\(\int \frac {(20-22 x+6 x^2) \log (3)+(12 x-3 x^2) \log (3) \log (4 x-x^2) \log (\log (4 x-x^2))+(8 x^2-2 x^3) \log (3) \log (4 x-x^2) \log ^2(\log (4 x-x^2))}{(-4 x+x^2) \log (4 x-x^2) \log ^2(\log (4 x-x^2))} \, dx\) [1195]

Optimal result

Integrand size = 119, antiderivative size = 29 \[ \int \frac {\left (20-22 x+6 x^2\right ) \log (3)+\left (12 x-3 x^2\right ) \log (3) \log \left (4 x-x^2\right ) \log \left (\log \left (4 x-x^2\right )\right )+\left (8 x^2-2 x^3\right ) \log (3) \log \left (4 x-x^2\right ) \log ^2\left (\log \left (4 x-x^2\right )\right )}{\left (-4 x+x^2\right ) \log \left (4 x-x^2\right ) \log ^2\left (\log \left (4 x-x^2\right )\right )} \, dx=\log (3) \left (-x^2+\frac {\left (-3+\frac {5}{x}\right ) x}{\log (\log ((4-x) x))}\right ) \] Output:

ln(3)*((5/x-3)/ln(ln((4-x)*x))*x-x^2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86 \[ \int \frac {\left (20-22 x+6 x^2\right ) \log (3)+\left (12 x-3 x^2\right ) \log (3) \log \left (4 x-x^2\right ) \log \left (\log \left (4 x-x^2\right )\right )+\left (8 x^2-2 x^3\right ) \log (3) \log \left (4 x-x^2\right ) \log ^2\left (\log \left (4 x-x^2\right )\right )}{\left (-4 x+x^2\right ) \log \left (4 x-x^2\right ) \log ^2\left (\log \left (4 x-x^2\right )\right )} \, dx=\log (3) \left (-x^2+\frac {5-3 x}{\log (\log (-((-4+x) x)))}\right ) \] Input:

Integrate[((20 - 22*x + 6*x^2)*Log[3] + (12*x - 3*x^2)*Log[3]*Log[4*x - x^ 
2]*Log[Log[4*x - x^2]] + (8*x^2 - 2*x^3)*Log[3]*Log[4*x - x^2]*Log[Log[4*x 
 - x^2]]^2)/((-4*x + x^2)*Log[4*x - x^2]*Log[Log[4*x - x^2]]^2),x]
 

Output:

Log[3]*(-x^2 + (5 - 3*x)/Log[Log[-((-4 + x)*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.

\(\displaystyle \int \frac {\left (12 x-3 x^2\right ) \log (3) \log \left (4 x-x^2\right ) \log \left (\log \left (4 x-x^2\right )\right )+\left (6 x^2-22 x+20\right ) \log (3)+\left (8 x^2-2 x^3\right ) \log (3) \log \left (4 x-x^2\right ) \log ^2\left (\log \left (4 x-x^2\right )\right )}{\left (x^2-4 x\right ) \log \left (4 x-x^2\right ) \log ^2\left (\log \left (4 x-x^2\right )\right )} \, dx\)

\(\Big \downarrow \) 2026

\(\displaystyle \int \frac {\left (12 x-3 x^2\right ) \log (3) \log \left (4 x-x^2\right ) \log \left (\log \left (4 x-x^2\right )\right )+\left (6 x^2-22 x+20\right ) \log (3)+\left (8 x^2-2 x^3\right ) \log (3) \log \left (4 x-x^2\right ) \log ^2\left (\log \left (4 x-x^2\right )\right )}{(x-4) x \log \left (4 x-x^2\right ) \log ^2\left (\log \left (4 x-x^2\right )\right )}dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {-\left (12 x-3 x^2\right ) \log (3) \log \left (4 x-x^2\right ) \log \left (\log \left (4 x-x^2\right )\right )-\left (6 x^2-22 x+20\right ) \log (3)-\left (\left (8 x^2-2 x^3\right ) \log (3) \log \left (4 x-x^2\right ) \log ^2\left (\log \left (4 x-x^2\right )\right )\right )}{(4-x) x \log ((4-x) x) \log ^2(\log ((4-x) x))}dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\log (3) \left (-6 x^2+22 x+(x-4) x \log (-((x-4) x)) \log (\log (-((x-4) x))) (2 x \log (\log (-((x-4) x)))+3)-20\right )}{(4-x) x \log ((4-x) x) \log ^2(\log ((4-x) x))}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \log (3) \int -\frac {6 x^2+(4-x) \log ((4-x) x) \log (\log ((4-x) x)) (2 x \log (\log ((4-x) x))+3) x-22 x+20}{(4-x) x \log ((4-x) x) \log ^2(\log ((4-x) x))}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -\log (3) \int \frac {6 x^2+(4-x) \log ((4-x) x) \log (\log ((4-x) x)) (2 x \log (\log ((4-x) x))+3) x-22 x+20}{(4-x) x \log ((4-x) x) \log ^2(\log ((4-x) x))}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\log (3) \int \left (2 x+\frac {3}{\log (\log ((4-x) x))}+\frac {2 \left (3 x^2-11 x+10\right )}{(4-x) \log ((4-x) x) \log ^2(\log ((4-x) x)) x}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\log (3) \left (-6 \int \frac {1}{\log ((4-x) x) \log ^2(\log ((4-x) x))}dx+7 \int \frac {1}{(4-x) \log ((4-x) x) \log ^2(\log ((4-x) x))}dx+5 \int \frac {1}{x \log ((4-x) x) \log ^2(\log ((4-x) x))}dx+3 \int \frac {1}{\log (\log ((4-x) x))}dx+x^2\right )\)

Input:

Int[((20 - 22*x + 6*x^2)*Log[3] + (12*x - 3*x^2)*Log[3]*Log[4*x - x^2]*Log 
[Log[4*x - x^2]] + (8*x^2 - 2*x^3)*Log[3]*Log[4*x - x^2]*Log[Log[4*x - x^2 
]]^2)/((-4*x + x^2)*Log[4*x - x^2]*Log[Log[4*x - x^2]]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 1.26 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.00

method	result	size
parallelrisch	\(-\frac {\ln \left (\ln \left (-x^{2}+4 x \right )\right ) \ln \left (3\right ) x^{2}+3 x \ln \left (3\right )-16 \ln \left (\ln \left (-x^{2}+4 x \right )\right ) \ln \left (3\right )-5 \ln \left (3\right )}{\ln \left (\ln \left (-x^{2}+4 x \right )\right )}\)	\(58\)

Input:

int(((-2*x^3+8*x^2)*ln(3)*ln(-x^2+4*x)*ln(ln(-x^2+4*x))^2+(-3*x^2+12*x)*ln 
(3)*ln(-x^2+4*x)*ln(ln(-x^2+4*x))+(6*x^2-22*x+20)*ln(3))/(x^2-4*x)/ln(-x^2 
+4*x)/ln(ln(-x^2+4*x))^2,x,method=_RETURNVERBOSE)
 

Output:

-(ln(ln(-x^2+4*x))*ln(3)*x^2+3*x*ln(3)-16*ln(ln(-x^2+4*x))*ln(3)-5*ln(3))/ 
ln(ln(-x^2+4*x))
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.41 \[ \int \frac {\left (20-22 x+6 x^2\right ) \log (3)+\left (12 x-3 x^2\right ) \log (3) \log \left (4 x-x^2\right ) \log \left (\log \left (4 x-x^2\right )\right )+\left (8 x^2-2 x^3\right ) \log (3) \log \left (4 x-x^2\right ) \log ^2\left (\log \left (4 x-x^2\right )\right )}{\left (-4 x+x^2\right ) \log \left (4 x-x^2\right ) \log ^2\left (\log \left (4 x-x^2\right )\right )} \, dx=-\frac {x^{2} \log \left (3\right ) \log \left (\log \left (-x^{2} + 4 \, x\right )\right ) + {\left (3 \, x - 5\right )} \log \left (3\right )}{\log \left (\log \left (-x^{2} + 4 \, x\right )\right )} \] Input:

integrate(((-2*x^3+8*x^2)*log(3)*log(-x^2+4*x)*log(log(-x^2+4*x))^2+(-3*x^ 
2+12*x)*log(3)*log(-x^2+4*x)*log(log(-x^2+4*x))+(6*x^2-22*x+20)*log(3))/(x 
^2-4*x)/log(-x^2+4*x)/log(log(-x^2+4*x))^2,x, algorithm="fricas")
 

Output:

-(x^2*log(3)*log(log(-x^2 + 4*x)) + (3*x - 5)*log(3))/log(log(-x^2 + 4*x))
 

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {\left (20-22 x+6 x^2\right ) \log (3)+\left (12 x-3 x^2\right ) \log (3) \log \left (4 x-x^2\right ) \log \left (\log \left (4 x-x^2\right )\right )+\left (8 x^2-2 x^3\right ) \log (3) \log \left (4 x-x^2\right ) \log ^2\left (\log \left (4 x-x^2\right )\right )}{\left (-4 x+x^2\right ) \log \left (4 x-x^2\right ) \log ^2\left (\log \left (4 x-x^2\right )\right )} \, dx=- x^{2} \log {\left (3 \right )} + \frac {- 3 x \log {\left (3 \right )} + 5 \log {\left (3 \right )}}{\log {\left (\log {\left (- x^{2} + 4 x \right )} \right )}} \] Input:

integrate(((-2*x**3+8*x**2)*ln(3)*ln(-x**2+4*x)*ln(ln(-x**2+4*x))**2+(-3*x 
**2+12*x)*ln(3)*ln(-x**2+4*x)*ln(ln(-x**2+4*x))+(6*x**2-22*x+20)*ln(3))/(x 
**2-4*x)/ln(-x**2+4*x)/ln(ln(-x**2+4*x))**2,x)
 

Output:

-x**2*log(3) + (-3*x*log(3) + 5*log(3))/log(log(-x**2 + 4*x))
 

Maxima [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.38 \[ \int \frac {\left (20-22 x+6 x^2\right ) \log (3)+\left (12 x-3 x^2\right ) \log (3) \log \left (4 x-x^2\right ) \log \left (\log \left (4 x-x^2\right )\right )+\left (8 x^2-2 x^3\right ) \log (3) \log \left (4 x-x^2\right ) \log ^2\left (\log \left (4 x-x^2\right )\right )}{\left (-4 x+x^2\right ) \log \left (4 x-x^2\right ) \log ^2\left (\log \left (4 x-x^2\right )\right )} \, dx=-\frac {x^{2} \log \left (3\right ) \log \left (\log \left (x\right ) + \log \left (-x + 4\right )\right ) + 3 \, x \log \left (3\right ) - 5 \, \log \left (3\right )}{\log \left (\log \left (x\right ) + \log \left (-x + 4\right )\right )} \] Input:

integrate(((-2*x^3+8*x^2)*log(3)*log(-x^2+4*x)*log(log(-x^2+4*x))^2+(-3*x^ 
2+12*x)*log(3)*log(-x^2+4*x)*log(log(-x^2+4*x))+(6*x^2-22*x+20)*log(3))/(x 
^2-4*x)/log(-x^2+4*x)/log(log(-x^2+4*x))^2,x, algorithm="maxima")
 

Output:

-(x^2*log(3)*log(log(x) + log(-x + 4)) + 3*x*log(3) - 5*log(3))/log(log(x) 
 + log(-x + 4))
 

Giac [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.14 \[ \int \frac {\left (20-22 x+6 x^2\right ) \log (3)+\left (12 x-3 x^2\right ) \log (3) \log \left (4 x-x^2\right ) \log \left (\log \left (4 x-x^2\right )\right )+\left (8 x^2-2 x^3\right ) \log (3) \log \left (4 x-x^2\right ) \log ^2\left (\log \left (4 x-x^2\right )\right )}{\left (-4 x+x^2\right ) \log \left (4 x-x^2\right ) \log ^2\left (\log \left (4 x-x^2\right )\right )} \, dx=-x^{2} \log \left (3\right ) - \frac {3 \, x \log \left (3\right ) - 5 \, \log \left (3\right )}{\log \left (\log \left (-x^{2} + 4 \, x\right )\right )} \] Input:

integrate(((-2*x^3+8*x^2)*log(3)*log(-x^2+4*x)*log(log(-x^2+4*x))^2+(-3*x^ 
2+12*x)*log(3)*log(-x^2+4*x)*log(log(-x^2+4*x))+(6*x^2-22*x+20)*log(3))/(x 
^2-4*x)/log(-x^2+4*x)/log(log(-x^2+4*x))^2,x, algorithm="giac")
 

Output:

-x^2*log(3) - (3*x*log(3) - 5*log(3))/log(log(-x^2 + 4*x))
 

Mupad [B] (verification not implemented)

Time = 1.71 (sec) , antiderivative size = 106, normalized size of antiderivative = 3.66 \[ \int \frac {\left (20-22 x+6 x^2\right ) \log (3)+\left (12 x-3 x^2\right ) \log (3) \log \left (4 x-x^2\right ) \log \left (\log \left (4 x-x^2\right )\right )+\left (8 x^2-2 x^3\right ) \log (3) \log \left (4 x-x^2\right ) \log ^2\left (\log \left (4 x-x^2\right )\right )}{\left (-4 x+x^2\right ) \log \left (4 x-x^2\right ) \log ^2\left (\log \left (4 x-x^2\right )\right )} \, dx=6\,\ln \left (x\,\left (x-4\right )\right )\,\ln \left (3\right )-x^2\,\ln \left (3\right )+\frac {5\,\ln \left (3\right )-3\,x\,\ln \left (3\right )+\frac {3\,x\,\ln \left (\ln \left (4\,x-x^2\right )\right )\,\ln \left (3\right )\,\ln \left (4\,x-x^2\right )\,\left (x-4\right )}{2\,\left (x-2\right )}}{\ln \left (\ln \left (4\,x-x^2\right )\right )}+\frac {\ln \left (4\,x-x^2\right )\,\left (12\,\ln \left (3\right )-\frac {3\,x^2\,\ln \left (3\right )}{2}\right )}{x-2} \] Input:

int(-(log(3)*(6*x^2 - 22*x + 20) + log(log(4*x - x^2))^2*log(3)*log(4*x - 
x^2)*(8*x^2 - 2*x^3) + log(log(4*x - x^2))*log(3)*log(4*x - x^2)*(12*x - 3 
*x^2))/(log(log(4*x - x^2))^2*log(4*x - x^2)*(4*x - x^2)),x)
 

Output:

6*log(x*(x - 4))*log(3) - x^2*log(3) + (5*log(3) - 3*x*log(3) + (3*x*log(l 
og(4*x - x^2))*log(3)*log(4*x - x^2)*(x - 4))/(2*(x - 2)))/log(log(4*x - x 
^2)) + (log(4*x - x^2)*(12*log(3) - (3*x^2*log(3))/2))/(x - 2)
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.28 \[ \int \frac {\left (20-22 x+6 x^2\right ) \log (3)+\left (12 x-3 x^2\right ) \log (3) \log \left (4 x-x^2\right ) \log \left (\log \left (4 x-x^2\right )\right )+\left (8 x^2-2 x^3\right ) \log (3) \log \left (4 x-x^2\right ) \log ^2\left (\log \left (4 x-x^2\right )\right )}{\left (-4 x+x^2\right ) \log \left (4 x-x^2\right ) \log ^2\left (\log \left (4 x-x^2\right )\right )} \, dx=\frac {\mathrm {log}\left (3\right ) \left (-\mathrm {log}\left (\mathrm {log}\left (-x^{2}+4 x \right )\right ) x^{2}-3 x +5\right )}{\mathrm {log}\left (\mathrm {log}\left (-x^{2}+4 x \right )\right )} \] Input:

int(((-2*x^3+8*x^2)*log(3)*log(-x^2+4*x)*log(log(-x^2+4*x))^2+(-3*x^2+12*x 
)*log(3)*log(-x^2+4*x)*log(log(-x^2+4*x))+(6*x^2-22*x+20)*log(3))/(x^2-4*x 
)/log(-x^2+4*x)/log(log(-x^2+4*x))^2,x)
 

Output:

(log(3)*( - log(log( - x**2 + 4*x))*x**2 - 3*x + 5))/log(log( - x**2 + 4*x 
))