Integrand size = 131, antiderivative size = 32 \[ \int \frac {-36+54 x+\left (-12+63 x-27 x^2\right ) \log (x)+\left (30-36 x+27 x^2\right ) \log ^2(x)}{\left (\left (6 x-9 x^2\right ) \log (x)+\left (10 x-21 x^2+9 x^3\right ) \log ^2(x)\right ) \log \left (\frac {(-6+9 x) \log ^2(x)}{-3 x+\left (-5 x+3 x^2\right ) \log (x)}\right ) \log ^2\left (\log \left (\frac {(-6+9 x) \log ^2(x)}{-3 x+\left (-5 x+3 x^2\right ) \log (x)}\right )\right )} \, dx=\frac {3}{\log \left (\log \left (\frac {3 \log (x)}{x+\frac {x+\frac {x}{\log (x)}}{\frac {2}{3}-x}}\right )\right )} \]
Time = 0.09 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \frac {-36+54 x+\left (-12+63 x-27 x^2\right ) \log (x)+\left (30-36 x+27 x^2\right ) \log ^2(x)}{\left (\left (6 x-9 x^2\right ) \log (x)+\left (10 x-21 x^2+9 x^3\right ) \log ^2(x)\right ) \log \left (\frac {(-6+9 x) \log ^2(x)}{-3 x+\left (-5 x+3 x^2\right ) \log (x)}\right ) \log ^2\left (\log \left (\frac {(-6+9 x) \log ^2(x)}{-3 x+\left (-5 x+3 x^2\right ) \log (x)}\right )\right )} \, dx=\frac {3}{\log \left (\log \left (\frac {3 (-2+3 x) \log ^2(x)}{x (-3+(-5+3 x) \log (x))}\right )\right )} \]
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]
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 |
\(\displaystyle -3 \int \left (\frac {9 \log ^2(x) x^2-9 \log (x) x^2-12 \log ^2(x) x+21 \log (x) x+18 x+10 \log ^2(x)-4 \log (x)-12}{2 x \log (x) (3 x \log (x)-5 \log (x)-3) \log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right ) \log ^2\left (\log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right )\right )}-\frac {3 \left (9 \log ^2(x) x^2-9 \log (x) x^2-12 \log ^2(x) x+21 \log (x) x+18 x+10 \log ^2(x)-4 \log (x)-12\right )}{2 (3 x-2) \log (x) (3 x \log (x)-5 \log (x)-3) \log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right ) \log ^2\left (\log \left (\frac {3 (2-3 x) \log ^2(x)}{x ((5-3 x) \log (x)+3)}\right )\right )}\right )dx\) |
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]
3.13.65.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 299.45 (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 )}\) | \(380\) |
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)
Time = 0.24 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.09 \[ \int \frac {-36+54 x+\left (-12+63 x-27 x^2\right ) \log (x)+\left (30-36 x+27 x^2\right ) \log ^2(x)}{\left (\left (6 x-9 x^2\right ) \log (x)+\left (10 x-21 x^2+9 x^3\right ) \log ^2(x)\right ) \log \left (\frac {(-6+9 x) \log ^2(x)}{-3 x+\left (-5 x+3 x^2\right ) \log (x)}\right ) \log ^2\left (\log \left (\frac {(-6+9 x) \log ^2(x)}{-3 x+\left (-5 x+3 x^2\right ) \log (x)}\right )\right )} \, dx=\frac {3}{\log \left (\log \left (\frac {3 \, {\left (3 \, x - 2\right )} \log \left (x\right )^{2}}{{\left (3 \, x^{2} - 5 \, x\right )} \log \left (x\right ) - 3 \, x}\right )\right )} \]
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=\
Time = 4.18 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.91 \[ \int \frac {-36+54 x+\left (-12+63 x-27 x^2\right ) \log (x)+\left (30-36 x+27 x^2\right ) \log ^2(x)}{\left (\left (6 x-9 x^2\right ) \log (x)+\left (10 x-21 x^2+9 x^3\right ) \log ^2(x)\right ) \log \left (\frac {(-6+9 x) \log ^2(x)}{-3 x+\left (-5 x+3 x^2\right ) \log (x)}\right ) \log ^2\left (\log \left (\frac {(-6+9 x) \log ^2(x)}{-3 x+\left (-5 x+3 x^2\right ) \log (x)}\right )\right )} \, dx=\frac {3}{\log {\left (\log {\left (\frac {\left (9 x - 6\right ) \log {\left (x \right )}^{2}}{- 3 x + \left (3 x^{2} - 5 x\right ) \log {\left (x \right )}} \right )} \right )}} \]
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)
Time = 0.37 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.12 \[ \int \frac {-36+54 x+\left (-12+63 x-27 x^2\right ) \log (x)+\left (30-36 x+27 x^2\right ) \log ^2(x)}{\left (\left (6 x-9 x^2\right ) \log (x)+\left (10 x-21 x^2+9 x^3\right ) \log ^2(x)\right ) \log \left (\frac {(-6+9 x) \log ^2(x)}{-3 x+\left (-5 x+3 x^2\right ) \log (x)}\right ) \log ^2\left (\log \left (\frac {(-6+9 x) \log ^2(x)}{-3 x+\left (-5 x+3 x^2\right ) \log (x)}\right )\right )} \, dx=\frac {3}{\log \left (\log \left (3\right ) - \log \left ({\left (3 \, x - 5\right )} \log \left (x\right ) - 3\right ) + \log \left (3 \, x - 2\right ) - \log \left (x\right ) + 2 \, \log \left (\log \left (x\right )\right )\right )} \]
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=\
Time = 0.66 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.22 \[ \int \frac {-36+54 x+\left (-12+63 x-27 x^2\right ) \log (x)+\left (30-36 x+27 x^2\right ) \log ^2(x)}{\left (\left (6 x-9 x^2\right ) \log (x)+\left (10 x-21 x^2+9 x^3\right ) \log ^2(x)\right ) \log \left (\frac {(-6+9 x) \log ^2(x)}{-3 x+\left (-5 x+3 x^2\right ) \log (x)}\right ) \log ^2\left (\log \left (\frac {(-6+9 x) \log ^2(x)}{-3 x+\left (-5 x+3 x^2\right ) \log (x)}\right )\right )} \, dx=\frac {3}{\log \left (\log \left (9 \, x \log \left (x\right )^{2} - 6 \, \log \left (x\right )^{2}\right ) - \log \left (3 \, x \log \left (x\right ) - 5 \, \log \left (x\right ) - 3\right ) - \log \left (x\right )\right )} \]
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=\
Time = 14.07 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.09 \[ \int \frac {-36+54 x+\left (-12+63 x-27 x^2\right ) \log (x)+\left (30-36 x+27 x^2\right ) \log ^2(x)}{\left (\left (6 x-9 x^2\right ) \log (x)+\left (10 x-21 x^2+9 x^3\right ) \log ^2(x)\right ) \log \left (\frac {(-6+9 x) \log ^2(x)}{-3 x+\left (-5 x+3 x^2\right ) \log (x)}\right ) \log ^2\left (\log \left (\frac {(-6+9 x) \log ^2(x)}{-3 x+\left (-5 x+3 x^2\right ) \log (x)}\right )\right )} \, dx=\frac {3}{\ln \left (\ln \left (-\frac {{\ln \left (x\right )}^2\,\left (9\,x-6\right )}{3\,x+\ln \left (x\right )\,\left (5\,x-3\,x^2\right )}\right )\right )} \]