\(\int \frac {(-36+12 x) \log (x) \log (3 x^2)+(-9+3 x-9 \log (x)) \log ^2(3 x^2)+(-12+7 x-x^2+(-12+x^2) \log (x)) \log ^4(3 x^2)}{81-54 x+9 x^2+(216-198 x+60 x^2-6 x^3) \log ^2(3 x^2)+(144-168 x+73 x^2-14 x^3+x^4) \log ^4(3 x^2)} \, dx\) [261]

Optimal result

Integrand size = 124, antiderivative size = 25 \[ \int \frac {(-36+12 x) \log (x) \log \left (3 x^2\right )+(-9+3 x-9 \log (x)) \log ^2\left (3 x^2\right )+\left (-12+7 x-x^2+\left (-12+x^2\right ) \log (x)\right ) \log ^4\left (3 x^2\right )}{81-54 x+9 x^2+\left (216-198 x+60 x^2-6 x^3\right ) \log ^2\left (3 x^2\right )+\left (144-168 x+73 x^2-14 x^3+x^4\right ) \log ^4\left (3 x^2\right )} \, dx=-\frac {x \log (x)}{(-3+x) \left (-4+x-\frac {3}{\log ^2\left (3 x^2\right )}\right )} \] Output:

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36 \[ \int \frac {(-36+12 x) \log (x) \log \left (3 x^2\right )+(-9+3 x-9 \log (x)) \log ^2\left (3 x^2\right )+\left (-12+7 x-x^2+\left (-12+x^2\right ) \log (x)\right ) \log ^4\left (3 x^2\right )}{81-54 x+9 x^2+\left (216-198 x+60 x^2-6 x^3\right ) \log ^2\left (3 x^2\right )+\left (144-168 x+73 x^2-14 x^3+x^4\right ) \log ^4\left (3 x^2\right )} \, dx=-\frac {x \log (x) \log ^2\left (3 x^2\right )}{(-3+x) \left (-3+(-4+x) \log ^2\left (3 x^2\right )\right )} \] Input:

Integrate[((-36 + 12*x)*Log[x]*Log[3*x^2] + (-9 + 3*x - 9*Log[x])*Log[3*x^ 
2]^2 + (-12 + 7*x - x^2 + (-12 + x^2)*Log[x])*Log[3*x^2]^4)/(81 - 54*x + 9 
*x^2 + (216 - 198*x + 60*x^2 - 6*x^3)*Log[3*x^2]^2 + (144 - 168*x + 73*x^2 
 - 14*x^3 + x^4)*Log[3*x^2]^4),x]
 

Output:

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

Rubi [F]

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

\(\displaystyle \int \frac {\left (-x^2+\left (x^2-12\right ) \log (x)+7 x-12\right ) \log ^4\left (3 x^2\right )+(3 x-9 \log (x)-9) \log ^2\left (3 x^2\right )+(12 x-36) \log (x) \log \left (3 x^2\right )}{9 x^2+\left (-6 x^3+60 x^2-198 x+216\right ) \log ^2\left (3 x^2\right )+\left (x^4-14 x^3+73 x^2-168 x+144\right ) \log ^4\left (3 x^2\right )-54 x+81} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {\left (-x^2+\left (x^2-12\right ) \log (x)+7 x-12\right ) \log ^4\left (3 x^2\right )+(3 x-9 \log (x)-9) \log ^2\left (3 x^2\right )+(12 x-36) \log (x) \log \left (3 x^2\right )}{(3-x)^2 \left (-x \log ^2\left (3 x^2\right )+4 \log ^2\left (3 x^2\right )+3\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {3 \left (-x^2+2 x^2 \log (x)+7 x-3 x \log (x)-12 \log (x)-12\right )}{(x-4)^2 (x-3)^2 \left (x \log ^2\left (3 x^2\right )-4 \log ^2\left (3 x^2\right )-3\right )}+\frac {3 \log (x) \left (4 x^2 \log \left (3 x^2\right )-32 x \log \left (3 x^2\right )+64 \log \left (3 x^2\right )+3 x\right )}{(x-4)^2 (x-3) \left (x \log ^2\left (3 x^2\right )-4 \log ^2\left (3 x^2\right )-3\right )^2}+\frac {-x^2+x^2 \log (x)+7 x-12 \log (x)-12}{(x-4)^2 (x-3)^2}\right )dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {3 \left (-x^2+2 x^2 \log (x)+7 x-3 x \log (x)-12 \log (x)-12\right )}{(x-4)^2 (x-3)^2 \left (x \log ^2\left (3 x^2\right )-4 \log ^2\left (3 x^2\right )-3\right )}+\frac {3 \log (x) \left (4 x^2 \log \left (3 x^2\right )-32 x \log \left (3 x^2\right )+64 \log \left (3 x^2\right )+3 x\right )}{(x-4)^2 (x-3) \left (x \log ^2\left (3 x^2\right )-4 \log ^2\left (3 x^2\right )-3\right )^2}+\frac {-x^2+x^2 \log (x)+7 x-12 \log (x)-12}{(x-4)^2 (x-3)^2}\right )dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {3 \left (-x^2+2 x^2 \log (x)+7 x-3 x \log (x)-12 \log (x)-12\right )}{(x-4)^2 (x-3)^2 \left (x \log ^2\left (3 x^2\right )-4 \log ^2\left (3 x^2\right )-3\right )}+\frac {3 \log (x) \left (4 x^2 \log \left (3 x^2\right )-32 x \log \left (3 x^2\right )+64 \log \left (3 x^2\right )+3 x\right )}{(x-4)^2 (x-3) \left (x \log ^2\left (3 x^2\right )-4 \log ^2\left (3 x^2\right )-3\right )^2}+\frac {-x^2+x^2 \log (x)+7 x-12 \log (x)-12}{(x-4)^2 (x-3)^2}\right )dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {3 \left (-x^2+2 x^2 \log (x)+7 x-3 x \log (x)-12 \log (x)-12\right )}{(x-4)^2 (x-3)^2 \left (x \log ^2\left (3 x^2\right )-4 \log ^2\left (3 x^2\right )-3\right )}+\frac {3 \log (x) \left (4 x^2 \log \left (3 x^2\right )-32 x \log \left (3 x^2\right )+64 \log \left (3 x^2\right )+3 x\right )}{(x-4)^2 (x-3) \left (x \log ^2\left (3 x^2\right )-4 \log ^2\left (3 x^2\right )-3\right )^2}+\frac {-x^2+x^2 \log (x)+7 x-12 \log (x)-12}{(x-4)^2 (x-3)^2}\right )dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {3 \left (-x^2+2 x^2 \log (x)+7 x-3 x \log (x)-12 \log (x)-12\right )}{(x-4)^2 (x-3)^2 \left (x \log ^2\left (3 x^2\right )-4 \log ^2\left (3 x^2\right )-3\right )}+\frac {3 \log (x) \left (4 x^2 \log \left (3 x^2\right )-32 x \log \left (3 x^2\right )+64 \log \left (3 x^2\right )+3 x\right )}{(x-4)^2 (x-3) \left (x \log ^2\left (3 x^2\right )-4 \log ^2\left (3 x^2\right )-3\right )^2}+\frac {-x^2+x^2 \log (x)+7 x-12 \log (x)-12}{(x-4)^2 (x-3)^2}\right )dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {3 \left (-x^2+2 x^2 \log (x)+7 x-3 x \log (x)-12 \log (x)-12\right )}{(x-4)^2 (x-3)^2 \left (x \log ^2\left (3 x^2\right )-4 \log ^2\left (3 x^2\right )-3\right )}+\frac {3 \log (x) \left (4 x^2 \log \left (3 x^2\right )-32 x \log \left (3 x^2\right )+64 \log \left (3 x^2\right )+3 x\right )}{(x-4)^2 (x-3) \left (x \log ^2\left (3 x^2\right )-4 \log ^2\left (3 x^2\right )-3\right )^2}+\frac {-x^2+x^2 \log (x)+7 x-12 \log (x)-12}{(x-4)^2 (x-3)^2}\right )dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {3 \left (-x^2+2 x^2 \log (x)+7 x-3 x \log (x)-12 \log (x)-12\right )}{(x-4)^2 (x-3)^2 \left (x \log ^2\left (3 x^2\right )-4 \log ^2\left (3 x^2\right )-3\right )}+\frac {3 \log (x) \left (4 x^2 \log \left (3 x^2\right )-32 x \log \left (3 x^2\right )+64 \log \left (3 x^2\right )+3 x\right )}{(x-4)^2 (x-3) \left (x \log ^2\left (3 x^2\right )-4 \log ^2\left (3 x^2\right )-3\right )^2}+\frac {-x^2+x^2 \log (x)+7 x-12 \log (x)-12}{(x-4)^2 (x-3)^2}\right )dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {3 \left (-x^2+2 x^2 \log (x)+7 x-3 x \log (x)-12 \log (x)-12\right )}{(x-4)^2 (x-3)^2 \left (x \log ^2\left (3 x^2\right )-4 \log ^2\left (3 x^2\right )-3\right )}+\frac {3 \log (x) \left (4 x^2 \log \left (3 x^2\right )-32 x \log \left (3 x^2\right )+64 \log \left (3 x^2\right )+3 x\right )}{(x-4)^2 (x-3) \left (x \log ^2\left (3 x^2\right )-4 \log ^2\left (3 x^2\right )-3\right )^2}+\frac {-x^2+x^2 \log (x)+7 x-12 \log (x)-12}{(x-4)^2 (x-3)^2}\right )dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {3 \left (-x^2+2 x^2 \log (x)+7 x-3 x \log (x)-12 \log (x)-12\right )}{(x-4)^2 (x-3)^2 \left (x \log ^2\left (3 x^2\right )-4 \log ^2\left (3 x^2\right )-3\right )}+\frac {3 \log (x) \left (4 x^2 \log \left (3 x^2\right )-32 x \log \left (3 x^2\right )+64 \log \left (3 x^2\right )+3 x\right )}{(x-4)^2 (x-3) \left (x \log ^2\left (3 x^2\right )-4 \log ^2\left (3 x^2\right )-3\right )^2}+\frac {-x^2+x^2 \log (x)+7 x-12 \log (x)-12}{(x-4)^2 (x-3)^2}\right )dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {3 \left (-x^2+2 x^2 \log (x)+7 x-3 x \log (x)-12 \log (x)-12\right )}{(x-4)^2 (x-3)^2 \left (x \log ^2\left (3 x^2\right )-4 \log ^2\left (3 x^2\right )-3\right )}+\frac {3 \log (x) \left (4 x^2 \log \left (3 x^2\right )-32 x \log \left (3 x^2\right )+64 \log \left (3 x^2\right )+3 x\right )}{(x-4)^2 (x-3) \left (x \log ^2\left (3 x^2\right )-4 \log ^2\left (3 x^2\right )-3\right )^2}+\frac {-x^2+x^2 \log (x)+7 x-12 \log (x)-12}{(x-4)^2 (x-3)^2}\right )dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {3 \left (-x^2+2 x^2 \log (x)+7 x-3 x \log (x)-12 \log (x)-12\right )}{(x-4)^2 (x-3)^2 \left (x \log ^2\left (3 x^2\right )-4 \log ^2\left (3 x^2\right )-3\right )}+\frac {3 \log (x) \left (4 x^2 \log \left (3 x^2\right )-32 x \log \left (3 x^2\right )+64 \log \left (3 x^2\right )+3 x\right )}{(x-4)^2 (x-3) \left (x \log ^2\left (3 x^2\right )-4 \log ^2\left (3 x^2\right )-3\right )^2}+\frac {-x^2+x^2 \log (x)+7 x-12 \log (x)-12}{(x-4)^2 (x-3)^2}\right )dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {3 \left (-x^2+2 x^2 \log (x)+7 x-3 x \log (x)-12 \log (x)-12\right )}{(x-4)^2 (x-3)^2 \left (x \log ^2\left (3 x^2\right )-4 \log ^2\left (3 x^2\right )-3\right )}+\frac {3 \log (x) \left (4 x^2 \log \left (3 x^2\right )-32 x \log \left (3 x^2\right )+64 \log \left (3 x^2\right )+3 x\right )}{(x-4)^2 (x-3) \left (x \log ^2\left (3 x^2\right )-4 \log ^2\left (3 x^2\right )-3\right )^2}+\frac {-x^2+x^2 \log (x)+7 x-12 \log (x)-12}{(x-4)^2 (x-3)^2}\right )dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {3 \left (-x^2+2 x^2 \log (x)+7 x-3 x \log (x)-12 \log (x)-12\right )}{(x-4)^2 (x-3)^2 \left (x \log ^2\left (3 x^2\right )-4 \log ^2\left (3 x^2\right )-3\right )}+\frac {3 \log (x) \left (4 x^2 \log \left (3 x^2\right )-32 x \log \left (3 x^2\right )+64 \log \left (3 x^2\right )+3 x\right )}{(x-4)^2 (x-3) \left (x \log ^2\left (3 x^2\right )-4 \log ^2\left (3 x^2\right )-3\right )^2}+\frac {-x^2+x^2 \log (x)+7 x-12 \log (x)-12}{(x-4)^2 (x-3)^2}\right )dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {3 \left (-x^2+2 x^2 \log (x)+7 x-3 x \log (x)-12 \log (x)-12\right )}{(x-4)^2 (x-3)^2 \left (x \log ^2\left (3 x^2\right )-4 \log ^2\left (3 x^2\right )-3\right )}+\frac {3 \log (x) \left (4 x^2 \log \left (3 x^2\right )-32 x \log \left (3 x^2\right )+64 \log \left (3 x^2\right )+3 x\right )}{(x-4)^2 (x-3) \left (x \log ^2\left (3 x^2\right )-4 \log ^2\left (3 x^2\right )-3\right )^2}+\frac {-x^2+x^2 \log (x)+7 x-12 \log (x)-12}{(x-4)^2 (x-3)^2}\right )dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {3 \left (-x^2+2 x^2 \log (x)+7 x-3 x \log (x)-12 \log (x)-12\right )}{(x-4)^2 (x-3)^2 \left (x \log ^2\left (3 x^2\right )-4 \log ^2\left (3 x^2\right )-3\right )}+\frac {3 \log (x) \left (4 x^2 \log \left (3 x^2\right )-32 x \log \left (3 x^2\right )+64 \log \left (3 x^2\right )+3 x\right )}{(x-4)^2 (x-3) \left (x \log ^2\left (3 x^2\right )-4 \log ^2\left (3 x^2\right )-3\right )^2}+\frac {-x^2+x^2 \log (x)+7 x-12 \log (x)-12}{(x-4)^2 (x-3)^2}\right )dx\)

Input:

Int[((-36 + 12*x)*Log[x]*Log[3*x^2] + (-9 + 3*x - 9*Log[x])*Log[3*x^2]^2 + 
 (-12 + 7*x - x^2 + (-12 + x^2)*Log[x])*Log[3*x^2]^4)/(81 - 54*x + 9*x^2 + 
 (216 - 198*x + 60*x^2 - 6*x^3)*Log[3*x^2]^2 + (144 - 168*x + 73*x^2 - 14* 
x^3 + x^4)*Log[3*x^2]^4),x]
 

Output:

$Aborted
 

Maple [C] (warning: unable to verify)

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

Time = 103.39 (sec) , antiderivative size = 498, normalized size of antiderivative = 19.92

\[-\frac {x \ln \left (x \right )}{x^{2}-7 x +12}-\frac {12 \ln \left (x \right ) x}{\left (x^{2}-7 x +12\right ) \left (-12+4 x \ln \left (3\right )^{2}-64 \ln \left (3\right ) \ln \left (x \right )+16 x \ln \left (3\right ) \ln \left (x \right )+16 x \ln \left (x \right )^{2}-16 \ln \left (3\right )^{2}-64 \ln \left (x \right )^{2}-8 i x \ln \left (x \right ) \pi \operatorname {csgn}\left (i x^{2}\right )^{3}-4 i x \ln \left (3\right ) \pi \operatorname {csgn}\left (i x^{2}\right )^{3}-64 i \ln \left (x \right ) \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+32 i \ln \left (x \right ) \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )+4 \pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{6}+16 i \ln \left (3\right ) \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+32 i \ln \left (x \right ) \pi \operatorname {csgn}\left (i x^{2}\right )^{3}-x \,\pi ^{2} \operatorname {csgn}\left (i x \right )^{4} \operatorname {csgn}\left (i x^{2}\right )^{2}+4 x \,\pi ^{2} \operatorname {csgn}\left (i x \right )^{3} \operatorname {csgn}\left (i x^{2}\right )^{3}-6 x \,\pi ^{2} \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )^{4}+4 x \,\pi ^{2} \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{5}+16 i \ln \left (3\right ) \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-32 i \ln \left (3\right ) \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+4 \pi ^{2} \operatorname {csgn}\left (i x \right )^{4} \operatorname {csgn}\left (i x^{2}\right )^{2}-16 \pi ^{2} \operatorname {csgn}\left (i x \right )^{3} \operatorname {csgn}\left (i x^{2}\right )^{3}+24 \pi ^{2} \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )^{4}-16 \pi ^{2} \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{5}-x \,\pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{6}-8 i x \ln \left (x \right ) \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )+16 i x \ln \left (x \right ) \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}-4 i x \ln \left (3\right ) \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )+8 i x \ln \left (3\right ) \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}\right )}\]

Input:

int((((x^2-12)*ln(x)-x^2+7*x-12)*ln(3*x^2)^4+(-9*ln(x)+3*x-9)*ln(3*x^2)^2+ 
(12*x-36)*ln(x)*ln(3*x^2))/((x^4-14*x^3+73*x^2-168*x+144)*ln(3*x^2)^4+(-6* 
x^3+60*x^2-198*x+216)*ln(3*x^2)^2+9*x^2-54*x+81),x)
 

Output:

-x/(x^2-7*x+12)*ln(x)-12*ln(x)*x/(x^2-7*x+12)/(-12+4*x*ln(3)^2-64*ln(3)*ln 
(x)+16*x*ln(3)*ln(x)+16*x*ln(x)^2-16*ln(3)^2-64*ln(x)^2-8*I*x*ln(x)*Pi*csg 
n(I*x)^2*csgn(I*x^2)-8*I*x*ln(x)*Pi*csgn(I*x^2)^3-4*I*x*ln(3)*Pi*csgn(I*x^ 
2)^3-64*I*ln(x)*Pi*csgn(I*x)*csgn(I*x^2)^2+4*Pi^2*csgn(I*x)^4*csgn(I*x^2)^ 
2-16*Pi^2*csgn(I*x)^3*csgn(I*x^2)^3+24*Pi^2*csgn(I*x)^2*csgn(I*x^2)^4-16*P 
i^2*csgn(I*x)*csgn(I*x^2)^5-x*Pi^2*csgn(I*x^2)^6+4*Pi^2*csgn(I*x^2)^6+16*I 
*ln(3)*Pi*csgn(I*x^2)^3+32*I*ln(x)*Pi*csgn(I*x^2)^3-x*Pi^2*csgn(I*x)^4*csg 
n(I*x^2)^2+4*x*Pi^2*csgn(I*x)^3*csgn(I*x^2)^3-6*x*Pi^2*csgn(I*x)^2*csgn(I* 
x^2)^4+4*x*Pi^2*csgn(I*x)*csgn(I*x^2)^5+16*I*x*ln(x)*Pi*csgn(I*x)*csgn(I*x 
^2)^2-4*I*x*ln(3)*Pi*csgn(I*x)^2*csgn(I*x^2)+8*I*x*ln(3)*Pi*csgn(I*x)*csgn 
(I*x^2)^2+32*I*ln(x)*Pi*csgn(I*x)^2*csgn(I*x^2)+16*I*ln(3)*Pi*csgn(I*x)^2* 
csgn(I*x^2)-32*I*ln(3)*Pi*csgn(I*x)*csgn(I*x^2)^2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (25) = 50\).

Time = 0.12 (sec) , antiderivative size = 75, normalized size of antiderivative = 3.00 \[ \int \frac {(-36+12 x) \log (x) \log \left (3 x^2\right )+(-9+3 x-9 \log (x)) \log ^2\left (3 x^2\right )+\left (-12+7 x-x^2+\left (-12+x^2\right ) \log (x)\right ) \log ^4\left (3 x^2\right )}{81-54 x+9 x^2+\left (216-198 x+60 x^2-6 x^3\right ) \log ^2\left (3 x^2\right )+\left (144-168 x+73 x^2-14 x^3+x^4\right ) \log ^4\left (3 x^2\right )} \, dx=-\frac {x \log \left (3\right )^{2} \log \left (x\right ) + 4 \, x \log \left (3\right ) \log \left (x\right )^{2} + 4 \, x \log \left (x\right )^{3}}{{\left (x^{2} - 7 \, x + 12\right )} \log \left (3\right )^{2} + 4 \, {\left (x^{2} - 7 \, x + 12\right )} \log \left (3\right ) \log \left (x\right ) + 4 \, {\left (x^{2} - 7 \, x + 12\right )} \log \left (x\right )^{2} - 3 \, x + 9} \] Input:

integrate((((x^2-12)*log(x)-x^2+7*x-12)*log(3*x^2)^4+(-9*log(x)+3*x-9)*log 
(3*x^2)^2+(12*x-36)*log(x)*log(3*x^2))/((x^4-14*x^3+73*x^2-168*x+144)*log( 
3*x^2)^4+(-6*x^3+60*x^2-198*x+216)*log(3*x^2)^2+9*x^2-54*x+81),x, algorith 
m="fricas")
 

Output:

-(x*log(3)^2*log(x) + 4*x*log(3)*log(x)^2 + 4*x*log(x)^3)/((x^2 - 7*x + 12 
)*log(3)^2 + 4*(x^2 - 7*x + 12)*log(3)*log(x) + 4*(x^2 - 7*x + 12)*log(x)^ 
2 - 3*x + 9)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (22) = 44\).

Time = 0.38 (sec) , antiderivative size = 117, normalized size of antiderivative = 4.68 \[ \int \frac {(-36+12 x) \log (x) \log \left (3 x^2\right )+(-9+3 x-9 \log (x)) \log ^2\left (3 x^2\right )+\left (-12+7 x-x^2+\left (-12+x^2\right ) \log (x)\right ) \log ^4\left (3 x^2\right )}{81-54 x+9 x^2+\left (216-198 x+60 x^2-6 x^3\right ) \log ^2\left (3 x^2\right )+\left (144-168 x+73 x^2-14 x^3+x^4\right ) \log ^4\left (3 x^2\right )} \, dx=- \frac {3 x \log {\left (x \right )}}{x^{3} \log {\left (3 \right )}^{2} - 11 x^{2} \log {\left (3 \right )}^{2} - 3 x^{2} + 21 x + 40 x \log {\left (3 \right )}^{2} + \left (4 x^{3} - 44 x^{2} + 160 x - 192\right ) \log {\left (x \right )}^{2} + \left (4 x^{3} \log {\left (3 \right )} - 44 x^{2} \log {\left (3 \right )} + 160 x \log {\left (3 \right )} - 192 \log {\left (3 \right )}\right ) \log {\left (x \right )} - 48 \log {\left (3 \right )}^{2} - 36} - \frac {x \log {\left (x \right )}}{x^{2} - 7 x + 12} \] Input:

integrate((((x**2-12)*ln(x)-x**2+7*x-12)*ln(3*x**2)**4+(-9*ln(x)+3*x-9)*ln 
(3*x**2)**2+(12*x-36)*ln(x)*ln(3*x**2))/((x**4-14*x**3+73*x**2-168*x+144)* 
ln(3*x**2)**4+(-6*x**3+60*x**2-198*x+216)*ln(3*x**2)**2+9*x**2-54*x+81),x)
 

Output:

-3*x*log(x)/(x**3*log(3)**2 - 11*x**2*log(3)**2 - 3*x**2 + 21*x + 40*x*log 
(3)**2 + (4*x**3 - 44*x**2 + 160*x - 192)*log(x)**2 + (4*x**3*log(3) - 44* 
x**2*log(3) + 160*x*log(3) - 192*log(3))*log(x) - 48*log(3)**2 - 36) - x*l 
og(x)/(x**2 - 7*x + 12)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 140 vs. \(2 (25) = 50\).

Time = 0.20 (sec) , antiderivative size = 140, normalized size of antiderivative = 5.60 \[ \int \frac {(-36+12 x) \log (x) \log \left (3 x^2\right )+(-9+3 x-9 \log (x)) \log ^2\left (3 x^2\right )+\left (-12+7 x-x^2+\left (-12+x^2\right ) \log (x)\right ) \log ^4\left (3 x^2\right )}{81-54 x+9 x^2+\left (216-198 x+60 x^2-6 x^3\right ) \log ^2\left (3 x^2\right )+\left (144-168 x+73 x^2-14 x^3+x^4\right ) \log ^4\left (3 x^2\right )} \, dx=-\frac {4 \, {\left (x^{2} - 4 \, x\right )} \log \left (x\right )^{3} + 4 \, {\left (x^{2} \log \left (3\right ) - 4 \, x \log \left (3\right )\right )} \log \left (x\right )^{2} + {\left (x^{2} \log \left (3\right )^{2} - 4 \, x \log \left (3\right )^{2}\right )} \log \left (x\right )}{x^{3} \log \left (3\right )^{2} - {\left (11 \, \log \left (3\right )^{2} + 3\right )} x^{2} + 4 \, {\left (x^{3} - 11 \, x^{2} + 40 \, x - 48\right )} \log \left (x\right )^{2} + {\left (40 \, \log \left (3\right )^{2} + 21\right )} x - 48 \, \log \left (3\right )^{2} + 4 \, {\left (x^{3} \log \left (3\right ) - 11 \, x^{2} \log \left (3\right ) + 40 \, x \log \left (3\right ) - 48 \, \log \left (3\right )\right )} \log \left (x\right ) - 36} \] Input:

integrate((((x^2-12)*log(x)-x^2+7*x-12)*log(3*x^2)^4+(-9*log(x)+3*x-9)*log 
(3*x^2)^2+(12*x-36)*log(x)*log(3*x^2))/((x^4-14*x^3+73*x^2-168*x+144)*log( 
3*x^2)^4+(-6*x^3+60*x^2-198*x+216)*log(3*x^2)^2+9*x^2-54*x+81),x, algorith 
m="maxima")
 

Output:

-(4*(x^2 - 4*x)*log(x)^3 + 4*(x^2*log(3) - 4*x*log(3))*log(x)^2 + (x^2*log 
(3)^2 - 4*x*log(3)^2)*log(x))/(x^3*log(3)^2 - (11*log(3)^2 + 3)*x^2 + 4*(x 
^3 - 11*x^2 + 40*x - 48)*log(x)^2 + (40*log(3)^2 + 21)*x - 48*log(3)^2 + 4 
*(x^3*log(3) - 11*x^2*log(3) + 40*x*log(3) - 48*log(3))*log(x) - 36)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 125 vs. \(2 (25) = 50\).

Time = 0.40 (sec) , antiderivative size = 125, normalized size of antiderivative = 5.00 \[ \int \frac {(-36+12 x) \log (x) \log \left (3 x^2\right )+(-9+3 x-9 \log (x)) \log ^2\left (3 x^2\right )+\left (-12+7 x-x^2+\left (-12+x^2\right ) \log (x)\right ) \log ^4\left (3 x^2\right )}{81-54 x+9 x^2+\left (216-198 x+60 x^2-6 x^3\right ) \log ^2\left (3 x^2\right )+\left (144-168 x+73 x^2-14 x^3+x^4\right ) \log ^4\left (3 x^2\right )} \, dx=-\frac {3 \, x \log \left (x\right )}{x^{3} \log \left (3\right )^{2} + 4 \, x^{3} \log \left (3\right ) \log \left (x\right ) + 4 \, x^{3} \log \left (x\right )^{2} - 11 \, x^{2} \log \left (3\right )^{2} - 44 \, x^{2} \log \left (3\right ) \log \left (x\right ) - 44 \, x^{2} \log \left (x\right )^{2} + 40 \, x \log \left (3\right )^{2} + 160 \, x \log \left (3\right ) \log \left (x\right ) + 160 \, x \log \left (x\right )^{2} - 3 \, x^{2} - 48 \, \log \left (3\right )^{2} - 192 \, \log \left (3\right ) \log \left (x\right ) - 192 \, \log \left (x\right )^{2} + 21 \, x - 36} - \frac {x \log \left (x\right )}{x^{2} - 7 \, x + 12} \] Input:

integrate((((x^2-12)*log(x)-x^2+7*x-12)*log(3*x^2)^4+(-9*log(x)+3*x-9)*log 
(3*x^2)^2+(12*x-36)*log(x)*log(3*x^2))/((x^4-14*x^3+73*x^2-168*x+144)*log( 
3*x^2)^4+(-6*x^3+60*x^2-198*x+216)*log(3*x^2)^2+9*x^2-54*x+81),x, algorith 
m="giac")
 

Output:

-3*x*log(x)/(x^3*log(3)^2 + 4*x^3*log(3)*log(x) + 4*x^3*log(x)^2 - 11*x^2* 
log(3)^2 - 44*x^2*log(3)*log(x) - 44*x^2*log(x)^2 + 40*x*log(3)^2 + 160*x* 
log(3)*log(x) + 160*x*log(x)^2 - 3*x^2 - 48*log(3)^2 - 192*log(3)*log(x) - 
 192*log(x)^2 + 21*x - 36) - x*log(x)/(x^2 - 7*x + 12)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(-36+12 x) \log (x) \log \left (3 x^2\right )+(-9+3 x-9 \log (x)) \log ^2\left (3 x^2\right )+\left (-12+7 x-x^2+\left (-12+x^2\right ) \log (x)\right ) \log ^4\left (3 x^2\right )}{81-54 x+9 x^2+\left (216-198 x+60 x^2-6 x^3\right ) \log ^2\left (3 x^2\right )+\left (144-168 x+73 x^2-14 x^3+x^4\right ) \log ^4\left (3 x^2\right )} \, dx=\int \frac {\left (7\,x+\ln \left (x\right )\,\left (x^2-12\right )-x^2-12\right )\,{\ln \left (3\,x^2\right )}^4+\left (3\,x-9\,\ln \left (x\right )-9\right )\,{\ln \left (3\,x^2\right )}^2+\ln \left (x\right )\,\left (12\,x-36\right )\,\ln \left (3\,x^2\right )}{9\,x^2-{\ln \left (3\,x^2\right )}^2\,\left (6\,x^3-60\,x^2+198\,x-216\right )-54\,x+{\ln \left (3\,x^2\right )}^4\,\left (x^4-14\,x^3+73\,x^2-168\,x+144\right )+81} \,d x \] Input:

int((log(3*x^2)^4*(7*x + log(x)*(x^2 - 12) - x^2 - 12) - log(3*x^2)^2*(9*l 
og(x) - 3*x + 9) + log(3*x^2)*log(x)*(12*x - 36))/(9*x^2 - log(3*x^2)^2*(1 
98*x - 60*x^2 + 6*x^3 - 216) - 54*x + log(3*x^2)^4*(73*x^2 - 168*x - 14*x^ 
3 + x^4 + 144) + 81),x)
 

Output:

int((log(3*x^2)^4*(7*x + log(x)*(x^2 - 12) - x^2 - 12) - log(3*x^2)^2*(9*l 
og(x) - 3*x + 9) + log(3*x^2)*log(x)*(12*x - 36))/(9*x^2 - log(3*x^2)^2*(1 
98*x - 60*x^2 + 6*x^3 - 216) - 54*x + log(3*x^2)^4*(73*x^2 - 168*x - 14*x^ 
3 + x^4 + 144) + 81), x)
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.12 \[ \int \frac {(-36+12 x) \log (x) \log \left (3 x^2\right )+(-9+3 x-9 \log (x)) \log ^2\left (3 x^2\right )+\left (-12+7 x-x^2+\left (-12+x^2\right ) \log (x)\right ) \log ^4\left (3 x^2\right )}{81-54 x+9 x^2+\left (216-198 x+60 x^2-6 x^3\right ) \log ^2\left (3 x^2\right )+\left (144-168 x+73 x^2-14 x^3+x^4\right ) \log ^4\left (3 x^2\right )} \, dx=-\frac {\mathrm {log}\left (3 x^{2}\right )^{2} \mathrm {log}\left (x \right ) x}{\mathrm {log}\left (3 x^{2}\right )^{2} x^{2}-7 \mathrm {log}\left (3 x^{2}\right )^{2} x +12 \mathrm {log}\left (3 x^{2}\right )^{2}-3 x +9} \] Input:

int((((x^2-12)*log(x)-x^2+7*x-12)*log(3*x^2)^4+(-9*log(x)+3*x-9)*log(3*x^2 
)^2+(12*x-36)*log(x)*log(3*x^2))/((x^4-14*x^3+73*x^2-168*x+144)*log(3*x^2) 
^4+(-6*x^3+60*x^2-198*x+216)*log(3*x^2)^2+9*x^2-54*x+81),x)
 

Output:

( - log(3*x**2)**2*log(x)*x)/(log(3*x**2)**2*x**2 - 7*log(3*x**2)**2*x + 1 
2*log(3*x**2)**2 - 3*x + 9)