\(\int \frac {320-5184 x+860 x^2+116 x^3+3 x^4+(-640 x+4928 x^2-712 x^3-108 x^4-3 x^5) \log (-4 x+x^2)+(-16 x+4 x^2+(16 x^2-4 x^3) \log (-4 x+x^2)) \log (x-x^2 \log (-4 x+x^2))}{4800 x-720 x^2-108 x^3-3 x^4+(-4800 x^2+720 x^3+108 x^4+3 x^5) \log (-4 x+x^2)} \, dx\) [174]

Optimal result

Integrand size = 151, antiderivative size = 28 \[ \int \frac {320-5184 x+860 x^2+116 x^3+3 x^4+\left (-640 x+4928 x^2-712 x^3-108 x^4-3 x^5\right ) \log \left (-4 x+x^2\right )+\left (-16 x+4 x^2+\left (16 x^2-4 x^3\right ) \log \left (-4 x+x^2\right )\right ) \log \left (x-x^2 \log \left (-4 x+x^2\right )\right )}{4800 x-720 x^2-108 x^3-3 x^4+\left (-4800 x^2+720 x^3+108 x^4+3 x^5\right ) \log \left (-4 x+x^2\right )} \, dx=-8-x+\frac {4 \log \left (x-x^2 \log ((-4+x) x)\right )}{3 (20+x)} \] Output:

4/3/(20+x)*ln(x-x^2*ln((-4+x)*x))-x-8
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {320-5184 x+860 x^2+116 x^3+3 x^4+\left (-640 x+4928 x^2-712 x^3-108 x^4-3 x^5\right ) \log \left (-4 x+x^2\right )+\left (-16 x+4 x^2+\left (16 x^2-4 x^3\right ) \log \left (-4 x+x^2\right )\right ) \log \left (x-x^2 \log \left (-4 x+x^2\right )\right )}{4800 x-720 x^2-108 x^3-3 x^4+\left (-4800 x^2+720 x^3+108 x^4+3 x^5\right ) \log \left (-4 x+x^2\right )} \, dx=\frac {1}{3} \left (-3 x+\frac {4 \log \left (x-x^2 \log ((-4+x) x)\right )}{20+x}\right ) \] Input:

Integrate[(320 - 5184*x + 860*x^2 + 116*x^3 + 3*x^4 + (-640*x + 4928*x^2 - 
 712*x^3 - 108*x^4 - 3*x^5)*Log[-4*x + x^2] + (-16*x + 4*x^2 + (16*x^2 - 4 
*x^3)*Log[-4*x + x^2])*Log[x - x^2*Log[-4*x + x^2]])/(4800*x - 720*x^2 - 1 
08*x^3 - 3*x^4 + (-4800*x^2 + 720*x^3 + 108*x^4 + 3*x^5)*Log[-4*x + x^2]), 
x]
 

Output:

(-3*x + (4*Log[x - x^2*Log[(-4 + x)*x]])/(20 + x))/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.

\(\displaystyle \int \frac {3 x^4+116 x^3+860 x^2+\left (4 x^2+\left (16 x^2-4 x^3\right ) \log \left (x^2-4 x\right )-16 x\right ) \log \left (x-x^2 \log \left (x^2-4 x\right )\right )+\left (-3 x^5-108 x^4-712 x^3+4928 x^2-640 x\right ) \log \left (x^2-4 x\right )-5184 x+320}{-3 x^4-108 x^3-720 x^2+\left (3 x^5+108 x^4+720 x^3-4800 x^2\right ) \log \left (x^2-4 x\right )+4800 x} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {3 x^4+116 x^3+860 x^2+\left (4 x^2+\left (16 x^2-4 x^3\right ) \log \left (x^2-4 x\right )-16 x\right ) \log \left (x-x^2 \log \left (x^2-4 x\right )\right )+\left (-3 x^5-108 x^4-712 x^3+4928 x^2-640 x\right ) \log \left (x^2-4 x\right )-5184 x+320}{3 (4-x) x (x+20)^2 (1-x \log ((x-4) x))}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \int \frac {3 x^4+116 x^3+860 x^2-5184 x-\left (3 x^5+108 x^4+712 x^3-4928 x^2+640 x\right ) \log \left (x^2-4 x\right )-4 \left (-x^2+4 x-\left (4 x^2-x^3\right ) \log \left (x^2-4 x\right )\right ) \log \left (x-x^2 \log \left (x^2-4 x\right )\right )+320}{(4-x) x (x+20)^2 (1-x \log (-((4-x) x)))}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {1}{3} \int \left (\frac {3 x^3}{(x-4) (x+20)^2 (x \log ((x-4) x)-1)}+\frac {116 x^2}{(x-4) (x+20)^2 (x \log ((x-4) x)-1)}+\frac {860 x}{(x-4) (x+20)^2 (x \log ((x-4) x)-1)}-\frac {4 \log \left (x-x^2 \log ((x-4) x)\right )}{(x+20)^2}-\frac {\left (3 x^2+60 x-8\right ) \log ((x-4) x)}{(x+20) (x \log ((x-4) x)-1)}-\frac {5184}{(x-4) (x+20)^2 (x \log ((x-4) x)-1)}+\frac {320}{(x-4) (x+20)^2 (x \log ((x-4) x)-1) x}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{3} \left (-4 \int \frac {\log \left (x-x^2 \log ((x-4) x)\right )}{(x+20)^2}dx+\frac {2}{3} \int \frac {1}{(x-4) (x \log ((x-4) x)-1)}dx+\frac {1}{5} \int \frac {1}{x (x \log ((x-4) x)-1)}dx+\frac {107}{15} \int \frac {1}{(x+20) (x \log ((x-4) x)-1)}dx-3 x+\frac {2 \log (x)}{5}-\frac {2}{5} \log (x+20)\right )\)

Input:

Int[(320 - 5184*x + 860*x^2 + 116*x^3 + 3*x^4 + (-640*x + 4928*x^2 - 712*x 
^3 - 108*x^4 - 3*x^5)*Log[-4*x + x^2] + (-16*x + 4*x^2 + (16*x^2 - 4*x^3)* 
Log[-4*x + x^2])*Log[x - x^2*Log[-4*x + x^2]])/(4800*x - 720*x^2 - 108*x^3 
 - 3*x^4 + (-4800*x^2 + 720*x^3 + 108*x^4 + 3*x^5)*Log[-4*x + x^2]),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 1.84 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29

method	result	size
parallelrisch	\(\frac {-19200-432 x^{2}-9600 x +576 \ln \left (-x^{2} \ln \left (x^{2}-4 x \right )+x \right )}{8640+432 x}\)	\(36\)

Input:

int((((-4*x^3+16*x^2)*ln(x^2-4*x)+4*x^2-16*x)*ln(-x^2*ln(x^2-4*x)+x)+(-3*x 
^5-108*x^4-712*x^3+4928*x^2-640*x)*ln(x^2-4*x)+3*x^4+116*x^3+860*x^2-5184* 
x+320)/((3*x^5+108*x^4+720*x^3-4800*x^2)*ln(x^2-4*x)-3*x^4-108*x^3-720*x^2 
+4800*x),x,method=_RETURNVERBOSE)
 

Output:

1/432*(-19200-432*x^2-9600*x+576*ln(-x^2*ln(x^2-4*x)+x))/(20+x)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {320-5184 x+860 x^2+116 x^3+3 x^4+\left (-640 x+4928 x^2-712 x^3-108 x^4-3 x^5\right ) \log \left (-4 x+x^2\right )+\left (-16 x+4 x^2+\left (16 x^2-4 x^3\right ) \log \left (-4 x+x^2\right )\right ) \log \left (x-x^2 \log \left (-4 x+x^2\right )\right )}{4800 x-720 x^2-108 x^3-3 x^4+\left (-4800 x^2+720 x^3+108 x^4+3 x^5\right ) \log \left (-4 x+x^2\right )} \, dx=-\frac {3 \, x^{2} + 60 \, x - 4 \, \log \left (-x^{2} \log \left (x^{2} - 4 \, x\right ) + x\right )}{3 \, {\left (x + 20\right )}} \] Input:

integrate((((-4*x^3+16*x^2)*log(x^2-4*x)+4*x^2-16*x)*log(-x^2*log(x^2-4*x) 
+x)+(-3*x^5-108*x^4-712*x^3+4928*x^2-640*x)*log(x^2-4*x)+3*x^4+116*x^3+860 
*x^2-5184*x+320)/((3*x^5+108*x^4+720*x^3-4800*x^2)*log(x^2-4*x)-3*x^4-108* 
x^3-720*x^2+4800*x),x, algorithm="fricas")
 

Output:

-1/3*(3*x^2 + 60*x - 4*log(-x^2*log(x^2 - 4*x) + x))/(x + 20)
 

Sympy [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.79 \[ \int \frac {320-5184 x+860 x^2+116 x^3+3 x^4+\left (-640 x+4928 x^2-712 x^3-108 x^4-3 x^5\right ) \log \left (-4 x+x^2\right )+\left (-16 x+4 x^2+\left (16 x^2-4 x^3\right ) \log \left (-4 x+x^2\right )\right ) \log \left (x-x^2 \log \left (-4 x+x^2\right )\right )}{4800 x-720 x^2-108 x^3-3 x^4+\left (-4800 x^2+720 x^3+108 x^4+3 x^5\right ) \log \left (-4 x+x^2\right )} \, dx=- x + \frac {4 \log {\left (- x^{2} \log {\left (x^{2} - 4 x \right )} + x \right )}}{3 x + 60} \] Input:

integrate((((-4*x**3+16*x**2)*ln(x**2-4*x)+4*x**2-16*x)*ln(-x**2*ln(x**2-4 
*x)+x)+(-3*x**5-108*x**4-712*x**3+4928*x**2-640*x)*ln(x**2-4*x)+3*x**4+116 
*x**3+860*x**2-5184*x+320)/((3*x**5+108*x**4+720*x**3-4800*x**2)*ln(x**2-4 
*x)-3*x**4-108*x**3-720*x**2+4800*x),x)
 

Output:

-x + 4*log(-x**2*log(x**2 - 4*x) + x)/(3*x + 60)
 

Maxima [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.32 \[ \int \frac {320-5184 x+860 x^2+116 x^3+3 x^4+\left (-640 x+4928 x^2-712 x^3-108 x^4-3 x^5\right ) \log \left (-4 x+x^2\right )+\left (-16 x+4 x^2+\left (16 x^2-4 x^3\right ) \log \left (-4 x+x^2\right )\right ) \log \left (x-x^2 \log \left (-4 x+x^2\right )\right )}{4800 x-720 x^2-108 x^3-3 x^4+\left (-4800 x^2+720 x^3+108 x^4+3 x^5\right ) \log \left (-4 x+x^2\right )} \, dx=-\frac {3 \, x^{2} + 60 \, x - 4 \, \log \left (-x \log \left (x - 4\right ) - x \log \left (x\right ) + 1\right ) - 4 \, \log \left (x\right )}{3 \, {\left (x + 20\right )}} \] Input:

integrate((((-4*x^3+16*x^2)*log(x^2-4*x)+4*x^2-16*x)*log(-x^2*log(x^2-4*x) 
+x)+(-3*x^5-108*x^4-712*x^3+4928*x^2-640*x)*log(x^2-4*x)+3*x^4+116*x^3+860 
*x^2-5184*x+320)/((3*x^5+108*x^4+720*x^3-4800*x^2)*log(x^2-4*x)-3*x^4-108* 
x^3-720*x^2+4800*x),x, algorithm="maxima")
 

Output:

-1/3*(3*x^2 + 60*x - 4*log(-x*log(x - 4) - x*log(x) + 1) - 4*log(x))/(x + 
20)
 

Giac [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {320-5184 x+860 x^2+116 x^3+3 x^4+\left (-640 x+4928 x^2-712 x^3-108 x^4-3 x^5\right ) \log \left (-4 x+x^2\right )+\left (-16 x+4 x^2+\left (16 x^2-4 x^3\right ) \log \left (-4 x+x^2\right )\right ) \log \left (x-x^2 \log \left (-4 x+x^2\right )\right )}{4800 x-720 x^2-108 x^3-3 x^4+\left (-4800 x^2+720 x^3+108 x^4+3 x^5\right ) \log \left (-4 x+x^2\right )} \, dx=-x + \frac {4 \, \log \left (-x \log \left (x^{2} - 4 \, x\right ) + 1\right )}{3 \, {\left (x + 20\right )}} + \frac {4 \, \log \left (x\right )}{3 \, {\left (x + 20\right )}} \] Input:

integrate((((-4*x^3+16*x^2)*log(x^2-4*x)+4*x^2-16*x)*log(-x^2*log(x^2-4*x) 
+x)+(-3*x^5-108*x^4-712*x^3+4928*x^2-640*x)*log(x^2-4*x)+3*x^4+116*x^3+860 
*x^2-5184*x+320)/((3*x^5+108*x^4+720*x^3-4800*x^2)*log(x^2-4*x)-3*x^4-108* 
x^3-720*x^2+4800*x),x, algorithm="giac")
 

Output:

-x + 4/3*log(-x*log(x^2 - 4*x) + 1)/(x + 20) + 4/3*log(x)/(x + 20)
 

Mupad [B] (verification not implemented)

Time = 2.71 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {320-5184 x+860 x^2+116 x^3+3 x^4+\left (-640 x+4928 x^2-712 x^3-108 x^4-3 x^5\right ) \log \left (-4 x+x^2\right )+\left (-16 x+4 x^2+\left (16 x^2-4 x^3\right ) \log \left (-4 x+x^2\right )\right ) \log \left (x-x^2 \log \left (-4 x+x^2\right )\right )}{4800 x-720 x^2-108 x^3-3 x^4+\left (-4800 x^2+720 x^3+108 x^4+3 x^5\right ) \log \left (-4 x+x^2\right )} \, dx=\frac {4\,\ln \left (x-x^2\,\ln \left (x^2-4\,x\right )\right )}{3\,\left (x+20\right )}-x \] Input:

int(-(log(x - x^2*log(x^2 - 4*x))*(log(x^2 - 4*x)*(16*x^2 - 4*x^3) - 16*x 
+ 4*x^2) - 5184*x + 860*x^2 + 116*x^3 + 3*x^4 - log(x^2 - 4*x)*(640*x - 49 
28*x^2 + 712*x^3 + 108*x^4 + 3*x^5) + 320)/(720*x^2 - log(x^2 - 4*x)*(720* 
x^3 - 4800*x^2 + 108*x^4 + 3*x^5) - 4800*x + 108*x^3 + 3*x^4),x)
 

Output:

(4*log(x - x^2*log(x^2 - 4*x)))/(3*(x + 20)) - x
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.64 \[ \int \frac {320-5184 x+860 x^2+116 x^3+3 x^4+\left (-640 x+4928 x^2-712 x^3-108 x^4-3 x^5\right ) \log \left (-4 x+x^2\right )+\left (-16 x+4 x^2+\left (16 x^2-4 x^3\right ) \log \left (-4 x+x^2\right )\right ) \log \left (x-x^2 \log \left (-4 x+x^2\right )\right )}{4800 x-720 x^2-108 x^3-3 x^4+\left (-4800 x^2+720 x^3+108 x^4+3 x^5\right ) \log \left (-4 x+x^2\right )} \, dx=\frac {-\mathrm {log}\left (-\mathrm {log}\left (x^{2}-4 x \right ) x^{2}+x \right ) x +\mathrm {log}\left (\mathrm {log}\left (x^{2}-4 x \right ) x -1\right ) x +20 \,\mathrm {log}\left (\mathrm {log}\left (x^{2}-4 x \right ) x -1\right )+\mathrm {log}\left (x \right ) x +20 \,\mathrm {log}\left (x \right )-15 x^{2}-300 x}{15 x +300} \] Input:

int((((-4*x^3+16*x^2)*log(x^2-4*x)+4*x^2-16*x)*log(-x^2*log(x^2-4*x)+x)+(- 
3*x^5-108*x^4-712*x^3+4928*x^2-640*x)*log(x^2-4*x)+3*x^4+116*x^3+860*x^2-5 
184*x+320)/((3*x^5+108*x^4+720*x^3-4800*x^2)*log(x^2-4*x)-3*x^4-108*x^3-72 
0*x^2+4800*x),x)
 

Output:

( - log( - log(x**2 - 4*x)*x**2 + x)*x + log(log(x**2 - 4*x)*x - 1)*x + 20 
*log(log(x**2 - 4*x)*x - 1) + log(x)*x + 20*log(x) - 15*x**2 - 300*x)/(15* 
(x + 20))