\(\int \frac {-8+(-x-2 x^2) \log (\frac {1}{2} (4 x^2-x^2 \log (3))) \log ^2(\log (\frac {1}{2} (4 x^2-x^2 \log (3))))}{16 x \log (\frac {1}{2} (4 x^2-x^2 \log (3)))+(-8 x-8 x^2-8 x^3) \log (\frac {1}{2} (4 x^2-x^2 \log (3))) \log (\log (\frac {1}{2} (4 x^2-x^2 \log (3))))+(x+2 x^2+3 x^3+2 x^4+x^5) \log (\frac {1}{2} (4 x^2-x^2 \log (3))) \log ^2(\log (\frac {1}{2} (4 x^2-x^2 \log (3))))} \, dx\) [165]

Optimal result

Integrand size = 188, antiderivative size = 28 \[ \int \frac {-8+\left (-x-2 x^2\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log ^2\left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )}{16 x \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )+\left (-8 x-8 x^2-8 x^3\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log \left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )+\left (x+2 x^2+3 x^3+2 x^4+x^5\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log ^2\left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )} \, dx=\frac {1}{1+x (1+x)-\frac {4}{\log \left (\log \left (\frac {1}{2} x^2 (4-\log (3))\right )\right )}} \] Output:

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

Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.36 \[ \int \frac {-8+\left (-x-2 x^2\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log ^2\left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )}{16 x \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )+\left (-8 x-8 x^2-8 x^3\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log \left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )+\left (x+2 x^2+3 x^3+2 x^4+x^5\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log ^2\left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )} \, dx=\frac {\log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )}{-4+\left (1+x+x^2\right ) \log \left (\log \left (-\frac {1}{2} x^2 (-4+\log (3))\right )\right )} \] Input:

Integrate[(-8 + (-x - 2*x^2)*Log[(4*x^2 - x^2*Log[3])/2]*Log[Log[(4*x^2 - 
x^2*Log[3])/2]]^2)/(16*x*Log[(4*x^2 - x^2*Log[3])/2] + (-8*x - 8*x^2 - 8*x 
^3)*Log[(4*x^2 - x^2*Log[3])/2]*Log[Log[(4*x^2 - x^2*Log[3])/2]] + (x + 2* 
x^2 + 3*x^3 + 2*x^4 + x^5)*Log[(4*x^2 - x^2*Log[3])/2]*Log[Log[(4*x^2 - x^ 
2*Log[3])/2]]^2),x]
 

Output:

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

\(\displaystyle \int \frac {\left (-2 x^2-x\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log ^2\left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )-8}{16 x \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )+\left (-8 x^3-8 x^2-8 x\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log \left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )+\left (x^5+2 x^4+3 x^3+2 x^2+x\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log ^2\left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )} \, dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

Input:

Int[(-8 + (-x - 2*x^2)*Log[(4*x^2 - x^2*Log[3])/2]*Log[Log[(4*x^2 - x^2*Lo 
g[3])/2]]^2)/(16*x*Log[(4*x^2 - x^2*Log[3])/2] + (-8*x - 8*x^2 - 8*x^3)*Lo 
g[(4*x^2 - x^2*Log[3])/2]*Log[Log[(4*x^2 - x^2*Log[3])/2]] + (x + 2*x^2 + 
3*x^3 + 2*x^4 + x^5)*Log[(4*x^2 - x^2*Log[3])/2]*Log[Log[(4*x^2 - x^2*Log[ 
3])/2]]^2),x]
 

Output:

$Aborted
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(55\) vs. \(2(26)=52\).

Time = 1.40 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.00

method	result	size
parallelrisch	\(\frac {\ln \left (\ln \left (-\frac {\left (-4+\ln \left (3\right )\right ) x^{2}}{2}\right )\right )}{\ln \left (\ln \left (-\frac {\left (-4+\ln \left (3\right )\right ) x^{2}}{2}\right )\right ) x^{2}+\ln \left (\ln \left (-\frac {\left (-4+\ln \left (3\right )\right ) x^{2}}{2}\right )\right ) x +\ln \left (\ln \left (-\frac {\left (-4+\ln \left (3\right )\right ) x^{2}}{2}\right )\right )-4}\)	\(56\)

Input:

int(((-2*x^2-x)*ln(-1/2*x^2*ln(3)+2*x^2)*ln(ln(-1/2*x^2*ln(3)+2*x^2))^2-8) 
/((x^5+2*x^4+3*x^3+2*x^2+x)*ln(-1/2*x^2*ln(3)+2*x^2)*ln(ln(-1/2*x^2*ln(3)+ 
2*x^2))^2+(-8*x^3-8*x^2-8*x)*ln(-1/2*x^2*ln(3)+2*x^2)*ln(ln(-1/2*x^2*ln(3) 
+2*x^2))+16*x*ln(-1/2*x^2*ln(3)+2*x^2)),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.50 \[ \int \frac {-8+\left (-x-2 x^2\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log ^2\left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )}{16 x \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )+\left (-8 x-8 x^2-8 x^3\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log \left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )+\left (x+2 x^2+3 x^3+2 x^4+x^5\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log ^2\left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )} \, dx=\frac {\log \left (\log \left (-\frac {1}{2} \, x^{2} \log \left (3\right ) + 2 \, x^{2}\right )\right )}{{\left (x^{2} + x + 1\right )} \log \left (\log \left (-\frac {1}{2} \, x^{2} \log \left (3\right ) + 2 \, x^{2}\right )\right ) - 4} \] Input:

integrate(((-2*x^2-x)*log(-1/2*x^2*log(3)+2*x^2)*log(log(-1/2*x^2*log(3)+2 
*x^2))^2-8)/((x^5+2*x^4+3*x^3+2*x^2+x)*log(-1/2*x^2*log(3)+2*x^2)*log(log( 
-1/2*x^2*log(3)+2*x^2))^2+(-8*x^3-8*x^2-8*x)*log(-1/2*x^2*log(3)+2*x^2)*lo 
g(log(-1/2*x^2*log(3)+2*x^2))+16*x*log(-1/2*x^2*log(3)+2*x^2)),x, algorith 
m="fricas")
 

Output:

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (22) = 44\).

Time = 0.14 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.93 \[ \int \frac {-8+\left (-x-2 x^2\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log ^2\left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )}{16 x \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )+\left (-8 x-8 x^2-8 x^3\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log \left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )+\left (x+2 x^2+3 x^3+2 x^4+x^5\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log ^2\left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )} \, dx=\frac {4}{- 4 x^{2} - 4 x + \left (x^{4} + 2 x^{3} + 3 x^{2} + 2 x + 1\right ) \log {\left (\log {\left (- \frac {x^{2} \log {\left (3 \right )}}{2} + 2 x^{2} \right )} \right )} - 4} + \frac {1}{x^{2} + x + 1} \] Input:

integrate(((-2*x**2-x)*ln(-1/2*x**2*ln(3)+2*x**2)*ln(ln(-1/2*x**2*ln(3)+2* 
x**2))**2-8)/((x**5+2*x**4+3*x**3+2*x**2+x)*ln(-1/2*x**2*ln(3)+2*x**2)*ln( 
ln(-1/2*x**2*ln(3)+2*x**2))**2+(-8*x**3-8*x**2-8*x)*ln(-1/2*x**2*ln(3)+2*x 
**2)*ln(ln(-1/2*x**2*ln(3)+2*x**2))+16*x*ln(-1/2*x**2*ln(3)+2*x**2)),x)
 

Output:

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

Maxima [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.64 \[ \int \frac {-8+\left (-x-2 x^2\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log ^2\left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )}{16 x \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )+\left (-8 x-8 x^2-8 x^3\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log \left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )+\left (x+2 x^2+3 x^3+2 x^4+x^5\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log ^2\left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )} \, dx=\frac {\log \left (-\log \left (2\right ) + 2 \, \log \left (x\right ) + \log \left (-\log \left (3\right ) + 4\right )\right )}{{\left (x^{2} + x + 1\right )} \log \left (-\log \left (2\right ) + 2 \, \log \left (x\right ) + \log \left (-\log \left (3\right ) + 4\right )\right ) - 4} \] Input:

integrate(((-2*x^2-x)*log(-1/2*x^2*log(3)+2*x^2)*log(log(-1/2*x^2*log(3)+2 
*x^2))^2-8)/((x^5+2*x^4+3*x^3+2*x^2+x)*log(-1/2*x^2*log(3)+2*x^2)*log(log( 
-1/2*x^2*log(3)+2*x^2))^2+(-8*x^3-8*x^2-8*x)*log(-1/2*x^2*log(3)+2*x^2)*lo 
g(log(-1/2*x^2*log(3)+2*x^2))+16*x*log(-1/2*x^2*log(3)+2*x^2)),x, algorith 
m="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 140 vs. \(2 (24) = 48\).

Time = 0.77 (sec) , antiderivative size = 140, normalized size of antiderivative = 5.00 \[ \int \frac {-8+\left (-x-2 x^2\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log ^2\left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )}{16 x \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )+\left (-8 x-8 x^2-8 x^3\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log \left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )+\left (x+2 x^2+3 x^3+2 x^4+x^5\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log ^2\left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )} \, dx=\frac {4}{x^{4} \log \left (-\log \left (2\right ) + \log \left (-x^{2} \log \left (3\right ) + 4 \, x^{2}\right )\right ) + 2 \, x^{3} \log \left (-\log \left (2\right ) + \log \left (-x^{2} \log \left (3\right ) + 4 \, x^{2}\right )\right ) + 3 \, x^{2} \log \left (-\log \left (2\right ) + \log \left (-x^{2} \log \left (3\right ) + 4 \, x^{2}\right )\right ) - 4 \, x^{2} + 2 \, x \log \left (-\log \left (2\right ) + \log \left (-x^{2} \log \left (3\right ) + 4 \, x^{2}\right )\right ) - 4 \, x + \log \left (-\log \left (2\right ) + \log \left (-x^{2} \log \left (3\right ) + 4 \, x^{2}\right )\right ) - 4} + \frac {1}{x^{2} + x + 1} \] Input:

integrate(((-2*x^2-x)*log(-1/2*x^2*log(3)+2*x^2)*log(log(-1/2*x^2*log(3)+2 
*x^2))^2-8)/((x^5+2*x^4+3*x^3+2*x^2+x)*log(-1/2*x^2*log(3)+2*x^2)*log(log( 
-1/2*x^2*log(3)+2*x^2))^2+(-8*x^3-8*x^2-8*x)*log(-1/2*x^2*log(3)+2*x^2)*lo 
g(log(-1/2*x^2*log(3)+2*x^2))+16*x*log(-1/2*x^2*log(3)+2*x^2)),x, algorith 
m="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {-8+\left (-x-2 x^2\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log ^2\left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )}{16 x \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )+\left (-8 x-8 x^2-8 x^3\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log \left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )+\left (x+2 x^2+3 x^3+2 x^4+x^5\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log ^2\left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )} \, dx=\int -\frac {\ln \left (2\,x^2-\frac {x^2\,\ln \left (3\right )}{2}\right )\,\left (2\,x^2+x\right )\,{\ln \left (\ln \left (2\,x^2-\frac {x^2\,\ln \left (3\right )}{2}\right )\right )}^2+8}{\ln \left (2\,x^2-\frac {x^2\,\ln \left (3\right )}{2}\right )\,\left (x^5+2\,x^4+3\,x^3+2\,x^2+x\right )\,{\ln \left (\ln \left (2\,x^2-\frac {x^2\,\ln \left (3\right )}{2}\right )\right )}^2-\ln \left (2\,x^2-\frac {x^2\,\ln \left (3\right )}{2}\right )\,\left (8\,x^3+8\,x^2+8\,x\right )\,\ln \left (\ln \left (2\,x^2-\frac {x^2\,\ln \left (3\right )}{2}\right )\right )+16\,x\,\ln \left (2\,x^2-\frac {x^2\,\ln \left (3\right )}{2}\right )} \,d x \] Input:

int(-(log(log(2*x^2 - (x^2*log(3))/2))^2*log(2*x^2 - (x^2*log(3))/2)*(x + 
2*x^2) + 8)/(16*x*log(2*x^2 - (x^2*log(3))/2) - log(log(2*x^2 - (x^2*log(3 
))/2))*log(2*x^2 - (x^2*log(3))/2)*(8*x + 8*x^2 + 8*x^3) + log(log(2*x^2 - 
 (x^2*log(3))/2))^2*log(2*x^2 - (x^2*log(3))/2)*(x + 2*x^2 + 3*x^3 + 2*x^4 
 + x^5)),x)
 

Output:

int(-(log(log(2*x^2 - (x^2*log(3))/2))^2*log(2*x^2 - (x^2*log(3))/2)*(x + 
2*x^2) + 8)/(16*x*log(2*x^2 - (x^2*log(3))/2) - log(log(2*x^2 - (x^2*log(3 
))/2))*log(2*x^2 - (x^2*log(3))/2)*(8*x + 8*x^2 + 8*x^3) + log(log(2*x^2 - 
 (x^2*log(3))/2))^2*log(2*x^2 - (x^2*log(3))/2)*(x + 2*x^2 + 3*x^3 + 2*x^4 
 + x^5)), x)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.54 \[ \int \frac {-8+\left (-x-2 x^2\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log ^2\left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )}{16 x \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )+\left (-8 x-8 x^2-8 x^3\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log \left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )+\left (x+2 x^2+3 x^3+2 x^4+x^5\right ) \log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right ) \log ^2\left (\log \left (\frac {1}{2} \left (4 x^2-x^2 \log (3)\right )\right )\right )} \, dx=\frac {\mathrm {log}\left (\mathrm {log}\left (-\frac {\mathrm {log}\left (3\right ) x^{2}}{2}+2 x^{2}\right )\right )}{\mathrm {log}\left (\mathrm {log}\left (-\frac {\mathrm {log}\left (3\right ) x^{2}}{2}+2 x^{2}\right )\right ) x^{2}+\mathrm {log}\left (\mathrm {log}\left (-\frac {\mathrm {log}\left (3\right ) x^{2}}{2}+2 x^{2}\right )\right ) x +\mathrm {log}\left (\mathrm {log}\left (-\frac {\mathrm {log}\left (3\right ) x^{2}}{2}+2 x^{2}\right )\right )-4} \] Input:

int(((-2*x^2-x)*log(-1/2*x^2*log(3)+2*x^2)*log(log(-1/2*x^2*log(3)+2*x^2)) 
^2-8)/((x^5+2*x^4+3*x^3+2*x^2+x)*log(-1/2*x^2*log(3)+2*x^2)*log(log(-1/2*x 
^2*log(3)+2*x^2))^2+(-8*x^3-8*x^2-8*x)*log(-1/2*x^2*log(3)+2*x^2)*log(log( 
-1/2*x^2*log(3)+2*x^2))+16*x*log(-1/2*x^2*log(3)+2*x^2)),x)
 

Output:

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