\(\int \frac {(-2 x^2-2 x^3+(18 x+38 x^2+4 x^3) \log (9+x) \log (\log (9+x))+(18+20 x-34 x^2-4 x^3) \log (9+x) \log ^2(\log (9+x))) \log (\frac {x+x^2+(2+x-x^2+\log (x)) \log (\log (9+x))}{\log (\log (9+x))})}{(9 x^2+10 x^3+x^4) \log (9+x) \log (\log (9+x))+(18 x+11 x^2-8 x^3-x^4+(9 x+x^2) \log (x)) \log (9+x) \log ^2(\log (9+x))} \, dx\) [340]

Optimal result

Integrand size = 161, antiderivative size = 24 \[ \int \frac {\left (-2 x^2-2 x^3+\left (18 x+38 x^2+4 x^3\right ) \log (9+x) \log (\log (9+x))+\left (18+20 x-34 x^2-4 x^3\right ) \log (9+x) \log ^2(\log (9+x))\right ) \log \left (\frac {x+x^2+\left (2+x-x^2+\log (x)\right ) \log (\log (9+x))}{\log (\log (9+x))}\right )}{\left (9 x^2+10 x^3+x^4\right ) \log (9+x) \log (\log (9+x))+\left (18 x+11 x^2-8 x^3-x^4+\left (9 x+x^2\right ) \log (x)\right ) \log (9+x) \log ^2(\log (9+x))} \, dx=\log ^2\left (\log (x)-(1+x) \left (-2+x-\frac {x}{\log (\log (9+x))}\right )\right ) \] Output:

ln(ln(x)-(-2+x-x/ln(ln(x+9)))*(1+x))^2
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.33 \[ \int \frac {\left (-2 x^2-2 x^3+\left (18 x+38 x^2+4 x^3\right ) \log (9+x) \log (\log (9+x))+\left (18+20 x-34 x^2-4 x^3\right ) \log (9+x) \log ^2(\log (9+x))\right ) \log \left (\frac {x+x^2+\left (2+x-x^2+\log (x)\right ) \log (\log (9+x))}{\log (\log (9+x))}\right )}{\left (9 x^2+10 x^3+x^4\right ) \log (9+x) \log (\log (9+x))+\left (18 x+11 x^2-8 x^3-x^4+\left (9 x+x^2\right ) \log (x)\right ) \log (9+x) \log ^2(\log (9+x))} \, dx=\log ^2\left (\frac {x+x^2+\left (2+x-x^2+\log (x)\right ) \log (\log (9+x))}{\log (\log (9+x))}\right ) \] Input:

Integrate[((-2*x^2 - 2*x^3 + (18*x + 38*x^2 + 4*x^3)*Log[9 + x]*Log[Log[9 
+ x]] + (18 + 20*x - 34*x^2 - 4*x^3)*Log[9 + x]*Log[Log[9 + x]]^2)*Log[(x 
+ x^2 + (2 + x - x^2 + Log[x])*Log[Log[9 + x]])/Log[Log[9 + x]]])/((9*x^2 
+ 10*x^3 + x^4)*Log[9 + x]*Log[Log[9 + x]] + (18*x + 11*x^2 - 8*x^3 - x^4 
+ (9*x + x^2)*Log[x])*Log[9 + x]*Log[Log[9 + x]]^2),x]
 

Output:

Log[(x + x^2 + (2 + x - x^2 + Log[x])*Log[Log[9 + x]])/Log[Log[9 + x]]]^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.

\(\displaystyle \int \frac {\left (-2 x^3-2 x^2+\left (-4 x^3-34 x^2+20 x+18\right ) \log (x+9) \log ^2(\log (x+9))+\left (4 x^3+38 x^2+18 x\right ) \log (x+9) \log (\log (x+9))\right ) \log \left (\frac {x^2+\left (-x^2+x+\log (x)+2\right ) \log (\log (x+9))+x}{\log (\log (x+9))}\right )}{\left (-x^4-8 x^3+11 x^2+\left (x^2+9 x\right ) \log (x)+18 x\right ) \log (x+9) \log ^2(\log (x+9))+\left (x^4+10 x^3+9 x^2\right ) \log (x+9) \log (\log (x+9))} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {\left (-2 x^3-2 x^2+\left (-4 x^3-34 x^2+20 x+18\right ) \log (x+9) \log ^2(\log (x+9))+\left (4 x^3+38 x^2+18 x\right ) \log (x+9) \log (\log (x+9))\right ) \log \left (\frac {x^2+\left (-x^2+x+\log (x)+2\right ) \log (\log (x+9))+x}{\log (\log (x+9))}\right )}{x (x+9) \log (x+9) \log (\log (x+9)) \left (x^2+x^2 (-\log (\log (x+9)))+x+x \log (\log (x+9))+\log (x) \log (\log (x+9))+2 \log (\log (x+9))\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {2 \left (x^3+2 x^3 \log (x+9) \log ^2(\log (x+9))-2 x^3 \log (x+9) \log (\log (x+9))+x^2+17 x^2 \log (x+9) \log ^2(\log (x+9))-19 x^2 \log (x+9) \log (\log (x+9))-10 x \log (x+9) \log ^2(\log (x+9))-9 \log (x+9) \log ^2(\log (x+9))-9 x \log (x+9) \log (\log (x+9))\right ) \log \left (\frac {x^2+\left (-x^2+x+\log (x)+2\right ) \log (\log (x+9))+x}{\log (\log (x+9))}\right )}{9 x \log (x+9) \log (\log (x+9)) \left (-x^2+x^2 \log (\log (x+9))-x-x \log (\log (x+9))-\log (x) \log (\log (x+9))-2 \log (\log (x+9))\right )}-\frac {2 \left (x^3+2 x^3 \log (x+9) \log ^2(\log (x+9))-2 x^3 \log (x+9) \log (\log (x+9))+x^2+17 x^2 \log (x+9) \log ^2(\log (x+9))-19 x^2 \log (x+9) \log (\log (x+9))-10 x \log (x+9) \log ^2(\log (x+9))-9 \log (x+9) \log ^2(\log (x+9))-9 x \log (x+9) \log (\log (x+9))\right ) \log \left (\frac {x^2+\left (-x^2+x+\log (x)+2\right ) \log (\log (x+9))+x}{\log (\log (x+9))}\right )}{9 (x+9) \log (x+9) \log (\log (x+9)) \left (-x^2+x^2 \log (\log (x+9))-x-x \log (\log (x+9))-\log (x) \log (\log (x+9))-2 \log (\log (x+9))\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {2 \left (-\left ((x+1) x^2\right )-\left (2 x^2+19 x+9\right ) \log (x+9) \log (\log (x+9)) ((x-1) \log (\log (x+9))-x)\right ) \log \left (\frac {x^2+\left (-x^2+x+\log (x)+2\right ) \log (\log (x+9))+x}{\log (\log (x+9))}\right )}{x (x+9) \log (x+9) \log (\log (x+9)) \left (x (x+1)-\left (x^2-x-\log (x)-2\right ) \log (\log (x+9))\right )}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \int -\frac {\left (x^2 (x+1)-\left (2 x^2+19 x+9\right ) \log (x+9) \log (\log (x+9)) (x+(1-x) \log (\log (x+9)))\right ) \log \left (\frac {x^2+x+\left (-x^2+x+\log (x)+2\right ) \log (\log (x+9))}{\log (\log (x+9))}\right )}{x (x+9) \log (x+9) \log (\log (x+9)) \left (x (x+1)+\left (-x^2+x+\log (x)+2\right ) \log (\log (x+9))\right )}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -2 \int \frac {\left (x^2 (x+1)-\left (2 x^2+19 x+9\right ) \log (x+9) \log (\log (x+9)) (x+(1-x) \log (\log (x+9)))\right ) \log \left (\frac {x^2+x+\left (-x^2+x+\log (x)+2\right ) \log (\log (x+9))}{\log (\log (x+9))}\right )}{x (x+9) \log (x+9) \log (\log (x+9)) \left (x (x+1)+\left (-x^2+x+\log (x)+2\right ) \log (\log (x+9))\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -2 \int \left (\frac {\left (2 \log (x+9) \log ^2(\log (x+9)) x^3-2 \log (x+9) \log (\log (x+9)) x^3+x^3+17 \log (x+9) \log ^2(\log (x+9)) x^2-19 \log (x+9) \log (\log (x+9)) x^2+x^2-10 \log (x+9) \log ^2(\log (x+9)) x-9 \log (x+9) \log (\log (x+9)) x-9 \log (x+9) \log ^2(\log (x+9))\right ) \log \left (\frac {x^2+x+\left (-x^2+x+\log (x)+2\right ) \log (\log (x+9))}{\log (\log (x+9))}\right )}{9 (x+9) \log (x+9) \log (\log (x+9)) \left (\log (\log (x+9)) x^2-x^2-\log (\log (x+9)) x-x-\log (x) \log (\log (x+9))-2 \log (\log (x+9))\right )}-\frac {\left (2 \log (x+9) \log ^2(\log (x+9)) x^3-2 \log (x+9) \log (\log (x+9)) x^3+x^3+17 \log (x+9) \log ^2(\log (x+9)) x^2-19 \log (x+9) \log (\log (x+9)) x^2+x^2-10 \log (x+9) \log ^2(\log (x+9)) x-9 \log (x+9) \log (\log (x+9)) x-9 \log (x+9) \log ^2(\log (x+9))\right ) \log \left (\frac {x^2+x+\left (-x^2+x+\log (x)+2\right ) \log (\log (x+9))}{\log (\log (x+9))}\right )}{9 x \log (x+9) \log (\log (x+9)) \left (\log (\log (x+9)) x^2-x^2-\log (\log (x+9)) x-x-\log (x) \log (\log (x+9))-2 \log (\log (x+9))\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -2 \int \frac {\left ((x+1) x^2+\left (2 x^2+19 x+9\right ) \log (x+9) \log (\log (x+9)) ((x-1) \log (\log (x+9))-x)\right ) \log \left (\frac {x^2+x+\left (-x^2+x+\log (x)+2\right ) \log (\log (x+9))}{\log (\log (x+9))}\right )}{x (x+9) \log (x+9) \log (\log (x+9)) \left (x (x+1)-\left (x^2-x-\log (x)-2\right ) \log (\log (x+9))\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -2 \int \left (\frac {\left (2 \log (x+9) \log ^2(\log (x+9)) x^3-2 \log (x+9) \log (\log (x+9)) x^3+x^3+17 \log (x+9) \log ^2(\log (x+9)) x^2-19 \log (x+9) \log (\log (x+9)) x^2+x^2-10 \log (x+9) \log ^2(\log (x+9)) x-9 \log (x+9) \log (\log (x+9)) x-9 \log (x+9) \log ^2(\log (x+9))\right ) \log \left (\frac {x^2+x+\left (-x^2+x+\log (x)+2\right ) \log (\log (x+9))}{\log (\log (x+9))}\right )}{9 (x+9) \log (x+9) \log (\log (x+9)) \left (\log (\log (x+9)) x^2-x^2-\log (\log (x+9)) x-x-\log (x) \log (\log (x+9))-2 \log (\log (x+9))\right )}-\frac {\left (2 \log (x+9) \log ^2(\log (x+9)) x^3-2 \log (x+9) \log (\log (x+9)) x^3+x^3+17 \log (x+9) \log ^2(\log (x+9)) x^2-19 \log (x+9) \log (\log (x+9)) x^2+x^2-10 \log (x+9) \log ^2(\log (x+9)) x-9 \log (x+9) \log (\log (x+9)) x-9 \log (x+9) \log ^2(\log (x+9))\right ) \log \left (\frac {x^2+x+\left (-x^2+x+\log (x)+2\right ) \log (\log (x+9))}{\log (\log (x+9))}\right )}{9 x \log (x+9) \log (\log (x+9)) \left (\log (\log (x+9)) x^2-x^2-\log (\log (x+9)) x-x-\log (x) \log (\log (x+9))-2 \log (\log (x+9))\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -2 \int \frac {\left ((x+1) x^2+\left (2 x^2+19 x+9\right ) \log (x+9) \log (\log (x+9)) ((x-1) \log (\log (x+9))-x)\right ) \log \left (\frac {x^2+x+\left (-x^2+x+\log (x)+2\right ) \log (\log (x+9))}{\log (\log (x+9))}\right )}{x (x+9) \log (x+9) \log (\log (x+9)) \left (x (x+1)-\left (x^2-x-\log (x)-2\right ) \log (\log (x+9))\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -2 \int \left (\frac {\left (2 \log (x+9) \log ^2(\log (x+9)) x^3-2 \log (x+9) \log (\log (x+9)) x^3+x^3+17 \log (x+9) \log ^2(\log (x+9)) x^2-19 \log (x+9) \log (\log (x+9)) x^2+x^2-10 \log (x+9) \log ^2(\log (x+9)) x-9 \log (x+9) \log (\log (x+9)) x-9 \log (x+9) \log ^2(\log (x+9))\right ) \log \left (\frac {x^2+x+\left (-x^2+x+\log (x)+2\right ) \log (\log (x+9))}{\log (\log (x+9))}\right )}{9 (x+9) \log (x+9) \log (\log (x+9)) \left (\log (\log (x+9)) x^2-x^2-\log (\log (x+9)) x-x-\log (x) \log (\log (x+9))-2 \log (\log (x+9))\right )}-\frac {\left (2 \log (x+9) \log ^2(\log (x+9)) x^3-2 \log (x+9) \log (\log (x+9)) x^3+x^3+17 \log (x+9) \log ^2(\log (x+9)) x^2-19 \log (x+9) \log (\log (x+9)) x^2+x^2-10 \log (x+9) \log ^2(\log (x+9)) x-9 \log (x+9) \log (\log (x+9)) x-9 \log (x+9) \log ^2(\log (x+9))\right ) \log \left (\frac {x^2+x+\left (-x^2+x+\log (x)+2\right ) \log (\log (x+9))}{\log (\log (x+9))}\right )}{9 x \log (x+9) \log (\log (x+9)) \left (\log (\log (x+9)) x^2-x^2-\log (\log (x+9)) x-x-\log (x) \log (\log (x+9))-2 \log (\log (x+9))\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -2 \int \frac {\left ((x+1) x^2+\left (2 x^2+19 x+9\right ) \log (x+9) \log (\log (x+9)) ((x-1) \log (\log (x+9))-x)\right ) \log \left (\frac {x^2+x+\left (-x^2+x+\log (x)+2\right ) \log (\log (x+9))}{\log (\log (x+9))}\right )}{x (x+9) \log (x+9) \log (\log (x+9)) \left (x (x+1)-\left (x^2-x-\log (x)-2\right ) \log (\log (x+9))\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -2 \int \left (\frac {\left (2 \log (x+9) \log ^2(\log (x+9)) x^3-2 \log (x+9) \log (\log (x+9)) x^3+x^3+17 \log (x+9) \log ^2(\log (x+9)) x^2-19 \log (x+9) \log (\log (x+9)) x^2+x^2-10 \log (x+9) \log ^2(\log (x+9)) x-9 \log (x+9) \log (\log (x+9)) x-9 \log (x+9) \log ^2(\log (x+9))\right ) \log \left (\frac {x^2+x+\left (-x^2+x+\log (x)+2\right ) \log (\log (x+9))}{\log (\log (x+9))}\right )}{9 (x+9) \log (x+9) \log (\log (x+9)) \left (\log (\log (x+9)) x^2-x^2-\log (\log (x+9)) x-x-\log (x) \log (\log (x+9))-2 \log (\log (x+9))\right )}-\frac {\left (2 \log (x+9) \log ^2(\log (x+9)) x^3-2 \log (x+9) \log (\log (x+9)) x^3+x^3+17 \log (x+9) \log ^2(\log (x+9)) x^2-19 \log (x+9) \log (\log (x+9)) x^2+x^2-10 \log (x+9) \log ^2(\log (x+9)) x-9 \log (x+9) \log (\log (x+9)) x-9 \log (x+9) \log ^2(\log (x+9))\right ) \log \left (\frac {x^2+x+\left (-x^2+x+\log (x)+2\right ) \log (\log (x+9))}{\log (\log (x+9))}\right )}{9 x \log (x+9) \log (\log (x+9)) \left (\log (\log (x+9)) x^2-x^2-\log (\log (x+9)) x-x-\log (x) \log (\log (x+9))-2 \log (\log (x+9))\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -2 \int \frac {\left ((x+1) x^2+\left (2 x^2+19 x+9\right ) \log (x+9) \log (\log (x+9)) ((x-1) \log (\log (x+9))-x)\right ) \log \left (\frac {x^2+x+\left (-x^2+x+\log (x)+2\right ) \log (\log (x+9))}{\log (\log (x+9))}\right )}{x (x+9) \log (x+9) \log (\log (x+9)) \left (x (x+1)-\left (x^2-x-\log (x)-2\right ) \log (\log (x+9))\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -2 \int \left (\frac {\left (2 \log (x+9) \log ^2(\log (x+9)) x^3-2 \log (x+9) \log (\log (x+9)) x^3+x^3+17 \log (x+9) \log ^2(\log (x+9)) x^2-19 \log (x+9) \log (\log (x+9)) x^2+x^2-10 \log (x+9) \log ^2(\log (x+9)) x-9 \log (x+9) \log (\log (x+9)) x-9 \log (x+9) \log ^2(\log (x+9))\right ) \log \left (\frac {x^2+x+\left (-x^2+x+\log (x)+2\right ) \log (\log (x+9))}{\log (\log (x+9))}\right )}{9 (x+9) \log (x+9) \log (\log (x+9)) \left (\log (\log (x+9)) x^2-x^2-\log (\log (x+9)) x-x-\log (x) \log (\log (x+9))-2 \log (\log (x+9))\right )}-\frac {\left (2 \log (x+9) \log ^2(\log (x+9)) x^3-2 \log (x+9) \log (\log (x+9)) x^3+x^3+17 \log (x+9) \log ^2(\log (x+9)) x^2-19 \log (x+9) \log (\log (x+9)) x^2+x^2-10 \log (x+9) \log ^2(\log (x+9)) x-9 \log (x+9) \log (\log (x+9)) x-9 \log (x+9) \log ^2(\log (x+9))\right ) \log \left (\frac {x^2+x+\left (-x^2+x+\log (x)+2\right ) \log (\log (x+9))}{\log (\log (x+9))}\right )}{9 x \log (x+9) \log (\log (x+9)) \left (\log (\log (x+9)) x^2-x^2-\log (\log (x+9)) x-x-\log (x) \log (\log (x+9))-2 \log (\log (x+9))\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -2 \int \frac {\left ((x+1) x^2+\left (2 x^2+19 x+9\right ) \log (x+9) \log (\log (x+9)) ((x-1) \log (\log (x+9))-x)\right ) \log \left (\frac {x^2+x+\left (-x^2+x+\log (x)+2\right ) \log (\log (x+9))}{\log (\log (x+9))}\right )}{x (x+9) \log (x+9) \log (\log (x+9)) \left (x (x+1)-\left (x^2-x-\log (x)-2\right ) \log (\log (x+9))\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -2 \int \left (\frac {\left (2 \log (x+9) \log ^2(\log (x+9)) x^3-2 \log (x+9) \log (\log (x+9)) x^3+x^3+17 \log (x+9) \log ^2(\log (x+9)) x^2-19 \log (x+9) \log (\log (x+9)) x^2+x^2-10 \log (x+9) \log ^2(\log (x+9)) x-9 \log (x+9) \log (\log (x+9)) x-9 \log (x+9) \log ^2(\log (x+9))\right ) \log \left (\frac {x^2+x+\left (-x^2+x+\log (x)+2\right ) \log (\log (x+9))}{\log (\log (x+9))}\right )}{9 (x+9) \log (x+9) \log (\log (x+9)) \left (\log (\log (x+9)) x^2-x^2-\log (\log (x+9)) x-x-\log (x) \log (\log (x+9))-2 \log (\log (x+9))\right )}-\frac {\left (2 \log (x+9) \log ^2(\log (x+9)) x^3-2 \log (x+9) \log (\log (x+9)) x^3+x^3+17 \log (x+9) \log ^2(\log (x+9)) x^2-19 \log (x+9) \log (\log (x+9)) x^2+x^2-10 \log (x+9) \log ^2(\log (x+9)) x-9 \log (x+9) \log (\log (x+9)) x-9 \log (x+9) \log ^2(\log (x+9))\right ) \log \left (\frac {x^2+x+\left (-x^2+x+\log (x)+2\right ) \log (\log (x+9))}{\log (\log (x+9))}\right )}{9 x \log (x+9) \log (\log (x+9)) \left (\log (\log (x+9)) x^2-x^2-\log (\log (x+9)) x-x-\log (x) \log (\log (x+9))-2 \log (\log (x+9))\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -2 \int \frac {\left ((x+1) x^2+\left (2 x^2+19 x+9\right ) \log (x+9) \log (\log (x+9)) ((x-1) \log (\log (x+9))-x)\right ) \log \left (\frac {x^2+x+\left (-x^2+x+\log (x)+2\right ) \log (\log (x+9))}{\log (\log (x+9))}\right )}{x (x+9) \log (x+9) \log (\log (x+9)) \left (x (x+1)-\left (x^2-x-\log (x)-2\right ) \log (\log (x+9))\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -2 \int \left (\frac {\left (2 \log (x+9) \log ^2(\log (x+9)) x^3-2 \log (x+9) \log (\log (x+9)) x^3+x^3+17 \log (x+9) \log ^2(\log (x+9)) x^2-19 \log (x+9) \log (\log (x+9)) x^2+x^2-10 \log (x+9) \log ^2(\log (x+9)) x-9 \log (x+9) \log (\log (x+9)) x-9 \log (x+9) \log ^2(\log (x+9))\right ) \log \left (\frac {x^2+x+\left (-x^2+x+\log (x)+2\right ) \log (\log (x+9))}{\log (\log (x+9))}\right )}{9 (x+9) \log (x+9) \log (\log (x+9)) \left (\log (\log (x+9)) x^2-x^2-\log (\log (x+9)) x-x-\log (x) \log (\log (x+9))-2 \log (\log (x+9))\right )}-\frac {\left (2 \log (x+9) \log ^2(\log (x+9)) x^3-2 \log (x+9) \log (\log (x+9)) x^3+x^3+17 \log (x+9) \log ^2(\log (x+9)) x^2-19 \log (x+9) \log (\log (x+9)) x^2+x^2-10 \log (x+9) \log ^2(\log (x+9)) x-9 \log (x+9) \log (\log (x+9)) x-9 \log (x+9) \log ^2(\log (x+9))\right ) \log \left (\frac {x^2+x+\left (-x^2+x+\log (x)+2\right ) \log (\log (x+9))}{\log (\log (x+9))}\right )}{9 x \log (x+9) \log (\log (x+9)) \left (\log (\log (x+9)) x^2-x^2-\log (\log (x+9)) x-x-\log (x) \log (\log (x+9))-2 \log (\log (x+9))\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -2 \int \frac {\left ((x+1) x^2+\left (2 x^2+19 x+9\right ) \log (x+9) \log (\log (x+9)) ((x-1) \log (\log (x+9))-x)\right ) \log \left (\frac {x^2+x+\left (-x^2+x+\log (x)+2\right ) \log (\log (x+9))}{\log (\log (x+9))}\right )}{x (x+9) \log (x+9) \log (\log (x+9)) \left (x (x+1)-\left (x^2-x-\log (x)-2\right ) \log (\log (x+9))\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -2 \int \left (\frac {\left (2 \log (x+9) \log ^2(\log (x+9)) x^3-2 \log (x+9) \log (\log (x+9)) x^3+x^3+17 \log (x+9) \log ^2(\log (x+9)) x^2-19 \log (x+9) \log (\log (x+9)) x^2+x^2-10 \log (x+9) \log ^2(\log (x+9)) x-9 \log (x+9) \log (\log (x+9)) x-9 \log (x+9) \log ^2(\log (x+9))\right ) \log \left (\frac {x^2+x+\left (-x^2+x+\log (x)+2\right ) \log (\log (x+9))}{\log (\log (x+9))}\right )}{9 (x+9) \log (x+9) \log (\log (x+9)) \left (\log (\log (x+9)) x^2-x^2-\log (\log (x+9)) x-x-\log (x) \log (\log (x+9))-2 \log (\log (x+9))\right )}-\frac {\left (2 \log (x+9) \log ^2(\log (x+9)) x^3-2 \log (x+9) \log (\log (x+9)) x^3+x^3+17 \log (x+9) \log ^2(\log (x+9)) x^2-19 \log (x+9) \log (\log (x+9)) x^2+x^2-10 \log (x+9) \log ^2(\log (x+9)) x-9 \log (x+9) \log (\log (x+9)) x-9 \log (x+9) \log ^2(\log (x+9))\right ) \log \left (\frac {x^2+x+\left (-x^2+x+\log (x)+2\right ) \log (\log (x+9))}{\log (\log (x+9))}\right )}{9 x \log (x+9) \log (\log (x+9)) \left (\log (\log (x+9)) x^2-x^2-\log (\log (x+9)) x-x-\log (x) \log (\log (x+9))-2 \log (\log (x+9))\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -2 \int \frac {\left ((x+1) x^2+\left (2 x^2+19 x+9\right ) \log (x+9) \log (\log (x+9)) ((x-1) \log (\log (x+9))-x)\right ) \log \left (\frac {x^2+x+\left (-x^2+x+\log (x)+2\right ) \log (\log (x+9))}{\log (\log (x+9))}\right )}{x (x+9) \log (x+9) \log (\log (x+9)) \left (x (x+1)-\left (x^2-x-\log (x)-2\right ) \log (\log (x+9))\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -2 \int \left (\frac {\left (2 \log (x+9) \log ^2(\log (x+9)) x^3-2 \log (x+9) \log (\log (x+9)) x^3+x^3+17 \log (x+9) \log ^2(\log (x+9)) x^2-19 \log (x+9) \log (\log (x+9)) x^2+x^2-10 \log (x+9) \log ^2(\log (x+9)) x-9 \log (x+9) \log (\log (x+9)) x-9 \log (x+9) \log ^2(\log (x+9))\right ) \log \left (\frac {x^2+x+\left (-x^2+x+\log (x)+2\right ) \log (\log (x+9))}{\log (\log (x+9))}\right )}{9 (x+9) \log (x+9) \log (\log (x+9)) \left (\log (\log (x+9)) x^2-x^2-\log (\log (x+9)) x-x-\log (x) \log (\log (x+9))-2 \log (\log (x+9))\right )}-\frac {\left (2 \log (x+9) \log ^2(\log (x+9)) x^3-2 \log (x+9) \log (\log (x+9)) x^3+x^3+17 \log (x+9) \log ^2(\log (x+9)) x^2-19 \log (x+9) \log (\log (x+9)) x^2+x^2-10 \log (x+9) \log ^2(\log (x+9)) x-9 \log (x+9) \log (\log (x+9)) x-9 \log (x+9) \log ^2(\log (x+9))\right ) \log \left (\frac {x^2+x+\left (-x^2+x+\log (x)+2\right ) \log (\log (x+9))}{\log (\log (x+9))}\right )}{9 x \log (x+9) \log (\log (x+9)) \left (\log (\log (x+9)) x^2-x^2-\log (\log (x+9)) x-x-\log (x) \log (\log (x+9))-2 \log (\log (x+9))\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -2 \int \frac {\left ((x+1) x^2+\left (2 x^2+19 x+9\right ) \log (x+9) \log (\log (x+9)) ((x-1) \log (\log (x+9))-x)\right ) \log \left (\frac {x^2+x+\left (-x^2+x+\log (x)+2\right ) \log (\log (x+9))}{\log (\log (x+9))}\right )}{x (x+9) \log (x+9) \log (\log (x+9)) \left (x (x+1)-\left (x^2-x-\log (x)-2\right ) \log (\log (x+9))\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -2 \int \left (\frac {\left (2 \log (x+9) \log ^2(\log (x+9)) x^3-2 \log (x+9) \log (\log (x+9)) x^3+x^3+17 \log (x+9) \log ^2(\log (x+9)) x^2-19 \log (x+9) \log (\log (x+9)) x^2+x^2-10 \log (x+9) \log ^2(\log (x+9)) x-9 \log (x+9) \log (\log (x+9)) x-9 \log (x+9) \log ^2(\log (x+9))\right ) \log \left (\frac {x^2+x+\left (-x^2+x+\log (x)+2\right ) \log (\log (x+9))}{\log (\log (x+9))}\right )}{9 (x+9) \log (x+9) \log (\log (x+9)) \left (\log (\log (x+9)) x^2-x^2-\log (\log (x+9)) x-x-\log (x) \log (\log (x+9))-2 \log (\log (x+9))\right )}-\frac {\left (2 \log (x+9) \log ^2(\log (x+9)) x^3-2 \log (x+9) \log (\log (x+9)) x^3+x^3+17 \log (x+9) \log ^2(\log (x+9)) x^2-19 \log (x+9) \log (\log (x+9)) x^2+x^2-10 \log (x+9) \log ^2(\log (x+9)) x-9 \log (x+9) \log (\log (x+9)) x-9 \log (x+9) \log ^2(\log (x+9))\right ) \log \left (\frac {x^2+x+\left (-x^2+x+\log (x)+2\right ) \log (\log (x+9))}{\log (\log (x+9))}\right )}{9 x \log (x+9) \log (\log (x+9)) \left (\log (\log (x+9)) x^2-x^2-\log (\log (x+9)) x-x-\log (x) \log (\log (x+9))-2 \log (\log (x+9))\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -2 \int \frac {\left ((x+1) x^2+\left (2 x^2+19 x+9\right ) \log (x+9) \log (\log (x+9)) ((x-1) \log (\log (x+9))-x)\right ) \log \left (\frac {x^2+x+\left (-x^2+x+\log (x)+2\right ) \log (\log (x+9))}{\log (\log (x+9))}\right )}{x (x+9) \log (x+9) \log (\log (x+9)) \left (x (x+1)-\left (x^2-x-\log (x)-2\right ) \log (\log (x+9))\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -2 \int \left (\frac {\left (2 \log (x+9) \log ^2(\log (x+9)) x^3-2 \log (x+9) \log (\log (x+9)) x^3+x^3+17 \log (x+9) \log ^2(\log (x+9)) x^2-19 \log (x+9) \log (\log (x+9)) x^2+x^2-10 \log (x+9) \log ^2(\log (x+9)) x-9 \log (x+9) \log (\log (x+9)) x-9 \log (x+9) \log ^2(\log (x+9))\right ) \log \left (\frac {x^2+x+\left (-x^2+x+\log (x)+2\right ) \log (\log (x+9))}{\log (\log (x+9))}\right )}{9 (x+9) \log (x+9) \log (\log (x+9)) \left (\log (\log (x+9)) x^2-x^2-\log (\log (x+9)) x-x-\log (x) \log (\log (x+9))-2 \log (\log (x+9))\right )}-\frac {\left (2 \log (x+9) \log ^2(\log (x+9)) x^3-2 \log (x+9) \log (\log (x+9)) x^3+x^3+17 \log (x+9) \log ^2(\log (x+9)) x^2-19 \log (x+9) \log (\log (x+9)) x^2+x^2-10 \log (x+9) \log ^2(\log (x+9)) x-9 \log (x+9) \log (\log (x+9)) x-9 \log (x+9) \log ^2(\log (x+9))\right ) \log \left (\frac {x^2+x+\left (-x^2+x+\log (x)+2\right ) \log (\log (x+9))}{\log (\log (x+9))}\right )}{9 x \log (x+9) \log (\log (x+9)) \left (\log (\log (x+9)) x^2-x^2-\log (\log (x+9)) x-x-\log (x) \log (\log (x+9))-2 \log (\log (x+9))\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -2 \int \frac {\left ((x+1) x^2+\left (2 x^2+19 x+9\right ) \log (x+9) \log (\log (x+9)) ((x-1) \log (\log (x+9))-x)\right ) \log \left (\frac {x^2+x+\left (-x^2+x+\log (x)+2\right ) \log (\log (x+9))}{\log (\log (x+9))}\right )}{x (x+9) \log (x+9) \log (\log (x+9)) \left (x (x+1)-\left (x^2-x-\log (x)-2\right ) \log (\log (x+9))\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -2 \int \left (\frac {\left (2 \log (x+9) \log ^2(\log (x+9)) x^3-2 \log (x+9) \log (\log (x+9)) x^3+x^3+17 \log (x+9) \log ^2(\log (x+9)) x^2-19 \log (x+9) \log (\log (x+9)) x^2+x^2-10 \log (x+9) \log ^2(\log (x+9)) x-9 \log (x+9) \log (\log (x+9)) x-9 \log (x+9) \log ^2(\log (x+9))\right ) \log \left (\frac {x^2+x+\left (-x^2+x+\log (x)+2\right ) \log (\log (x+9))}{\log (\log (x+9))}\right )}{9 (x+9) \log (x+9) \log (\log (x+9)) \left (\log (\log (x+9)) x^2-x^2-\log (\log (x+9)) x-x-\log (x) \log (\log (x+9))-2 \log (\log (x+9))\right )}-\frac {\left (2 \log (x+9) \log ^2(\log (x+9)) x^3-2 \log (x+9) \log (\log (x+9)) x^3+x^3+17 \log (x+9) \log ^2(\log (x+9)) x^2-19 \log (x+9) \log (\log (x+9)) x^2+x^2-10 \log (x+9) \log ^2(\log (x+9)) x-9 \log (x+9) \log (\log (x+9)) x-9 \log (x+9) \log ^2(\log (x+9))\right ) \log \left (\frac {x^2+x+\left (-x^2+x+\log (x)+2\right ) \log (\log (x+9))}{\log (\log (x+9))}\right )}{9 x \log (x+9) \log (\log (x+9)) \left (\log (\log (x+9)) x^2-x^2-\log (\log (x+9)) x-x-\log (x) \log (\log (x+9))-2 \log (\log (x+9))\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -2 \int \frac {\left ((x+1) x^2+\left (2 x^2+19 x+9\right ) \log (x+9) \log (\log (x+9)) ((x-1) \log (\log (x+9))-x)\right ) \log \left (\frac {x^2+x+\left (-x^2+x+\log (x)+2\right ) \log (\log (x+9))}{\log (\log (x+9))}\right )}{x (x+9) \log (x+9) \log (\log (x+9)) \left (x (x+1)-\left (x^2-x-\log (x)-2\right ) \log (\log (x+9))\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -2 \int \left (\frac {\left (2 \log (x+9) \log ^2(\log (x+9)) x^3-2 \log (x+9) \log (\log (x+9)) x^3+x^3+17 \log (x+9) \log ^2(\log (x+9)) x^2-19 \log (x+9) \log (\log (x+9)) x^2+x^2-10 \log (x+9) \log ^2(\log (x+9)) x-9 \log (x+9) \log (\log (x+9)) x-9 \log (x+9) \log ^2(\log (x+9))\right ) \log \left (\frac {x^2+x+\left (-x^2+x+\log (x)+2\right ) \log (\log (x+9))}{\log (\log (x+9))}\right )}{9 (x+9) \log (x+9) \log (\log (x+9)) \left (\log (\log (x+9)) x^2-x^2-\log (\log (x+9)) x-x-\log (x) \log (\log (x+9))-2 \log (\log (x+9))\right )}-\frac {\left (2 \log (x+9) \log ^2(\log (x+9)) x^3-2 \log (x+9) \log (\log (x+9)) x^3+x^3+17 \log (x+9) \log ^2(\log (x+9)) x^2-19 \log (x+9) \log (\log (x+9)) x^2+x^2-10 \log (x+9) \log ^2(\log (x+9)) x-9 \log (x+9) \log (\log (x+9)) x-9 \log (x+9) \log ^2(\log (x+9))\right ) \log \left (\frac {x^2+x+\left (-x^2+x+\log (x)+2\right ) \log (\log (x+9))}{\log (\log (x+9))}\right )}{9 x \log (x+9) \log (\log (x+9)) \left (\log (\log (x+9)) x^2-x^2-\log (\log (x+9)) x-x-\log (x) \log (\log (x+9))-2 \log (\log (x+9))\right )}\right )dx\)

Input:

Int[((-2*x^2 - 2*x^3 + (18*x + 38*x^2 + 4*x^3)*Log[9 + x]*Log[Log[9 + x]] 
+ (18 + 20*x - 34*x^2 - 4*x^3)*Log[9 + x]*Log[Log[9 + x]]^2)*Log[(x + x^2 
+ (2 + x - x^2 + Log[x])*Log[Log[9 + x]])/Log[Log[9 + x]]])/((9*x^2 + 10*x 
^3 + x^4)*Log[9 + x]*Log[Log[9 + x]] + (18*x + 11*x^2 - 8*x^3 - x^4 + (9*x 
 + x^2)*Log[x])*Log[9 + x]*Log[Log[9 + x]]^2),x]
 

Output:

$Aborted
 

Maple [F]

\[\int \frac {\left (\left (-4 x^{3}-34 x^{2}+20 x +18\right ) \ln \left (x +9\right ) \ln \left (\ln \left (x +9\right )\right )^{2}+\left (4 x^{3}+38 x^{2}+18 x \right ) \ln \left (x +9\right ) \ln \left (\ln \left (x +9\right )\right )-2 x^{3}-2 x^{2}\right ) \ln \left (\frac {\left (\ln \left (x \right )-x^{2}+x +2\right ) \ln \left (\ln \left (x +9\right )\right )+x^{2}+x}{\ln \left (\ln \left (x +9\right )\right )}\right )}{\left (\left (x^{2}+9 x \right ) \ln \left (x \right )-x^{4}-8 x^{3}+11 x^{2}+18 x \right ) \ln \left (x +9\right ) \ln \left (\ln \left (x +9\right )\right )^{2}+\left (x^{4}+10 x^{3}+9 x^{2}\right ) \ln \left (x +9\right ) \ln \left (\ln \left (x +9\right )\right )}d x\]

Input:

int(((-4*x^3-34*x^2+20*x+18)*ln(x+9)*ln(ln(x+9))^2+(4*x^3+38*x^2+18*x)*ln( 
x+9)*ln(ln(x+9))-2*x^3-2*x^2)*ln(((ln(x)-x^2+x+2)*ln(ln(x+9))+x^2+x)/ln(ln 
(x+9)))/(((x^2+9*x)*ln(x)-x^4-8*x^3+11*x^2+18*x)*ln(x+9)*ln(ln(x+9))^2+(x^ 
4+10*x^3+9*x^2)*ln(x+9)*ln(ln(x+9))),x)
 

Output:

int(((-4*x^3-34*x^2+20*x+18)*ln(x+9)*ln(ln(x+9))^2+(4*x^3+38*x^2+18*x)*ln( 
x+9)*ln(ln(x+9))-2*x^3-2*x^2)*ln(((ln(x)-x^2+x+2)*ln(ln(x+9))+x^2+x)/ln(ln 
(x+9)))/(((x^2+9*x)*ln(x)-x^4-8*x^3+11*x^2+18*x)*ln(x+9)*ln(ln(x+9))^2+(x^ 
4+10*x^3+9*x^2)*ln(x+9)*ln(ln(x+9))),x)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.46 \[ \int \frac {\left (-2 x^2-2 x^3+\left (18 x+38 x^2+4 x^3\right ) \log (9+x) \log (\log (9+x))+\left (18+20 x-34 x^2-4 x^3\right ) \log (9+x) \log ^2(\log (9+x))\right ) \log \left (\frac {x+x^2+\left (2+x-x^2+\log (x)\right ) \log (\log (9+x))}{\log (\log (9+x))}\right )}{\left (9 x^2+10 x^3+x^4\right ) \log (9+x) \log (\log (9+x))+\left (18 x+11 x^2-8 x^3-x^4+\left (9 x+x^2\right ) \log (x)\right ) \log (9+x) \log ^2(\log (9+x))} \, dx=\log \left (\frac {x^{2} - {\left (x^{2} - x - \log \left (x\right ) - 2\right )} \log \left (\log \left (x + 9\right )\right ) + x}{\log \left (\log \left (x + 9\right )\right )}\right )^{2} \] Input:

integrate(((-4*x^3-34*x^2+20*x+18)*log(x+9)*log(log(x+9))^2+(4*x^3+38*x^2+ 
18*x)*log(x+9)*log(log(x+9))-2*x^3-2*x^2)*log(((log(x)-x^2+x+2)*log(log(x+ 
9))+x^2+x)/log(log(x+9)))/(((x^2+9*x)*log(x)-x^4-8*x^3+11*x^2+18*x)*log(x+ 
9)*log(log(x+9))^2+(x^4+10*x^3+9*x^2)*log(x+9)*log(log(x+9))),x, algorithm 
="fricas")
 

Output:

log((x^2 - (x^2 - x - log(x) - 2)*log(log(x + 9)) + x)/log(log(x + 9)))^2
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (-2 x^2-2 x^3+\left (18 x+38 x^2+4 x^3\right ) \log (9+x) \log (\log (9+x))+\left (18+20 x-34 x^2-4 x^3\right ) \log (9+x) \log ^2(\log (9+x))\right ) \log \left (\frac {x+x^2+\left (2+x-x^2+\log (x)\right ) \log (\log (9+x))}{\log (\log (9+x))}\right )}{\left (9 x^2+10 x^3+x^4\right ) \log (9+x) \log (\log (9+x))+\left (18 x+11 x^2-8 x^3-x^4+\left (9 x+x^2\right ) \log (x)\right ) \log (9+x) \log ^2(\log (9+x))} \, dx=\text {Timed out} \] Input:

integrate(((-4*x**3-34*x**2+20*x+18)*ln(x+9)*ln(ln(x+9))**2+(4*x**3+38*x** 
2+18*x)*ln(x+9)*ln(ln(x+9))-2*x**3-2*x**2)*ln(((ln(x)-x**2+x+2)*ln(ln(x+9) 
)+x**2+x)/ln(ln(x+9)))/(((x**2+9*x)*ln(x)-x**4-8*x**3+11*x**2+18*x)*ln(x+9 
)*ln(ln(x+9))**2+(x**4+10*x**3+9*x**2)*ln(x+9)*ln(ln(x+9))),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (-2 x^2-2 x^3+\left (18 x+38 x^2+4 x^3\right ) \log (9+x) \log (\log (9+x))+\left (18+20 x-34 x^2-4 x^3\right ) \log (9+x) \log ^2(\log (9+x))\right ) \log \left (\frac {x+x^2+\left (2+x-x^2+\log (x)\right ) \log (\log (9+x))}{\log (\log (9+x))}\right )}{\left (9 x^2+10 x^3+x^4\right ) \log (9+x) \log (\log (9+x))+\left (18 x+11 x^2-8 x^3-x^4+\left (9 x+x^2\right ) \log (x)\right ) \log (9+x) \log ^2(\log (9+x))} \, dx=\int { \frac {2 \, {\left ({\left (2 \, x^{3} + 17 \, x^{2} - 10 \, x - 9\right )} \log \left (x + 9\right ) \log \left (\log \left (x + 9\right )\right )^{2} + x^{3} - {\left (2 \, x^{3} + 19 \, x^{2} + 9 \, x\right )} \log \left (x + 9\right ) \log \left (\log \left (x + 9\right )\right ) + x^{2}\right )} \log \left (\frac {x^{2} - {\left (x^{2} - x - \log \left (x\right ) - 2\right )} \log \left (\log \left (x + 9\right )\right ) + x}{\log \left (\log \left (x + 9\right )\right )}\right )}{{\left (x^{4} + 8 \, x^{3} - 11 \, x^{2} - {\left (x^{2} + 9 \, x\right )} \log \left (x\right ) - 18 \, x\right )} \log \left (x + 9\right ) \log \left (\log \left (x + 9\right )\right )^{2} - {\left (x^{4} + 10 \, x^{3} + 9 \, x^{2}\right )} \log \left (x + 9\right ) \log \left (\log \left (x + 9\right )\right )} \,d x } \] Input:

integrate(((-4*x^3-34*x^2+20*x+18)*log(x+9)*log(log(x+9))^2+(4*x^3+38*x^2+ 
18*x)*log(x+9)*log(log(x+9))-2*x^3-2*x^2)*log(((log(x)-x^2+x+2)*log(log(x+ 
9))+x^2+x)/log(log(x+9)))/(((x^2+9*x)*log(x)-x^4-8*x^3+11*x^2+18*x)*log(x+ 
9)*log(log(x+9))^2+(x^4+10*x^3+9*x^2)*log(x+9)*log(log(x+9))),x, algorithm 
="maxima")
 

Output:

2*integrate(((2*x^3 + 17*x^2 - 10*x - 9)*log(x + 9)*log(log(x + 9))^2 + x^ 
3 - (2*x^3 + 19*x^2 + 9*x)*log(x + 9)*log(log(x + 9)) + x^2)*log((x^2 - (x 
^2 - x - log(x) - 2)*log(log(x + 9)) + x)/log(log(x + 9)))/((x^4 + 8*x^3 - 
 11*x^2 - (x^2 + 9*x)*log(x) - 18*x)*log(x + 9)*log(log(x + 9))^2 - (x^4 + 
 10*x^3 + 9*x^2)*log(x + 9)*log(log(x + 9))), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 148 vs. \(2 (24) = 48\).

Time = 0.50 (sec) , antiderivative size = 148, normalized size of antiderivative = 6.17 \[ \int \frac {\left (-2 x^2-2 x^3+\left (18 x+38 x^2+4 x^3\right ) \log (9+x) \log (\log (9+x))+\left (18+20 x-34 x^2-4 x^3\right ) \log (9+x) \log ^2(\log (9+x))\right ) \log \left (\frac {x+x^2+\left (2+x-x^2+\log (x)\right ) \log (\log (9+x))}{\log (\log (9+x))}\right )}{\left (9 x^2+10 x^3+x^4\right ) \log (9+x) \log (\log (9+x))+\left (18 x+11 x^2-8 x^3-x^4+\left (9 x+x^2\right ) \log (x)\right ) \log (9+x) \log ^2(\log (9+x))} \, dx=-\log \left (x^{2} \log \left (\log \left (x + 9\right )\right ) - x^{2} - x \log \left (\log \left (x + 9\right )\right ) - \log \left (x\right ) \log \left (\log \left (x + 9\right )\right ) - x - 2 \, \log \left (\log \left (x + 9\right )\right )\right )^{2} + 2 \, {\left (\log \left (x^{2} \log \left (\log \left (x + 9\right )\right ) - x^{2} - x \log \left (\log \left (x + 9\right )\right ) - \log \left (x\right ) \log \left (\log \left (x + 9\right )\right ) - x - 2 \, \log \left (\log \left (x + 9\right )\right )\right ) - \log \left (\log \left (\log \left (x + 9\right )\right )\right )\right )} \log \left (-x^{2} \log \left (\log \left (x + 9\right )\right ) + x^{2} + x \log \left (\log \left (x + 9\right )\right ) + \log \left (x\right ) \log \left (\log \left (x + 9\right )\right ) + x + 2 \, \log \left (\log \left (x + 9\right )\right )\right ) + \log \left (\log \left (\log \left (x + 9\right )\right )\right )^{2} \] Input:

integrate(((-4*x^3-34*x^2+20*x+18)*log(x+9)*log(log(x+9))^2+(4*x^3+38*x^2+ 
18*x)*log(x+9)*log(log(x+9))-2*x^3-2*x^2)*log(((log(x)-x^2+x+2)*log(log(x+ 
9))+x^2+x)/log(log(x+9)))/(((x^2+9*x)*log(x)-x^4-8*x^3+11*x^2+18*x)*log(x+ 
9)*log(log(x+9))^2+(x^4+10*x^3+9*x^2)*log(x+9)*log(log(x+9))),x, algorithm 
="giac")
 

Output:

-log(x^2*log(log(x + 9)) - x^2 - x*log(log(x + 9)) - log(x)*log(log(x + 9) 
) - x - 2*log(log(x + 9)))^2 + 2*(log(x^2*log(log(x + 9)) - x^2 - x*log(lo 
g(x + 9)) - log(x)*log(log(x + 9)) - x - 2*log(log(x + 9))) - log(log(log( 
x + 9))))*log(-x^2*log(log(x + 9)) + x^2 + x*log(log(x + 9)) + log(x)*log( 
log(x + 9)) + x + 2*log(log(x + 9))) + log(log(log(x + 9)))^2
 

Mupad [B] (verification not implemented)

Time = 4.53 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.33 \[ \int \frac {\left (-2 x^2-2 x^3+\left (18 x+38 x^2+4 x^3\right ) \log (9+x) \log (\log (9+x))+\left (18+20 x-34 x^2-4 x^3\right ) \log (9+x) \log ^2(\log (9+x))\right ) \log \left (\frac {x+x^2+\left (2+x-x^2+\log (x)\right ) \log (\log (9+x))}{\log (\log (9+x))}\right )}{\left (9 x^2+10 x^3+x^4\right ) \log (9+x) \log (\log (9+x))+\left (18 x+11 x^2-8 x^3-x^4+\left (9 x+x^2\right ) \log (x)\right ) \log (9+x) \log ^2(\log (9+x))} \, dx={\ln \left (\frac {x+\ln \left (\ln \left (x+9\right )\right )\,\left (x+\ln \left (x\right )-x^2+2\right )+x^2}{\ln \left (\ln \left (x+9\right )\right )}\right )}^2 \] Input:

int(-(log((x + log(log(x + 9))*(x + log(x) - x^2 + 2) + x^2)/log(log(x + 9 
)))*(2*x^2 + 2*x^3 - log(x + 9)*log(log(x + 9))^2*(20*x - 34*x^2 - 4*x^3 + 
 18) - log(x + 9)*log(log(x + 9))*(18*x + 38*x^2 + 4*x^3)))/(log(x + 9)*lo 
g(log(x + 9))^2*(18*x + log(x)*(9*x + x^2) + 11*x^2 - 8*x^3 - x^4) + log(x 
 + 9)*log(log(x + 9))*(9*x^2 + 10*x^3 + x^4)),x)
 

Output:

log((x + log(log(x + 9))*(x + log(x) - x^2 + 2) + x^2)/log(log(x + 9)))^2
 

Reduce [B] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.00 \[ \int \frac {\left (-2 x^2-2 x^3+\left (18 x+38 x^2+4 x^3\right ) \log (9+x) \log (\log (9+x))+\left (18+20 x-34 x^2-4 x^3\right ) \log (9+x) \log ^2(\log (9+x))\right ) \log \left (\frac {x+x^2+\left (2+x-x^2+\log (x)\right ) \log (\log (9+x))}{\log (\log (9+x))}\right )}{\left (9 x^2+10 x^3+x^4\right ) \log (9+x) \log (\log (9+x))+\left (18 x+11 x^2-8 x^3-x^4+\left (9 x+x^2\right ) \log (x)\right ) \log (9+x) \log ^2(\log (9+x))} \, dx=\mathrm {log}\left (\frac {\mathrm {log}\left (\mathrm {log}\left (x +9\right )\right ) \mathrm {log}\left (x \right )-\mathrm {log}\left (\mathrm {log}\left (x +9\right )\right ) x^{2}+\mathrm {log}\left (\mathrm {log}\left (x +9\right )\right ) x +2 \,\mathrm {log}\left (\mathrm {log}\left (x +9\right )\right )+x^{2}+x}{\mathrm {log}\left (\mathrm {log}\left (x +9\right )\right )}\right )^{2} \] Input:

int(((-4*x^3-34*x^2+20*x+18)*log(x+9)*log(log(x+9))^2+(4*x^3+38*x^2+18*x)* 
log(x+9)*log(log(x+9))-2*x^3-2*x^2)*log(((log(x)-x^2+x+2)*log(log(x+9))+x^ 
2+x)/log(log(x+9)))/(((x^2+9*x)*log(x)-x^4-8*x^3+11*x^2+18*x)*log(x+9)*log 
(log(x+9))^2+(x^4+10*x^3+9*x^2)*log(x+9)*log(log(x+9))),x)
 

Output:

log((log(log(x + 9))*log(x) - log(log(x + 9))*x**2 + log(log(x + 9))*x + 2 
*log(log(x + 9)) + x**2 + x)/log(log(x + 9)))**2