\(\int \frac {(6-30 x+5 x^2) \log (x)+(150 x^2-25 x^3+(30 x-5 x^2) \log (\frac {6-x}{x})) \log ^2(x)+(30 x-5 x^2+(6-x) \log (\frac {6-x}{x})) \log (5 x+\log (\frac {6-x}{x}))}{(150 x^3-25 x^4+(30 x^2-5 x^3) \log (\frac {6-x}{x})) \log ^2(x)+(-30 x^2+5 x^3+(-6 x+x^2) \log (\frac {6-x}{x})) \log (x) \log (5 x+\log (\frac {6-x}{x}))} \, dx\) [744]

Optimal result

Integrand size = 180, antiderivative size = 25 \[ \int \frac {\left (6-30 x+5 x^2\right ) \log (x)+\left (150 x^2-25 x^3+\left (30 x-5 x^2\right ) \log \left (\frac {6-x}{x}\right )\right ) \log ^2(x)+\left (30 x-5 x^2+(6-x) \log \left (\frac {6-x}{x}\right )\right ) \log \left (5 x+\log \left (\frac {6-x}{x}\right )\right )}{\left (150 x^3-25 x^4+\left (30 x^2-5 x^3\right ) \log \left (\frac {6-x}{x}\right )\right ) \log ^2(x)+\left (-30 x^2+5 x^3+\left (-6 x+x^2\right ) \log \left (\frac {6-x}{x}\right )\right ) \log (x) \log \left (5 x+\log \left (\frac {6-x}{x}\right )\right )} \, dx=\log \left (-5 x+\frac {\log \left (5 x+\log \left (\frac {6-x}{x}\right )\right )}{\log (x)}\right ) \] Output:

ln(ln(ln((6-x)/x)+5*x)/ln(x)-5*x)
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {\left (6-30 x+5 x^2\right ) \log (x)+\left (150 x^2-25 x^3+\left (30 x-5 x^2\right ) \log \left (\frac {6-x}{x}\right )\right ) \log ^2(x)+\left (30 x-5 x^2+(6-x) \log \left (\frac {6-x}{x}\right )\right ) \log \left (5 x+\log \left (\frac {6-x}{x}\right )\right )}{\left (150 x^3-25 x^4+\left (30 x^2-5 x^3\right ) \log \left (\frac {6-x}{x}\right )\right ) \log ^2(x)+\left (-30 x^2+5 x^3+\left (-6 x+x^2\right ) \log \left (\frac {6-x}{x}\right )\right ) \log (x) \log \left (5 x+\log \left (\frac {6-x}{x}\right )\right )} \, dx=-\log (\log (x))+\log \left (5 x \log (x)-\log \left (5 x+\log \left (-1+\frac {6}{x}\right )\right )\right ) \] Input:

Integrate[((6 - 30*x + 5*x^2)*Log[x] + (150*x^2 - 25*x^3 + (30*x - 5*x^2)* 
Log[(6 - x)/x])*Log[x]^2 + (30*x - 5*x^2 + (6 - x)*Log[(6 - x)/x])*Log[5*x 
 + Log[(6 - x)/x]])/((150*x^3 - 25*x^4 + (30*x^2 - 5*x^3)*Log[(6 - x)/x])* 
Log[x]^2 + (-30*x^2 + 5*x^3 + (-6*x + x^2)*Log[(6 - x)/x])*Log[x]*Log[5*x 
+ Log[(6 - x)/x]]),x]
 

Output:

-Log[Log[x]] + Log[5*x*Log[x] - Log[5*x + Log[-1 + 6/x]]]
 

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.

\(\displaystyle \int \frac {\left (5 x^2-30 x+6\right ) \log (x)+\left (-5 x^2+30 x+(6-x) \log \left (\frac {6-x}{x}\right )\right ) \log \left (5 x+\log \left (\frac {6-x}{x}\right )\right )+\left (-25 x^3+150 x^2+\left (30 x-5 x^2\right ) \log \left (\frac {6-x}{x}\right )\right ) \log ^2(x)}{\left (5 x^3-30 x^2+\left (x^2-6 x\right ) \log \left (\frac {6-x}{x}\right )\right ) \log \left (5 x+\log \left (\frac {6-x}{x}\right )\right ) \log (x)+\left (-25 x^4+150 x^3+\left (30 x^2-5 x^3\right ) \log \left (\frac {6-x}{x}\right )\right ) \log ^2(x)} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {\left (5 x^2-30 x+6\right ) \log (x)+\left (-5 x^2+30 x+(6-x) \log \left (\frac {6-x}{x}\right )\right ) \log \left (5 x+\log \left (\frac {6-x}{x}\right )\right )+\left (-25 x^3+150 x^2+\left (30 x-5 x^2\right ) \log \left (\frac {6-x}{x}\right )\right ) \log ^2(x)}{(6-x) x \left (5 x+\log \left (\frac {6}{x}-1\right )\right ) \log (x) \left (5 x \log (x)-\log \left (5 x+\log \left (\frac {6}{x}-1\right )\right )\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {25 x^3 \log (x)-5 x^2+5 x^2 \log \left (\frac {6}{x}-1\right ) \log (x)-150 x^2 \log (x)+30 x-30 x \log \left (\frac {6}{x}-1\right ) \log (x)-6}{(x-6) x \left (5 x+\log \left (\frac {6}{x}-1\right )\right ) \left (5 x \log (x)-\log \left (5 x+\log \left (\frac {6}{x}-1\right )\right )\right )}+\frac {\log \left (5 x+\log \left (\frac {6}{x}-1\right )\right )}{x \log (x) \left (5 x \log (x)-\log \left (5 x+\log \left (\frac {6}{x}-1\right )\right )\right )}\right )dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \left (\frac {25 x^3 \log (x)-5 x^2+5 x^2 \log \left (\frac {6}{x}-1\right ) \log (x)-150 x^2 \log (x)+30 x-30 x \log \left (\frac {6}{x}-1\right ) \log (x)-6}{(x-6) x \left (5 x+\log \left (\frac {6}{x}-1\right )\right ) \left (5 x \log (x)-\log \left (5 x+\log \left (\frac {6}{x}-1\right )\right )\right )}+\frac {\log \left (5 x+\log \left (\frac {6}{x}-1\right )\right )}{x \log (x) \left (5 x \log (x)-\log \left (5 x+\log \left (\frac {6}{x}-1\right )\right )\right )}\right )dx\)

Input:

Int[((6 - 30*x + 5*x^2)*Log[x] + (150*x^2 - 25*x^3 + (30*x - 5*x^2)*Log[(6 
 - x)/x])*Log[x]^2 + (30*x - 5*x^2 + (6 - x)*Log[(6 - x)/x])*Log[5*x + Log 
[(6 - x)/x]])/((150*x^3 - 25*x^4 + (30*x^2 - 5*x^3)*Log[(6 - x)/x])*Log[x] 
^2 + (-30*x^2 + 5*x^3 + (-6*x + x^2)*Log[(6 - x)/x])*Log[x]*Log[5*x + Log[ 
(6 - x)/x]]),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 25.29 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16

method	result	size
parallelrisch	\(-\ln \left (\ln \left (x \right )\right )+\ln \left (x \ln \left (x \right )-\frac {\ln \left (\ln \left (-\frac {-6+x}{x}\right )+5 x \right )}{5}\right )\)	\(29\)
default	\(-\ln \left (\ln \left (x \right )\right )+\ln \left (-5 x \ln \left (x \right )+\ln \left (i \pi -\ln \left (x \right )+\ln \left (-6+x \right )-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (-6+x \right )}{x}\right ) \left (-\operatorname {csgn}\left (\frac {i \left (-6+x \right )}{x}\right )+\operatorname {csgn}\left (\frac {i}{x}\right )\right ) \left (-\operatorname {csgn}\left (\frac {i \left (-6+x \right )}{x}\right )+\operatorname {csgn}\left (i \left (-6+x \right )\right )\right )}{2}+i \pi \operatorname {csgn}\left (\frac {i \left (-6+x \right )}{x}\right )^{2} \left (\operatorname {csgn}\left (\frac {i \left (-6+x \right )}{x}\right )-1\right )+5 x \right )\right )\)	\(113\)
risch	\(-\ln \left (\ln \left (x \right )\right )+\ln \left (-5 x \ln \left (x \right )+\ln \left (i \pi -\ln \left (x \right )+\ln \left (-6+x \right )-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (-6+x \right )}{x}\right ) \left (-\operatorname {csgn}\left (\frac {i \left (-6+x \right )}{x}\right )+\operatorname {csgn}\left (\frac {i}{x}\right )\right ) \left (-\operatorname {csgn}\left (\frac {i \left (-6+x \right )}{x}\right )+\operatorname {csgn}\left (i \left (-6+x \right )\right )\right )}{2}+i \pi \operatorname {csgn}\left (\frac {i \left (-6+x \right )}{x}\right )^{2} \left (\operatorname {csgn}\left (\frac {i \left (-6+x \right )}{x}\right )-1\right )+5 x \right )\right )\)	\(113\)

Input:

int((((-x+6)*ln((-x+6)/x)-5*x^2+30*x)*ln(ln((-x+6)/x)+5*x)+((-5*x^2+30*x)* 
ln((-x+6)/x)-25*x^3+150*x^2)*ln(x)^2+(5*x^2-30*x+6)*ln(x))/(((x^2-6*x)*ln( 
(-x+6)/x)+5*x^3-30*x^2)*ln(x)*ln(ln((-x+6)/x)+5*x)+((-5*x^3+30*x^2)*ln((-x 
+6)/x)-25*x^4+150*x^3)*ln(x)^2),x,method=_RETURNVERBOSE)
 

Output:

-ln(ln(x))+ln(x*ln(x)-1/5*ln(ln(-(-6+x)/x)+5*x))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {\left (6-30 x+5 x^2\right ) \log (x)+\left (150 x^2-25 x^3+\left (30 x-5 x^2\right ) \log \left (\frac {6-x}{x}\right )\right ) \log ^2(x)+\left (30 x-5 x^2+(6-x) \log \left (\frac {6-x}{x}\right )\right ) \log \left (5 x+\log \left (\frac {6-x}{x}\right )\right )}{\left (150 x^3-25 x^4+\left (30 x^2-5 x^3\right ) \log \left (\frac {6-x}{x}\right )\right ) \log ^2(x)+\left (-30 x^2+5 x^3+\left (-6 x+x^2\right ) \log \left (\frac {6-x}{x}\right )\right ) \log (x) \log \left (5 x+\log \left (\frac {6-x}{x}\right )\right )} \, dx=\log \left (-5 \, x \log \left (x\right ) + \log \left (5 \, x + \log \left (-\frac {x - 6}{x}\right )\right )\right ) - \log \left (\log \left (x\right )\right ) \] Input:

integrate((((6-x)*log((6-x)/x)-5*x^2+30*x)*log(log((6-x)/x)+5*x)+((-5*x^2+ 
30*x)*log((6-x)/x)-25*x^3+150*x^2)*log(x)^2+(5*x^2-30*x+6)*log(x))/(((x^2- 
6*x)*log((6-x)/x)+5*x^3-30*x^2)*log(x)*log(log((6-x)/x)+5*x)+((-5*x^3+30*x 
^2)*log((6-x)/x)-25*x^4+150*x^3)*log(x)^2),x, algorithm="fricas")
 

Output:

log(-5*x*log(x) + log(5*x + log(-(x - 6)/x))) - log(log(x))
 

Sympy [A] (verification not implemented)

Time = 0.66 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {\left (6-30 x+5 x^2\right ) \log (x)+\left (150 x^2-25 x^3+\left (30 x-5 x^2\right ) \log \left (\frac {6-x}{x}\right )\right ) \log ^2(x)+\left (30 x-5 x^2+(6-x) \log \left (\frac {6-x}{x}\right )\right ) \log \left (5 x+\log \left (\frac {6-x}{x}\right )\right )}{\left (150 x^3-25 x^4+\left (30 x^2-5 x^3\right ) \log \left (\frac {6-x}{x}\right )\right ) \log ^2(x)+\left (-30 x^2+5 x^3+\left (-6 x+x^2\right ) \log \left (\frac {6-x}{x}\right )\right ) \log (x) \log \left (5 x+\log \left (\frac {6-x}{x}\right )\right )} \, dx=\log {\left (- 5 x \log {\left (x \right )} + \log {\left (5 x + \log {\left (\frac {6 - x}{x} \right )} \right )} \right )} - \log {\left (\log {\left (x \right )} \right )} \] Input:

integrate((((6-x)*ln((6-x)/x)-5*x**2+30*x)*ln(ln((6-x)/x)+5*x)+((-5*x**2+3 
0*x)*ln((6-x)/x)-25*x**3+150*x**2)*ln(x)**2+(5*x**2-30*x+6)*ln(x))/(((x**2 
-6*x)*ln((6-x)/x)+5*x**3-30*x**2)*ln(x)*ln(ln((6-x)/x)+5*x)+((-5*x**3+30*x 
**2)*ln((6-x)/x)-25*x**4+150*x**3)*ln(x)**2),x)
 

Output:

log(-5*x*log(x) + log(5*x + log((6 - x)/x))) - log(log(x))
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {\left (6-30 x+5 x^2\right ) \log (x)+\left (150 x^2-25 x^3+\left (30 x-5 x^2\right ) \log \left (\frac {6-x}{x}\right )\right ) \log ^2(x)+\left (30 x-5 x^2+(6-x) \log \left (\frac {6-x}{x}\right )\right ) \log \left (5 x+\log \left (\frac {6-x}{x}\right )\right )}{\left (150 x^3-25 x^4+\left (30 x^2-5 x^3\right ) \log \left (\frac {6-x}{x}\right )\right ) \log ^2(x)+\left (-30 x^2+5 x^3+\left (-6 x+x^2\right ) \log \left (\frac {6-x}{x}\right )\right ) \log (x) \log \left (5 x+\log \left (\frac {6-x}{x}\right )\right )} \, dx=\log \left (-5 \, x \log \left (x\right ) + \log \left (5 \, x - \log \left (x\right ) + \log \left (-x + 6\right )\right )\right ) - \log \left (\log \left (x\right )\right ) \] Input:

integrate((((6-x)*log((6-x)/x)-5*x^2+30*x)*log(log((6-x)/x)+5*x)+((-5*x^2+ 
30*x)*log((6-x)/x)-25*x^3+150*x^2)*log(x)^2+(5*x^2-30*x+6)*log(x))/(((x^2- 
6*x)*log((6-x)/x)+5*x^3-30*x^2)*log(x)*log(log((6-x)/x)+5*x)+((-5*x^3+30*x 
^2)*log((6-x)/x)-25*x^4+150*x^3)*log(x)^2),x, algorithm="maxima")
 

Output:

log(-5*x*log(x) + log(5*x - log(x) + log(-x + 6))) - log(log(x))
 

Giac [A] (verification not implemented)

Time = 0.43 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {\left (6-30 x+5 x^2\right ) \log (x)+\left (150 x^2-25 x^3+\left (30 x-5 x^2\right ) \log \left (\frac {6-x}{x}\right )\right ) \log ^2(x)+\left (30 x-5 x^2+(6-x) \log \left (\frac {6-x}{x}\right )\right ) \log \left (5 x+\log \left (\frac {6-x}{x}\right )\right )}{\left (150 x^3-25 x^4+\left (30 x^2-5 x^3\right ) \log \left (\frac {6-x}{x}\right )\right ) \log ^2(x)+\left (-30 x^2+5 x^3+\left (-6 x+x^2\right ) \log \left (\frac {6-x}{x}\right )\right ) \log (x) \log \left (5 x+\log \left (\frac {6-x}{x}\right )\right )} \, dx=\log \left (-5 \, x \log \left (x\right ) + \log \left (5 \, x - \log \left (x\right ) + \log \left (-x + 6\right )\right )\right ) - \log \left (\log \left (x\right )\right ) \] Input:

integrate((((6-x)*log((6-x)/x)-5*x^2+30*x)*log(log((6-x)/x)+5*x)+((-5*x^2+ 
30*x)*log((6-x)/x)-25*x^3+150*x^2)*log(x)^2+(5*x^2-30*x+6)*log(x))/(((x^2- 
6*x)*log((6-x)/x)+5*x^3-30*x^2)*log(x)*log(log((6-x)/x)+5*x)+((-5*x^3+30*x 
^2)*log((6-x)/x)-25*x^4+150*x^3)*log(x)^2),x, algorithm="giac")
 

Output:

log(-5*x*log(x) + log(5*x - log(x) + log(-x + 6))) - log(log(x))
 

Mupad [B] (verification not implemented)

Time = 4.45 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {\left (6-30 x+5 x^2\right ) \log (x)+\left (150 x^2-25 x^3+\left (30 x-5 x^2\right ) \log \left (\frac {6-x}{x}\right )\right ) \log ^2(x)+\left (30 x-5 x^2+(6-x) \log \left (\frac {6-x}{x}\right )\right ) \log \left (5 x+\log \left (\frac {6-x}{x}\right )\right )}{\left (150 x^3-25 x^4+\left (30 x^2-5 x^3\right ) \log \left (\frac {6-x}{x}\right )\right ) \log ^2(x)+\left (-30 x^2+5 x^3+\left (-6 x+x^2\right ) \log \left (\frac {6-x}{x}\right )\right ) \log (x) \log \left (5 x+\log \left (\frac {6-x}{x}\right )\right )} \, dx=\ln \left (\ln \left (5\,x+\ln \left (-\frac {x-6}{x}\right )\right )-5\,x\,\ln \left (x\right )\right )-\ln \left (\ln \left (x\right )\right ) \] Input:

int((log(x)*(5*x^2 - 30*x + 6) - log(5*x + log(-(x - 6)/x))*(log(-(x - 6)/ 
x)*(x - 6) - 30*x + 5*x^2) + log(x)^2*(log(-(x - 6)/x)*(30*x - 5*x^2) + 15 
0*x^2 - 25*x^3))/(log(x)^2*(log(-(x - 6)/x)*(30*x^2 - 5*x^3) + 150*x^3 - 2 
5*x^4) - log(5*x + log(-(x - 6)/x))*log(x)*(log(-(x - 6)/x)*(6*x - x^2) + 
30*x^2 - 5*x^3)),x)
 

Output:

log(log(5*x + log(-(x - 6)/x)) - 5*x*log(x)) - log(log(x))
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {\left (6-30 x+5 x^2\right ) \log (x)+\left (150 x^2-25 x^3+\left (30 x-5 x^2\right ) \log \left (\frac {6-x}{x}\right )\right ) \log ^2(x)+\left (30 x-5 x^2+(6-x) \log \left (\frac {6-x}{x}\right )\right ) \log \left (5 x+\log \left (\frac {6-x}{x}\right )\right )}{\left (150 x^3-25 x^4+\left (30 x^2-5 x^3\right ) \log \left (\frac {6-x}{x}\right )\right ) \log ^2(x)+\left (-30 x^2+5 x^3+\left (-6 x+x^2\right ) \log \left (\frac {6-x}{x}\right )\right ) \log (x) \log \left (5 x+\log \left (\frac {6-x}{x}\right )\right )} \, dx=-\mathrm {log}\left (\mathrm {log}\left (x \right )\right )+\mathrm {log}\left (\mathrm {log}\left (\mathrm {log}\left (\frac {-x +6}{x}\right )+5 x \right )-5 \,\mathrm {log}\left (x \right ) x \right ) \] Input:

int((((6-x)*log((6-x)/x)-5*x^2+30*x)*log(log((6-x)/x)+5*x)+((-5*x^2+30*x)* 
log((6-x)/x)-25*x^3+150*x^2)*log(x)^2+(5*x^2-30*x+6)*log(x))/(((x^2-6*x)*l 
og((6-x)/x)+5*x^3-30*x^2)*log(x)*log(log((6-x)/x)+5*x)+((-5*x^3+30*x^2)*lo 
g((6-x)/x)-25*x^4+150*x^3)*log(x)^2),x)
 

Output:

 - log(log(x)) + log(log(log(( - x + 6)/x) + 5*x) - 5*log(x)*x)