Integrand size = 211, antiderivative size = 26 \[ \int \frac {12+2 x+\left (732-1248 x+582 x^2-27 x^3-27 x^4\right ) \log (x)+\left (-444+466 x-72 x^2-27 x^3\right ) \log (x) \log \left (\frac {6+x}{x \log (x)}\right )+\left (90-39 x-9 x^2\right ) \log (x) \log ^2\left (\frac {6+x}{x \log (x)}\right )+(-6-x) \log (x) \log ^3\left (\frac {6+x}{x \log (x)}\right )}{\left (-750+1225 x-585 x^2+27 x^3+27 x^4\right ) \log (x)+\left (450-465 x+72 x^2+27 x^3\right ) \log (x) \log \left (\frac {6+x}{x \log (x)}\right )+\left (-90+39 x+9 x^2\right ) \log (x) \log ^2\left (\frac {6+x}{x \log (x)}\right )+(6+x) \log (x) \log ^3\left (\frac {6+x}{x \log (x)}\right )} \, dx=-x+\frac {x}{\left (-5+3 x+\log \left (\frac {1+\frac {6}{x}}{\log (x)}\right )\right )^2} \]
Time = 0.13 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \frac {12+2 x+\left (732-1248 x+582 x^2-27 x^3-27 x^4\right ) \log (x)+\left (-444+466 x-72 x^2-27 x^3\right ) \log (x) \log \left (\frac {6+x}{x \log (x)}\right )+\left (90-39 x-9 x^2\right ) \log (x) \log ^2\left (\frac {6+x}{x \log (x)}\right )+(-6-x) \log (x) \log ^3\left (\frac {6+x}{x \log (x)}\right )}{\left (-750+1225 x-585 x^2+27 x^3+27 x^4\right ) \log (x)+\left (450-465 x+72 x^2+27 x^3\right ) \log (x) \log \left (\frac {6+x}{x \log (x)}\right )+\left (-90+39 x+9 x^2\right ) \log (x) \log ^2\left (\frac {6+x}{x \log (x)}\right )+(6+x) \log (x) \log ^3\left (\frac {6+x}{x \log (x)}\right )} \, dx=-x+\frac {x}{\left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^2} \]
Integrate[(12 + 2*x + (732 - 1248*x + 582*x^2 - 27*x^3 - 27*x^4)*Log[x] + (-444 + 466*x - 72*x^2 - 27*x^3)*Log[x]*Log[(6 + x)/(x*Log[x])] + (90 - 39 *x - 9*x^2)*Log[x]*Log[(6 + x)/(x*Log[x])]^2 + (-6 - x)*Log[x]*Log[(6 + x) /(x*Log[x])]^3)/((-750 + 1225*x - 585*x^2 + 27*x^3 + 27*x^4)*Log[x] + (450 - 465*x + 72*x^2 + 27*x^3)*Log[x]*Log[(6 + x)/(x*Log[x])] + (-90 + 39*x + 9*x^2)*Log[x]*Log[(6 + x)/(x*Log[x])]^2 + (6 + x)*Log[x]*Log[(6 + x)/(x*L og[x])]^3),x]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (-9 x^2-39 x+90\right ) \log (x) \log ^2\left (\frac {x+6}{x \log (x)}\right )+\left (-27 x^3-72 x^2+466 x-444\right ) \log (x) \log \left (\frac {x+6}{x \log (x)}\right )+\left (-27 x^4-27 x^3+582 x^2-1248 x+732\right ) \log (x)+2 x+(-x-6) \log (x) \log ^3\left (\frac {x+6}{x \log (x)}\right )+12}{\left (9 x^2+39 x-90\right ) \log (x) \log ^2\left (\frac {x+6}{x \log (x)}\right )+\left (27 x^3+72 x^2-465 x+450\right ) \log (x) \log \left (\frac {x+6}{x \log (x)}\right )+\left (27 x^4+27 x^3-585 x^2+1225 x-750\right ) \log (x)+(x+6) \log (x) \log ^3\left (\frac {x+6}{x \log (x)}\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-\left (-9 x^2-39 x+90\right ) \log (x) \log ^2\left (\frac {x+6}{x \log (x)}\right )-\left (-27 x^3-72 x^2+466 x-444\right ) \log (x) \log \left (\frac {x+6}{x \log (x)}\right )-\left (-27 x^4-27 x^3+582 x^2-1248 x+732\right ) \log (x)-2 x-\left ((-x-6) \log (x) \log ^3\left (\frac {x+6}{x \log (x)}\right )\right )-12}{(x+6) \log (x) \left (-3 x-\log \left (\frac {x+6}{x \log (x)}\right )+5\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {2 \left (3 x^2 \log (x)-x+18 x \log (x)-6 \log (x)-6\right )}{(x+6) \log (x) \left (3 x+\log \left (\frac {x+6}{x \log (x)}\right )-5\right )^3}+\frac {1}{\left (3 x+\log \left (\frac {x+6}{x \log (x)}\right )-5\right )^2}-1\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -6 \int \frac {x}{\left (3 x+\log \left (\frac {x+6}{x \log (x)}\right )-5\right )^3}dx+12 \int \frac {1}{(x+6) \left (3 x+\log \left (\frac {x+6}{x \log (x)}\right )-5\right )^3}dx+2 \int \frac {1}{\log (x) \left (3 x+\log \left (\frac {x+6}{x \log (x)}\right )-5\right )^3}dx+\int \frac {1}{\left (3 x+\log \left (\frac {x+6}{x \log (x)}\right )-5\right )^2}dx-x\) |
Int[(12 + 2*x + (732 - 1248*x + 582*x^2 - 27*x^3 - 27*x^4)*Log[x] + (-444 + 466*x - 72*x^2 - 27*x^3)*Log[x]*Log[(6 + x)/(x*Log[x])] + (90 - 39*x - 9 *x^2)*Log[x]*Log[(6 + x)/(x*Log[x])]^2 + (-6 - x)*Log[x]*Log[(6 + x)/(x*Lo g[x])]^3)/((-750 + 1225*x - 585*x^2 + 27*x^3 + 27*x^4)*Log[x] + (450 - 465 *x + 72*x^2 + 27*x^3)*Log[x]*Log[(6 + x)/(x*Log[x])] + (-90 + 39*x + 9*x^2 )*Log[x]*Log[(6 + x)/(x*Log[x])]^2 + (6 + x)*Log[x]*Log[(6 + x)/(x*Log[x]) ]^3),x]
3.21.15.3.1 Defintions of rubi rules used
Leaf count of result is larger than twice the leaf count of optimal. \(109\) vs. \(2(26)=52\).
Time = 7.27 (sec) , antiderivative size = 110, normalized size of antiderivative = 4.23
| method | result | size |
| norman | \(\frac {76 x -\frac {10 \ln \left (\frac {6+x}{x \ln \left (x \right )}\right )^{2}}{3}+\frac {100 \ln \left (\frac {6+x}{x \ln \left (x \right )}\right )}{3}-10 \ln \left (\frac {6+x}{x \ln \left (x \right )}\right ) x -9 x^{3}-6 \ln \left (\frac {6+x}{x \ln \left (x \right )}\right ) x^{2}-\ln \left (\frac {6+x}{x \ln \left (x \right )}\right )^{2} x -\frac {250}{3}}{\left (\ln \left (\frac {6+x}{x \ln \left (x \right )}\right )+3 x -5\right )^{2}}\) | \(110\) |
| parallelrisch | \(\frac {300-6 \ln \left (\frac {6+x}{x \ln \left (x \right )}\right ) x^{2}+82 \ln \left (\frac {6+x}{x \ln \left (x \right )}\right ) x +12 \ln \left (\frac {6+x}{x \ln \left (x \right )}\right )^{2}-\ln \left (\frac {6+x}{x \ln \left (x \right )}\right )^{2} x -384 x -120 \ln \left (\frac {6+x}{x \ln \left (x \right )}\right )-9 x^{3}+138 x^{2}}{9 x^{2}+6 \ln \left (\frac {6+x}{x \ln \left (x \right )}\right ) x +\ln \left (\frac {6+x}{x \ln \left (x \right )}\right )^{2}-30 x -10 \ln \left (\frac {6+x}{x \ln \left (x \right )}\right )+25}\) | \(151\) |
| risch | \(-x +\frac {4 x}{\left (2 \ln \left (6+x \right )-2 \ln \left (\ln \left (x \right )\right )-10-2 \ln \left (x \right )-i \pi \operatorname {csgn}\left (\frac {i \left (6+x \right )}{\ln \left (x \right )}\right )^{3}-i \pi \operatorname {csgn}\left (\frac {i \left (6+x \right )}{\ln \left (x \right ) x}\right )^{3}+i \pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (\frac {i \left (6+x \right )}{\ln \left (x \right ) x}\right )^{2}+i \pi \,\operatorname {csgn}\left (i \left (6+x \right )\right ) \operatorname {csgn}\left (\frac {i \left (6+x \right )}{\ln \left (x \right )}\right )^{2}+i \pi \operatorname {csgn}\left (\frac {i \left (6+x \right )}{\ln \left (x \right )}\right )^{2} \operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right )+i \pi \,\operatorname {csgn}\left (\frac {i \left (6+x \right )}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (\frac {i \left (6+x \right )}{\ln \left (x \right ) x}\right )^{2}-i \pi \,\operatorname {csgn}\left (i \left (6+x \right )\right ) \operatorname {csgn}\left (\frac {i \left (6+x \right )}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right )-i \pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (\frac {i \left (6+x \right )}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (\frac {i \left (6+x \right )}{\ln \left (x \right ) x}\right )+6 x \right )^{2}}\) | \(240\) |
| default | \(\text {Expression too large to display}\) | \(4298\) |
| parts | \(\text {Expression too large to display}\) | \(4298\) |
int(((-x-6)*ln(x)*ln((6+x)/x/ln(x))^3+(-9*x^2-39*x+90)*ln(x)*ln((6+x)/x/ln (x))^2+(-27*x^3-72*x^2+466*x-444)*ln(x)*ln((6+x)/x/ln(x))+(-27*x^4-27*x^3+ 582*x^2-1248*x+732)*ln(x)+2*x+12)/((6+x)*ln(x)*ln((6+x)/x/ln(x))^3+(9*x^2+ 39*x-90)*ln(x)*ln((6+x)/x/ln(x))^2+(27*x^3+72*x^2-465*x+450)*ln(x)*ln((6+x )/x/ln(x))+(27*x^4+27*x^3-585*x^2+1225*x-750)*ln(x)),x,method=_RETURNVERBO SE)
(76*x-10/3*ln((6+x)/x/ln(x))^2+100/3*ln((6+x)/x/ln(x))-10*ln((6+x)/x/ln(x) )*x-9*x^3-6*ln((6+x)/x/ln(x))*x^2-ln((6+x)/x/ln(x))^2*x-250/3)/(ln((6+x)/x /ln(x))+3*x-5)^2
Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (26) = 52\).
Time = 0.25 (sec) , antiderivative size = 100, normalized size of antiderivative = 3.85 \[ \int \frac {12+2 x+\left (732-1248 x+582 x^2-27 x^3-27 x^4\right ) \log (x)+\left (-444+466 x-72 x^2-27 x^3\right ) \log (x) \log \left (\frac {6+x}{x \log (x)}\right )+\left (90-39 x-9 x^2\right ) \log (x) \log ^2\left (\frac {6+x}{x \log (x)}\right )+(-6-x) \log (x) \log ^3\left (\frac {6+x}{x \log (x)}\right )}{\left (-750+1225 x-585 x^2+27 x^3+27 x^4\right ) \log (x)+\left (450-465 x+72 x^2+27 x^3\right ) \log (x) \log \left (\frac {6+x}{x \log (x)}\right )+\left (-90+39 x+9 x^2\right ) \log (x) \log ^2\left (\frac {6+x}{x \log (x)}\right )+(6+x) \log (x) \log ^3\left (\frac {6+x}{x \log (x)}\right )} \, dx=-\frac {9 \, x^{3} + x \log \left (\frac {x + 6}{x \log \left (x\right )}\right )^{2} - 30 \, x^{2} + 2 \, {\left (3 \, x^{2} - 5 \, x\right )} \log \left (\frac {x + 6}{x \log \left (x\right )}\right ) + 24 \, x}{9 \, x^{2} + 2 \, {\left (3 \, x - 5\right )} \log \left (\frac {x + 6}{x \log \left (x\right )}\right ) + \log \left (\frac {x + 6}{x \log \left (x\right )}\right )^{2} - 30 \, x + 25} \]
integrate(((-x-6)*log(x)*log((6+x)/x/log(x))^3+(-9*x^2-39*x+90)*log(x)*log ((6+x)/x/log(x))^2+(-27*x^3-72*x^2+466*x-444)*log(x)*log((6+x)/x/log(x))+( -27*x^4-27*x^3+582*x^2-1248*x+732)*log(x)+2*x+12)/((6+x)*log(x)*log((6+x)/ x/log(x))^3+(9*x^2+39*x-90)*log(x)*log((6+x)/x/log(x))^2+(27*x^3+72*x^2-46 5*x+450)*log(x)*log((6+x)/x/log(x))+(27*x^4+27*x^3-585*x^2+1225*x-750)*log (x)),x, algorithm=\
-(9*x^3 + x*log((x + 6)/(x*log(x)))^2 - 30*x^2 + 2*(3*x^2 - 5*x)*log((x + 6)/(x*log(x))) + 24*x)/(9*x^2 + 2*(3*x - 5)*log((x + 6)/(x*log(x))) + log( (x + 6)/(x*log(x)))^2 - 30*x + 25)
Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (19) = 38\).
Time = 0.19 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.50 \[ \int \frac {12+2 x+\left (732-1248 x+582 x^2-27 x^3-27 x^4\right ) \log (x)+\left (-444+466 x-72 x^2-27 x^3\right ) \log (x) \log \left (\frac {6+x}{x \log (x)}\right )+\left (90-39 x-9 x^2\right ) \log (x) \log ^2\left (\frac {6+x}{x \log (x)}\right )+(-6-x) \log (x) \log ^3\left (\frac {6+x}{x \log (x)}\right )}{\left (-750+1225 x-585 x^2+27 x^3+27 x^4\right ) \log (x)+\left (450-465 x+72 x^2+27 x^3\right ) \log (x) \log \left (\frac {6+x}{x \log (x)}\right )+\left (-90+39 x+9 x^2\right ) \log (x) \log ^2\left (\frac {6+x}{x \log (x)}\right )+(6+x) \log (x) \log ^3\left (\frac {6+x}{x \log (x)}\right )} \, dx=- x + \frac {x}{9 x^{2} - 30 x + \left (6 x - 10\right ) \log {\left (\frac {x + 6}{x \log {\left (x \right )}} \right )} + \log {\left (\frac {x + 6}{x \log {\left (x \right )}} \right )}^{2} + 25} \]
integrate(((-x-6)*ln(x)*ln((6+x)/x/ln(x))**3+(-9*x**2-39*x+90)*ln(x)*ln((6 +x)/x/ln(x))**2+(-27*x**3-72*x**2+466*x-444)*ln(x)*ln((6+x)/x/ln(x))+(-27* x**4-27*x**3+582*x**2-1248*x+732)*ln(x)+2*x+12)/((6+x)*ln(x)*ln((6+x)/x/ln (x))**3+(9*x**2+39*x-90)*ln(x)*ln((6+x)/x/ln(x))**2+(27*x**3+72*x**2-465*x +450)*ln(x)*ln((6+x)/x/ln(x))+(27*x**4+27*x**3-585*x**2+1225*x-750)*ln(x)) ,x)
-x + x/(9*x**2 - 30*x + (6*x - 10)*log((x + 6)/(x*log(x))) + log((x + 6)/( x*log(x)))**2 + 25)
Leaf count of result is larger than twice the leaf count of optimal. 165 vs. \(2 (26) = 52\).
Time = 0.28 (sec) , antiderivative size = 165, normalized size of antiderivative = 6.35 \[ \int \frac {12+2 x+\left (732-1248 x+582 x^2-27 x^3-27 x^4\right ) \log (x)+\left (-444+466 x-72 x^2-27 x^3\right ) \log (x) \log \left (\frac {6+x}{x \log (x)}\right )+\left (90-39 x-9 x^2\right ) \log (x) \log ^2\left (\frac {6+x}{x \log (x)}\right )+(-6-x) \log (x) \log ^3\left (\frac {6+x}{x \log (x)}\right )}{\left (-750+1225 x-585 x^2+27 x^3+27 x^4\right ) \log (x)+\left (450-465 x+72 x^2+27 x^3\right ) \log (x) \log \left (\frac {6+x}{x \log (x)}\right )+\left (-90+39 x+9 x^2\right ) \log (x) \log ^2\left (\frac {6+x}{x \log (x)}\right )+(6+x) \log (x) \log ^3\left (\frac {6+x}{x \log (x)}\right )} \, dx=-\frac {9 \, x^{3} + x \log \left (x + 6\right )^{2} + x \log \left (x\right )^{2} + x \log \left (\log \left (x\right )\right )^{2} - 30 \, x^{2} + 2 \, {\left (3 \, x^{2} - x \log \left (x\right ) - x \log \left (\log \left (x\right )\right ) - 5 \, x\right )} \log \left (x + 6\right ) - 2 \, {\left (3 \, x^{2} - 5 \, x\right )} \log \left (x\right ) - 2 \, {\left (3 \, x^{2} - x \log \left (x\right ) - 5 \, x\right )} \log \left (\log \left (x\right )\right ) + 24 \, x}{9 \, x^{2} + 2 \, {\left (3 \, x - \log \left (x\right ) - \log \left (\log \left (x\right )\right ) - 5\right )} \log \left (x + 6\right ) + \log \left (x + 6\right )^{2} - 2 \, {\left (3 \, x - 5\right )} \log \left (x\right ) + \log \left (x\right )^{2} - 2 \, {\left (3 \, x - \log \left (x\right ) - 5\right )} \log \left (\log \left (x\right )\right ) + \log \left (\log \left (x\right )\right )^{2} - 30 \, x + 25} \]
integrate(((-x-6)*log(x)*log((6+x)/x/log(x))^3+(-9*x^2-39*x+90)*log(x)*log ((6+x)/x/log(x))^2+(-27*x^3-72*x^2+466*x-444)*log(x)*log((6+x)/x/log(x))+( -27*x^4-27*x^3+582*x^2-1248*x+732)*log(x)+2*x+12)/((6+x)*log(x)*log((6+x)/ x/log(x))^3+(9*x^2+39*x-90)*log(x)*log((6+x)/x/log(x))^2+(27*x^3+72*x^2-46 5*x+450)*log(x)*log((6+x)/x/log(x))+(27*x^4+27*x^3-585*x^2+1225*x-750)*log (x)),x, algorithm=\
-(9*x^3 + x*log(x + 6)^2 + x*log(x)^2 + x*log(log(x))^2 - 30*x^2 + 2*(3*x^ 2 - x*log(x) - x*log(log(x)) - 5*x)*log(x + 6) - 2*(3*x^2 - 5*x)*log(x) - 2*(3*x^2 - x*log(x) - 5*x)*log(log(x)) + 24*x)/(9*x^2 + 2*(3*x - log(x) - log(log(x)) - 5)*log(x + 6) + log(x + 6)^2 - 2*(3*x - 5)*log(x) + log(x)^2 - 2*(3*x - log(x) - 5)*log(log(x)) + log(log(x))^2 - 30*x + 25)
Leaf count of result is larger than twice the leaf count of optimal. 466 vs. \(2 (26) = 52\).
Time = 1.93 (sec) , antiderivative size = 466, normalized size of antiderivative = 17.92 \[ \int \frac {12+2 x+\left (732-1248 x+582 x^2-27 x^3-27 x^4\right ) \log (x)+\left (-444+466 x-72 x^2-27 x^3\right ) \log (x) \log \left (\frac {6+x}{x \log (x)}\right )+\left (90-39 x-9 x^2\right ) \log (x) \log ^2\left (\frac {6+x}{x \log (x)}\right )+(-6-x) \log (x) \log ^3\left (\frac {6+x}{x \log (x)}\right )}{\left (-750+1225 x-585 x^2+27 x^3+27 x^4\right ) \log (x)+\left (450-465 x+72 x^2+27 x^3\right ) \log (x) \log \left (\frac {6+x}{x \log (x)}\right )+\left (-90+39 x+9 x^2\right ) \log (x) \log ^2\left (\frac {6+x}{x \log (x)}\right )+(6+x) \log (x) \log ^3\left (\frac {6+x}{x \log (x)}\right )} \, dx =\text {Too large to display} \]
integrate(((-x-6)*log(x)*log((6+x)/x/log(x))^3+(-9*x^2-39*x+90)*log(x)*log ((6+x)/x/log(x))^2+(-27*x^3-72*x^2+466*x-444)*log(x)*log((6+x)/x/log(x))+( -27*x^4-27*x^3+582*x^2-1248*x+732)*log(x)+2*x+12)/((6+x)*log(x)*log((6+x)/ x/log(x))^3+(9*x^2+39*x-90)*log(x)*log((6+x)/x/log(x))^2+(27*x^3+72*x^2-46 5*x+450)*log(x)*log((6+x)/x/log(x))+(27*x^4+27*x^3-585*x^2+1225*x-750)*log (x)),x, algorithm=\
-x + (3*x^3*log(x) + 18*x^2*log(x) - x^2 - 6*x*log(x) - 6*x)/(27*x^4*log(x ) + 18*x^3*log(x + 6)*log(x) + 3*x^2*log(x + 6)^2*log(x) - 18*x^3*log(x)^2 - 6*x^2*log(x + 6)*log(x)^2 + 3*x^2*log(x)^3 - 18*x^3*log(x)*log(log(x)) - 6*x^2*log(x + 6)*log(x)*log(log(x)) + 6*x^2*log(x)^2*log(log(x)) + 3*x^2 *log(x)*log(log(x))^2 + 72*x^3*log(x) + 78*x^2*log(x + 6)*log(x) + 18*x*lo g(x + 6)^2*log(x) - 78*x^2*log(x)^2 - 36*x*log(x + 6)*log(x)^2 + 18*x*log( x)^3 - 78*x^2*log(x)*log(log(x)) - 36*x*log(x + 6)*log(x)*log(log(x)) + 36 *x*log(x)^2*log(log(x)) + 18*x*log(x)*log(log(x))^2 - 9*x^3 - 6*x^2*log(x + 6) - x*log(x + 6)^2 - 513*x^2*log(x) - 214*x*log(x + 6)*log(x) - 6*log(x + 6)^2*log(x) + 215*x*log(x)^2 + 12*log(x + 6)*log(x)^2 - 6*log(x)^3 + 6* x^2*log(log(x)) + 2*x*log(x + 6)*log(log(x)) + 214*x*log(x)*log(log(x)) + 12*log(x + 6)*log(x)*log(log(x)) - 12*log(x)^2*log(log(x)) - x*log(log(x)) ^2 - 6*log(x)*log(log(x))^2 - 24*x^2 - 26*x*log(x + 6) - 6*log(x + 6)^2 + 656*x*log(x) + 72*log(x + 6)*log(x) - 66*log(x)^2 + 26*x*log(log(x)) + 12* log(x + 6)*log(log(x)) - 72*log(x)*log(log(x)) - 6*log(log(x))^2 + 155*x + 60*log(x + 6) - 210*log(x) - 60*log(log(x)) - 150)
Timed out. \[ \int \frac {12+2 x+\left (732-1248 x+582 x^2-27 x^3-27 x^4\right ) \log (x)+\left (-444+466 x-72 x^2-27 x^3\right ) \log (x) \log \left (\frac {6+x}{x \log (x)}\right )+\left (90-39 x-9 x^2\right ) \log (x) \log ^2\left (\frac {6+x}{x \log (x)}\right )+(-6-x) \log (x) \log ^3\left (\frac {6+x}{x \log (x)}\right )}{\left (-750+1225 x-585 x^2+27 x^3+27 x^4\right ) \log (x)+\left (450-465 x+72 x^2+27 x^3\right ) \log (x) \log \left (\frac {6+x}{x \log (x)}\right )+\left (-90+39 x+9 x^2\right ) \log (x) \log ^2\left (\frac {6+x}{x \log (x)}\right )+(6+x) \log (x) \log ^3\left (\frac {6+x}{x \log (x)}\right )} \, dx=\int -\frac {\ln \left (x\right )\,\left (x+6\right )\,{\ln \left (\frac {x+6}{x\,\ln \left (x\right )}\right )}^3+\ln \left (x\right )\,\left (9\,x^2+39\,x-90\right )\,{\ln \left (\frac {x+6}{x\,\ln \left (x\right )}\right )}^2+\ln \left (x\right )\,\left (27\,x^3+72\,x^2-466\,x+444\right )\,\ln \left (\frac {x+6}{x\,\ln \left (x\right )}\right )-2\,x+\ln \left (x\right )\,\left (27\,x^4+27\,x^3-582\,x^2+1248\,x-732\right )-12}{\ln \left (x\right )\,\left (x+6\right )\,{\ln \left (\frac {x+6}{x\,\ln \left (x\right )}\right )}^3+\ln \left (x\right )\,\left (9\,x^2+39\,x-90\right )\,{\ln \left (\frac {x+6}{x\,\ln \left (x\right )}\right )}^2+\ln \left (x\right )\,\left (27\,x^3+72\,x^2-465\,x+450\right )\,\ln \left (\frac {x+6}{x\,\ln \left (x\right )}\right )+\ln \left (x\right )\,\left (27\,x^4+27\,x^3-585\,x^2+1225\,x-750\right )} \,d x \]
int(-(log(x)*(1248*x - 582*x^2 + 27*x^3 + 27*x^4 - 732) - 2*x + log((x + 6 )/(x*log(x)))^3*log(x)*(x + 6) + log((x + 6)/(x*log(x)))*log(x)*(72*x^2 - 466*x + 27*x^3 + 444) + log((x + 6)/(x*log(x)))^2*log(x)*(39*x + 9*x^2 - 9 0) - 12)/(log(x)*(1225*x - 585*x^2 + 27*x^3 + 27*x^4 - 750) + log((x + 6)/ (x*log(x)))^3*log(x)*(x + 6) + log((x + 6)/(x*log(x)))*log(x)*(72*x^2 - 46 5*x + 27*x^3 + 450) + log((x + 6)/(x*log(x)))^2*log(x)*(39*x + 9*x^2 - 90) ),x)
int(-(log(x)*(1248*x - 582*x^2 + 27*x^3 + 27*x^4 - 732) - 2*x + log((x + 6 )/(x*log(x)))^3*log(x)*(x + 6) + log((x + 6)/(x*log(x)))*log(x)*(72*x^2 - 466*x + 27*x^3 + 444) + log((x + 6)/(x*log(x)))^2*log(x)*(39*x + 9*x^2 - 9 0) - 12)/(log(x)*(1225*x - 585*x^2 + 27*x^3 + 27*x^4 - 750) + log((x + 6)/ (x*log(x)))^3*log(x)*(x + 6) + log((x + 6)/(x*log(x)))*log(x)*(72*x^2 - 46 5*x + 27*x^3 + 450) + log((x + 6)/(x*log(x)))^2*log(x)*(39*x + 9*x^2 - 90) ), x)