \log (x) (3 x \log (x)-5 \log (x)-3) \log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right ) \log ^2\left (\log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right )\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -3 \int \frac {\left (-9 x^2+12 x-10\right ) \log ^2(x)+\left (9 x^2-21 x+4\right ) \log (x)-18 x+12}{(2-3 x) x \log (x) (3-(3 x-5) \log (x)) \log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right ) \log ^2\left (\log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right )\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -3 \int \left (\frac {9 \log ^2(x) x^2-9 \log (x) x^2-12 \log ^2(x) x+21 \log (x) x+18 x+10 \log ^2(x)-4 \log (x)-12}{2 x \log (x) (3 x \log (x)-5 \log (x)-3) \log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right ) \log ^2\left (\log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right )\right )}-\frac {3 \left (9 \log ^2(x) x^2-9 \log (x) x^2-12 \log ^2(x) x+21 \log (x) x+18 x+10 \log ^2(x)-4 \log (x)-12\right )}{2 (3 x-2) \log (x) (3 x \log (x)-5 \log (x)-3) \log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right ) \log ^2\left (\log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right )\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -3 \int \frac {\left (-9 x^2+12 x-10\right ) \log ^2(x)+\left (9 x^2-21 x+4\right ) \log (x)-18 x+12}{(2-3 x) x \log (x) (3-(3 x-5) \log (x)) \log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right ) \log ^2\left (\log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right )\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -3 \int \left (\frac {9 \log ^2(x) x^2-9 \log (x) x^2-12 \log ^2(x) x+21 \log (x) x+18 x+10 \log ^2(x)-4 \log (x)-12}{2 x \log (x) (3 x \log (x)-5 \log (x)-3) \log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right ) \log ^2\left (\log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right )\right )}-\frac {3 \left (9 \log ^2(x) x^2-9 \log (x) x^2-12 \log ^2(x) x+21 \log (x) x+18 x+10 \log ^2(x)-4 \log (x)-12\right )}{2 (3 x-2) \log (x) (3 x \log (x)-5 \log (x)-3) \log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right ) \log ^2\left (\log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right )\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -3 \int \frac {\left (-9 x^2+12 x-10\right ) \log ^2(x)+\left (9 x^2-21 x+4\right ) \log (x)-18 x+12}{(2-3 x) x \log (x) (3-(3 x-5) \log (x)) \log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right ) \log ^2\left (\log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right )\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -3 \int \left (\frac {9 \log ^2(x) x^2-9 \log (x) x^2-12 \log ^2(x) x+21 \log (x) x+18 x+10 \log ^2(x)-4 \log (x)-12}{2 x \log (x) (3 x \log (x)-5 \log (x)-3) \log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right ) \log ^2\left (\log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right )\right )}-\frac {3 \left (9 \log ^2(x) x^2-9 \log (x) x^2-12 \log ^2(x) x+21 \log (x) x+18 x+10 \log ^2(x)-4 \log (x)-12\right )}{2 (3 x-2) \log (x) (3 x \log (x)-5 \log (x)-3) \log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right ) \log ^2\left (\log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right )\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -3 \int \frac {\left (-9 x^2+12 x-10\right ) \log ^2(x)+\left (9 x^2-21 x+4\right ) \log (x)-18 x+12}{(2-3 x) x \log (x) (3-(3 x-5) \log (x)) \log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right ) \log ^2\left (\log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right )\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -3 \int \left (\frac {9 \log ^2(x) x^2-9 \log (x) x^2-12 \log ^2(x) x+21 \log (x) x+18 x+10 \log ^2(x)-4 \log (x)-12}{2 x \log (x) (3 x \log (x)-5 \log (x)-3) \log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right ) \log ^2\left (\log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right )\right )}-\frac {3 \left (9 \log ^2(x) x^2-9 \log (x) x^2-12 \log ^2(x) x+21 \log (x) x+18 x+10 \log ^2(x)-4 \log (x)-12\right )}{2 (3 x-2) \log (x) (3 x \log (x)-5 \log (x)-3) \log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right ) \log ^2\left (\log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right )\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -3 \int \frac {\left (-9 x^2+12 x-10\right ) \log ^2(x)+\left (9 x^2-21 x+4\right ) \log (x)-18 x+12}{(2-3 x) x \log (x) (3-(3 x-5) \log (x)) \log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right ) \log ^2\left (\log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right )\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -3 \int \left (\frac {9 \log ^2(x) x^2-9 \log (x) x^2-12 \log ^2(x) x+21 \log (x) x+18 x+10 \log ^2(x)-4 \log (x)-12}{2 x \log (x) (3 x \log (x)-5 \log (x)-3) \log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right ) \log ^2\left (\log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right )\right )}-\frac {3 \left (9 \log ^2(x) x^2-9 \log (x) x^2-12 \log ^2(x) x+21 \log (x) x+18 x+10 \log ^2(x)-4 \log (x)-12\right )}{2 (3 x-2) \log (x) (3 x \log (x)-5 \log (x)-3) \log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right ) \log ^2\left (\log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right )\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -3 \int \frac {\left (-9 x^2+12 x-10\right ) \log ^2(x)+\left (9 x^2-21 x+4\right ) \log (x)-18 x+12}{(2-3 x) x \log (x) (3-(3 x-5) \log (x)) \log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right ) \log ^2\left (\log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right )\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -3 \int \left (\frac {9 \log ^2(x) x^2-9 \log (x) x^2-12 \log ^2(x) x+21 \log (x) x+18 x+10 \log ^2(x)-4 \log (x)-12}{2 x \log (x) (3 x \log (x)-5 \log (x)-3) \log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right ) \log ^2\left (\log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right )\right )}-\frac {3 \left (9 \log ^2(x) x^2-9 \log (x) x^2-12 \log ^2(x) x+21 \log (x) x+18 x+10 \log ^2(x)-4 \log (x)-12\right )}{2 (3 x-2) \log (x) (3 x \log (x)-5 \log (x)-3) \log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right ) \log ^2\left (\log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right )\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -3 \int \frac {\left (-9 x^2+12 x-10\right ) \log ^2(x)+\left (9 x^2-21 x+4\right ) \log (x)-18 x+12}{(2-3 x) x \log (x) (3-(3 x-5) \log (x)) \log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right ) \log ^2\left (\log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right )\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -3 \int \left (\frac {9 \log ^2(x) x^2-9 \log (x) x^2-12 \log ^2(x) x+21 \log (x) x+18 x+10 \log ^2(x)-4 \log (x)-12}{2 x \log (x) (3 x \log (x)-5 \log (x)-3) \log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right ) \log ^2\left (\log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right )\right )}-\frac {3 \left (9 \log ^2(x) x^2-9 \log (x) x^2-12 \log ^2(x) x+21 \log (x) x+18 x+10 \log ^2(x)-4 \log (x)-12\right )}{2 (3 x-2) \log (x) (3 x \log (x)-5 \log (x)-3) \log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right ) \log ^2\left (\log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right )\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -3 \int \frac {\left (-9 x^2+12 x-10\right ) \log ^2(x)+\left (9 x^2-21 x+4\right ) \log (x)-18 x+12}{(2-3 x) x \log (x) (3-(3 x-5) \log (x)) \log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right ) \log ^2\left (\log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right )\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -3 \int \left (\frac {9 \log ^2(x) x^2-9 \log (x) x^2-12 \log ^2(x) x+21 \log (x) x+18 x+10 \log ^2(x)-4 \log (x)-12}{2 x \log (x) (3 x \log (x)-5 \log (x)-3) \log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right ) \log ^2\left (\log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right )\right )}-\frac {3 \left (9 \log ^2(x) x^2-9 \log (x) x^2-12 \log ^2(x) x+21 \log (x) x+18 x+10 \log ^2(x)-4 \log (x)-12\right )}{2 (3 x-2) \log (x) (3 x \log (x)-5 \log (x)-3) \log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right ) \log ^2\left (\log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right )\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -3 \int \frac {\left (-9 x^2+12 x-10\right ) \log ^2(x)+\left (9 x^2-21 x+4\right ) \log (x)-18 x+12}{(2-3 x) x \log (x) (3-(3 x-5) \log (x)) \log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right ) \log ^2\left (\log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right )\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -3 \int \left (\frac {9 \log ^2(x) x^2-9 \log (x) x^2-12 \log ^2(x) x+21 \log (x) x+18 x+10 \log ^2(x)-4 \log (x)-12}{2 x \log (x) (3 x \log (x)-5 \log (x)-3) \log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right ) \log ^2\left (\log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right )\right )}-\frac {3 \left (9 \log ^2(x) x^2-9 \log (x) x^2-12 \log ^2(x) x+21 \log (x) x+18 x+10 \log ^2(x)-4 \log (x)-12\right )}{2 (3 x-2) \log (x) (3 x \log (x)-5 \log (x)-3) \log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right ) \log ^2\left (\log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right )\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -3 \int \frac {\left (-9 x^2+12 x-10\right ) \log ^2(x)+\left (9 x^2-21 x+4\right ) \log (x)-18 x+12}{(2-3 x) x \log (x) (3-(3 x-5) \log (x)) \log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right ) \log ^2\left (\log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right )\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -3 \int \left (\frac {9 \log ^2(x) x^2-9 \log (x) x^2-12 \log ^2(x) x+21 \log (x) x+18 x+10 \log ^2(x)-4 \log (x)-12}{2 x \log (x) (3 x \log (x)-5 \log (x)-3) \log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right ) \log ^2\left (\log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right )\right )}-\frac {3 \left (9 \log ^2(x) x^2-9 \log (x) x^2-12 \log ^2(x) x+21 \log (x) x+18 x+10 \log ^2(x)-4 \log (x)-12\right )}{2 (3 x-2) \log (x) (3 x \log (x)-5 \log (x)-3) \log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right ) \log ^2\left (\log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right )\right )}\right )dx\)

Input:

Int[(-36 + 54*x + (-12 + 63*x - 27*x^2)*Log[x] + (30 - 36*x + 27*x^2)*Log[ 
x]^2)/(((6*x - 9*x^2)*Log[x] + (10*x - 21*x^2 + 9*x^3)*Log[x]^2)*Log[((-6 
+ 9*x)*Log[x]^2)/(-3*x + (-5*x + 3*x^2)*Log[x])]*Log[Log[((-6 + 9*x)*Log[x 
]^2)/(-3*x + (-5*x + 3*x^2)*Log[x])]]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 284.78 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.03

method	result	size
parallelrisch	\(\frac {3}{\ln \left (\ln \left (\frac {\left (9 x -6\right ) \ln \left (x \right )^{2}}{x \left (3 x \ln \left (x \right )-5 \ln \left (x \right )-3\right )}\right )\right )}\)	\(33\)
default	\(\frac {3}{\ln \left (\ln \left (3\right )+2 \ln \left (\ln \left (x \right )\right )-\ln \left (x \right )-\ln \left (-1+\left (x -\frac {5}{3}\right ) \ln \left (x \right )\right )+\ln \left (-\frac {2}{3}+x \right )-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (-\frac {2}{3}+x \right )}{-1+\left (x -\frac {5}{3}\right ) \ln \left (x \right )}\right ) \left (-\operatorname {csgn}\left (\frac {i \left (-\frac {2}{3}+x \right )}{-1+\left (x -\frac {5}{3}\right ) \ln \left (x \right )}\right )+\operatorname {csgn}\left (\frac {i}{-1+\left (x -\frac {5}{3}\right ) \ln \left (x \right )}\right )\right ) \left (-\operatorname {csgn}\left (\frac {i \left (-\frac {2}{3}+x \right )}{-1+\left (x -\frac {5}{3}\right ) \ln \left (x \right )}\right )+\operatorname {csgn}\left (i \left (-\frac {2}{3}+x \right )\right )\right )}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (-\frac {2}{3}+x \right )}{x \left (-1+\left (x -\frac {5}{3}\right ) \ln \left (x \right )\right )}\right ) \left (-\operatorname {csgn}\left (\frac {i \left (-\frac {2}{3}+x \right )}{x \left (-1+\left (x -\frac {5}{3}\right ) \ln \left (x \right )\right )}\right )+\operatorname {csgn}\left (\frac {i}{x}\right )\right ) \left (-\operatorname {csgn}\left (\frac {i \left (-\frac {2}{3}+x \right )}{x \left (-1+\left (x -\frac {5}{3}\right ) \ln \left (x \right )\right )}\right )+\operatorname {csgn}\left (\frac {i \left (-\frac {2}{3}+x \right )}{-1+\left (x -\frac {5}{3}\right ) \ln \left (x \right )}\right )\right )}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \ln \left (x \right )^{2}\right ) {\left (-\operatorname {csgn}\left (i \ln \left (x \right )^{2}\right )+\operatorname {csgn}\left (i \ln \left (x \right )\right )\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \ln \left (x \right )^{2} \left (-\frac {2}{3}+x \right )}{\left (-1+\left (x -\frac {5}{3}\right ) \ln \left (x \right )\right ) x}\right ) \left (-\operatorname {csgn}\left (\frac {i \ln \left (x \right )^{2} \left (-\frac {2}{3}+x \right )}{\left (-1+\left (x -\frac {5}{3}\right ) \ln \left (x \right )\right ) x}\right )+\operatorname {csgn}\left (i \ln \left (x \right )^{2}\right )\right ) \left (-\operatorname {csgn}\left (\frac {i \ln \left (x \right )^{2} \left (-\frac {2}{3}+x \right )}{\left (-1+\left (x -\frac {5}{3}\right ) \ln \left (x \right )\right ) x}\right )+\operatorname {csgn}\left (\frac {i \left (-\frac {2}{3}+x \right )}{x \left (-1+\left (x -\frac {5}{3}\right ) \ln \left (x \right )\right )}\right )\right )}{2}\right )}\)	\(350\)
risch	\(\frac {3}{\ln \left (\ln \left (3\right )-\ln \left (x \right )+2 \ln \left (\ln \left (x \right )\right )+\ln \left (-\frac {2}{3}+x \right )-\ln \left (-1+x \ln \left (x \right )-\frac {5 \ln \left (x \right )}{3}\right )-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (-\frac {2}{3}+x \right )}{-1+x \ln \left (x \right )-\frac {5 \ln \left (x \right )}{3}}\right ) \left (-\operatorname {csgn}\left (\frac {i \left (-\frac {2}{3}+x \right )}{-1+x \ln \left (x \right )-\frac {5 \ln \left (x \right )}{3}}\right )+\operatorname {csgn}\left (i \left (-\frac {2}{3}+x \right )\right )\right ) \left (-\operatorname {csgn}\left (\frac {i \left (-\frac {2}{3}+x \right )}{-1+x \ln \left (x \right )-\frac {5 \ln \left (x \right )}{3}}\right )+\operatorname {csgn}\left (\frac {i}{-1+x \ln \left (x \right )-\frac {5 \ln \left (x \right )}{3}}\right )\right )}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \ln \left (x \right )^{2}\right ) {\left (-\operatorname {csgn}\left (i \ln \left (x \right )^{2}\right )+\operatorname {csgn}\left (i \ln \left (x \right )\right )\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \ln \left (x \right )^{2} \left (-\frac {2}{3}+x \right )}{-1+x \ln \left (x \right )-\frac {5 \ln \left (x \right )}{3}}\right ) \left (-\operatorname {csgn}\left (\frac {i \ln \left (x \right )^{2} \left (-\frac {2}{3}+x \right )}{-1+x \ln \left (x \right )-\frac {5 \ln \left (x \right )}{3}}\right )+\operatorname {csgn}\left (i \ln \left (x \right )^{2}\right )\right ) \left (-\operatorname {csgn}\left (\frac {i \ln \left (x \right )^{2} \left (-\frac {2}{3}+x \right )}{-1+x \ln \left (x \right )-\frac {5 \ln \left (x \right )}{3}}\right )+\operatorname {csgn}\left (\frac {i \left (-\frac {2}{3}+x \right )}{-1+x \ln \left (x \right )-\frac {5 \ln \left (x \right )}{3}}\right )\right )}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \ln \left (x \right )^{2} \left (-\frac {2}{3}+x \right )}{\left (-1+x \ln \left (x \right )-\frac {5 \ln \left (x \right )}{3}\right ) x}\right ) \left (-\operatorname {csgn}\left (\frac {i \ln \left (x \right )^{2} \left (-\frac {2}{3}+x \right )}{\left (-1+x \ln \left (x \right )-\frac {5 \ln \left (x \right )}{3}\right ) x}\right )+\operatorname {csgn}\left (\frac {i}{x}\right )\right ) \left (-\operatorname {csgn}\left (\frac {i \ln \left (x \right )^{2} \left (-\frac {2}{3}+x \right )}{\left (-1+x \ln \left (x \right )-\frac {5 \ln \left (x \right )}{3}\right ) x}\right )+\operatorname {csgn}\left (\frac {i \ln \left (x \right )^{2} \left (-\frac {2}{3}+x \right )}{-1+x \ln \left (x \right )-\frac {5 \ln \left (x \right )}{3}}\right )\right )}{2}\right )}\)	\(380\)

Input:

int(((27*x^2-36*x+30)*ln(x)^2+(-27*x^2+63*x-12)*ln(x)+54*x-36)/((9*x^3-21* 
x^2+10*x)*ln(x)^2+(-9*x^2+6*x)*ln(x))/ln((9*x-6)*ln(x)^2/((3*x^2-5*x)*ln(x 
)-3*x))/ln(ln((9*x-6)*ln(x)^2/((3*x^2-5*x)*ln(x)-3*x)))^2,x,method=_RETURN 
VERBOSE)
 

Output:

3/ln(ln((9*x-6)*ln(x)^2/x/(3*x*ln(x)-5*ln(x)-3)))
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.09 \[ \int \frac {-36+54 x+\left (-12+63 x-27 x^2\right ) \log (x)+\left (30-36 x+27 x^2\right ) \log ^2(x)}{\left (\left (6 x-9 x^2\right ) \log (x)+\left (10 x-21 x^2+9 x^3\right ) \log ^2(x)\right ) \log \left (\frac {(-6+9 x) \log ^2(x)}{-3 x+\left (-5 x+3 x^2\right ) \log (x)}\right ) \log ^2\left (\log \left (\frac {(-6+9 x) \log ^2(x)}{-3 x+\left (-5 x+3 x^2\right ) \log (x)}\right )\right )} \, dx=\frac {3}{\log \left (\log \left (\frac {3 \, {\left (3 \, x - 2\right )} \log \left (x\right )^{2}}{{\left (3 \, x^{2} - 5 \, x\right )} \log \left (x\right ) - 3 \, x}\right )\right )} \] Input:

integrate(((27*x^2-36*x+30)*log(x)^2+(-27*x^2+63*x-12)*log(x)+54*x-36)/((9 
*x^3-21*x^2+10*x)*log(x)^2+(-9*x^2+6*x)*log(x))/log((9*x-6)*log(x)^2/((3*x 
^2-5*x)*log(x)-3*x))/log(log((9*x-6)*log(x)^2/((3*x^2-5*x)*log(x)-3*x)))^2 
,x, algorithm="fricas")
 

Output:

3/log(log(3*(3*x - 2)*log(x)^2/((3*x^2 - 5*x)*log(x) - 3*x)))
 

Sympy [A] (verification not implemented)

Time = 4.92 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.91 \[ \int \frac {-36+54 x+\left (-12+63 x-27 x^2\right ) \log (x)+\left (30-36 x+27 x^2\right ) \log ^2(x)}{\left (\left (6 x-9 x^2\right ) \log (x)+\left (10 x-21 x^2+9 x^3\right ) \log ^2(x)\right ) \log \left (\frac {(-6+9 x) \log ^2(x)}{-3 x+\left (-5 x+3 x^2\right ) \log (x)}\right ) \log ^2\left (\log \left (\frac {(-6+9 x) \log ^2(x)}{-3 x+\left (-5 x+3 x^2\right ) \log (x)}\right )\right )} \, dx=\frac {3}{\log {\left (\log {\left (\frac {\left (9 x - 6\right ) \log {\left (x \right )}^{2}}{- 3 x + \left (3 x^{2} - 5 x\right ) \log {\left (x \right )}} \right )} \right )}} \] Input:

integrate(((27*x**2-36*x+30)*ln(x)**2+(-27*x**2+63*x-12)*ln(x)+54*x-36)/(( 
9*x**3-21*x**2+10*x)*ln(x)**2+(-9*x**2+6*x)*ln(x))/ln((9*x-6)*ln(x)**2/((3 
*x**2-5*x)*ln(x)-3*x))/ln(ln((9*x-6)*ln(x)**2/((3*x**2-5*x)*ln(x)-3*x)))** 
2,x)
 

Output:

3/log(log((9*x - 6)*log(x)**2/(-3*x + (3*x**2 - 5*x)*log(x))))
 

Maxima [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.12 \[ \int \frac {-36+54 x+\left (-12+63 x-27 x^2\right ) \log (x)+\left (30-36 x+27 x^2\right ) \log ^2(x)}{\left (\left (6 x-9 x^2\right ) \log (x)+\left (10 x-21 x^2+9 x^3\right ) \log ^2(x)\right ) \log \left (\frac {(-6+9 x) \log ^2(x)}{-3 x+\left (-5 x+3 x^2\right ) \log (x)}\right ) \log ^2\left (\log \left (\frac {(-6+9 x) \log ^2(x)}{-3 x+\left (-5 x+3 x^2\right ) \log (x)}\right )\right )} \, dx=\frac {3}{\log \left (\log \left (3\right ) - \log \left ({\left (3 \, x - 5\right )} \log \left (x\right ) - 3\right ) + \log \left (3 \, x - 2\right ) - \log \left (x\right ) + 2 \, \log \left (\log \left (x\right )\right )\right )} \] Input:

integrate(((27*x^2-36*x+30)*log(x)^2+(-27*x^2+63*x-12)*log(x)+54*x-36)/((9 
*x^3-21*x^2+10*x)*log(x)^2+(-9*x^2+6*x)*log(x))/log((9*x-6)*log(x)^2/((3*x 
^2-5*x)*log(x)-3*x))/log(log((9*x-6)*log(x)^2/((3*x^2-5*x)*log(x)-3*x)))^2 
,x, algorithm="maxima")
 

Output:

3/log(log(3) - log((3*x - 5)*log(x) - 3) + log(3*x - 2) - log(x) + 2*log(l 
og(x)))
 

Giac [A] (verification not implemented)

Time = 0.56 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.22 \[ \int \frac {-36+54 x+\left (-12+63 x-27 x^2\right ) \log (x)+\left (30-36 x+27 x^2\right ) \log ^2(x)}{\left (\left (6 x-9 x^2\right ) \log (x)+\left (10 x-21 x^2+9 x^3\right ) \log ^2(x)\right ) \log \left (\frac {(-6+9 x) \log ^2(x)}{-3 x+\left (-5 x+3 x^2\right ) \log (x)}\right ) \log ^2\left (\log \left (\frac {(-6+9 x) \log ^2(x)}{-3 x+\left (-5 x+3 x^2\right ) \log (x)}\right )\right )} \, dx=\frac {3}{\log \left (\log \left (9 \, x \log \left (x\right )^{2} - 6 \, \log \left (x\right )^{2}\right ) - \log \left (3 \, x \log \left (x\right ) - 5 \, \log \left (x\right ) - 3\right ) - \log \left (x\right )\right )} \] Input:

integrate(((27*x^2-36*x+30)*log(x)^2+(-27*x^2+63*x-12)*log(x)+54*x-36)/((9 
*x^3-21*x^2+10*x)*log(x)^2+(-9*x^2+6*x)*log(x))/log((9*x-6)*log(x)^2/((3*x 
^2-5*x)*log(x)-3*x))/log(log((9*x-6)*log(x)^2/((3*x^2-5*x)*log(x)-3*x)))^2 
,x, algorithm="giac")
 

Output:

3/log(log(9*x*log(x)^2 - 6*log(x)^2) - log(3*x*log(x) - 5*log(x) - 3) - lo 
g(x))
 

Mupad [B] (verification not implemented)

Time = 5.20 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.09 \[ \int \frac {-36+54 x+\left (-12+63 x-27 x^2\right ) \log (x)+\left (30-36 x+27 x^2\right ) \log ^2(x)}{\left (\left (6 x-9 x^2\right ) \log (x)+\left (10 x-21 x^2+9 x^3\right ) \log ^2(x)\right ) \log \left (\frac {(-6+9 x) \log ^2(x)}{-3 x+\left (-5 x+3 x^2\right ) \log (x)}\right ) \log ^2\left (\log \left (\frac {(-6+9 x) \log ^2(x)}{-3 x+\left (-5 x+3 x^2\right ) \log (x)}\right )\right )} \, dx=\frac {3}{\ln \left (\ln \left (-\frac {{\ln \left (x\right )}^2\,\left (9\,x-6\right )}{3\,x+\ln \left (x\right )\,\left (5\,x-3\,x^2\right )}\right )\right )} \] Input:

int((54*x + log(x)^2*(27*x^2 - 36*x + 30) - log(x)*(27*x^2 - 63*x + 12) - 
36)/(log(-(log(x)^2*(9*x - 6))/(3*x + log(x)*(5*x - 3*x^2)))*log(log(-(log 
(x)^2*(9*x - 6))/(3*x + log(x)*(5*x - 3*x^2))))^2*(log(x)^2*(10*x - 21*x^2 
 + 9*x^3) + log(x)*(6*x - 9*x^2))),x)
 

Output:

3/log(log(-(log(x)^2*(9*x - 6))/(3*x + log(x)*(5*x - 3*x^2))))
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.22 \[ \int \frac {-36+54 x+\left (-12+63 x-27 x^2\right ) \log (x)+\left (30-36 x+27 x^2\right ) \log ^2(x)}{\left (\left (6 x-9 x^2\right ) \log (x)+\left (10 x-21 x^2+9 x^3\right ) \log ^2(x)\right ) \log \left (\frac {(-6+9 x) \log ^2(x)}{-3 x+\left (-5 x+3 x^2\right ) \log (x)}\right ) \log ^2\left (\log \left (\frac {(-6+9 x) \log ^2(x)}{-3 x+\left (-5 x+3 x^2\right ) \log (x)}\right )\right )} \, dx=\frac {3}{\mathrm {log}\left (\mathrm {log}\left (\frac {9 \mathrm {log}\left (x \right )^{2} x -6 \mathrm {log}\left (x \right )^{2}}{3 \,\mathrm {log}\left (x \right ) x^{2}-5 \,\mathrm {log}\left (x \right ) x -3 x}\right )\right )} \] Input:

int(((27*x^2-36*x+30)*log(x)^2+(-27*x^2+63*x-12)*log(x)+54*x-36)/((9*x^3-2 
1*x^2+10*x)*log(x)^2+(-9*x^2+6*x)*log(x))/log((9*x-6)*log(x)^2/((3*x^2-5*x 
)*log(x)-3*x))/log(log((9*x-6)*log(x)^2/((3*x^2-5*x)*log(x)-3*x)))^2,x)
 

Output:

3/log(log((9*log(x)**2*x - 6*log(x)**2)/(3*log(x)*x**2 - 5*log(x)*x - 3*x) 
))