\(\int \frac {-2400 x+1200 x^2+(-3750+1875 x+4800 x^2-1800 x^3) \log (x)+(2400-1200 x+(-6000 x+2400 x^2) \log (x)) \log ((2-x) \log (x))+(1200-600 x) \log (x) \log ^2((2-x) \log (x))}{(-5000 x^2+2500 x^3+3200 x^4-1600 x^5-512 x^6+256 x^7) \log (x)+(-6400 x^3+3200 x^4+2048 x^5-1024 x^6) \log (x) \log ((2-x) \log (x))+(3200 x^2-1600 x^3-3072 x^4+1536 x^5) \log (x) \log ^2((2-x) \log (x))+(2048 x^3-1024 x^4) \log (x) \log ^3((2-x) \log (x))+(-512 x^2+256 x^3) \log (x) \log ^4((2-x) \log (x))} \, dx\) [607]

Optimal result

Integrand size = 229, antiderivative size = 30 \[ \int \frac {-2400 x+1200 x^2+\left (-3750+1875 x+4800 x^2-1800 x^3\right ) \log (x)+\left (2400-1200 x+\left (-6000 x+2400 x^2\right ) \log (x)\right ) \log ((2-x) \log (x))+(1200-600 x) \log (x) \log ^2((2-x) \log (x))}{\left (-5000 x^2+2500 x^3+3200 x^4-1600 x^5-512 x^6+256 x^7\right ) \log (x)+\left (-6400 x^3+3200 x^4+2048 x^5-1024 x^6\right ) \log (x) \log ((2-x) \log (x))+\left (3200 x^2-1600 x^3-3072 x^4+1536 x^5\right ) \log (x) \log ^2((2-x) \log (x))+\left (2048 x^3-1024 x^4\right ) \log (x) \log ^3((2-x) \log (x))+\left (-512 x^2+256 x^3\right ) \log (x) \log ^4((2-x) \log (x))} \, dx=\frac {3}{2 x \left (-2+\frac {16}{25} (x-\log ((2-x) \log (x)))^2\right )} \] Output:

3/2/x/(4*(1/5*x-1/5*ln((2-x)*ln(x)))*(4/5*x-4/5*ln((2-x)*ln(x)))-2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.30 \[ \int \frac {-2400 x+1200 x^2+\left (-3750+1875 x+4800 x^2-1800 x^3\right ) \log (x)+\left (2400-1200 x+\left (-6000 x+2400 x^2\right ) \log (x)\right ) \log ((2-x) \log (x))+(1200-600 x) \log (x) \log ^2((2-x) \log (x))}{\left (-5000 x^2+2500 x^3+3200 x^4-1600 x^5-512 x^6+256 x^7\right ) \log (x)+\left (-6400 x^3+3200 x^4+2048 x^5-1024 x^6\right ) \log (x) \log ((2-x) \log (x))+\left (3200 x^2-1600 x^3-3072 x^4+1536 x^5\right ) \log (x) \log ^2((2-x) \log (x))+\left (2048 x^3-1024 x^4\right ) \log (x) \log ^3((2-x) \log (x))+\left (-512 x^2+256 x^3\right ) \log (x) \log ^4((2-x) \log (x))} \, dx=\frac {75}{4 x \left (-25+8 x^2-16 x \log (-((-2+x) \log (x)))+8 \log ^2(-((-2+x) \log (x)))\right )} \] Input:

Integrate[(-2400*x + 1200*x^2 + (-3750 + 1875*x + 4800*x^2 - 1800*x^3)*Log 
[x] + (2400 - 1200*x + (-6000*x + 2400*x^2)*Log[x])*Log[(2 - x)*Log[x]] + 
(1200 - 600*x)*Log[x]*Log[(2 - x)*Log[x]]^2)/((-5000*x^2 + 2500*x^3 + 3200 
*x^4 - 1600*x^5 - 512*x^6 + 256*x^7)*Log[x] + (-6400*x^3 + 3200*x^4 + 2048 
*x^5 - 1024*x^6)*Log[x]*Log[(2 - x)*Log[x]] + (3200*x^2 - 1600*x^3 - 3072* 
x^4 + 1536*x^5)*Log[x]*Log[(2 - x)*Log[x]]^2 + (2048*x^3 - 1024*x^4)*Log[x 
]*Log[(2 - x)*Log[x]]^3 + (-512*x^2 + 256*x^3)*Log[x]*Log[(2 - x)*Log[x]]^ 
4),x]
 

Output:

75/(4*x*(-25 + 8*x^2 - 16*x*Log[-((-2 + x)*Log[x])] + 8*Log[-((-2 + x)*Log 
[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.

\(\displaystyle \int \frac {1200 x^2+\left (\left (2400 x^2-6000 x\right ) \log (x)-1200 x+2400\right ) \log ((2-x) \log (x))+\left (-1800 x^3+4800 x^2+1875 x-3750\right ) \log (x)-2400 x+(1200-600 x) \log (x) \log ^2((2-x) \log (x))}{\left (2048 x^3-1024 x^4\right ) \log (x) \log ^3((2-x) \log (x))+\left (256 x^3-512 x^2\right ) \log (x) \log ^4((2-x) \log (x))+\left (-1024 x^6+2048 x^5+3200 x^4-6400 x^3\right ) \log (x) \log ((2-x) \log (x))+\left (1536 x^5-3072 x^4-1600 x^3+3200 x^2\right ) \log (x) \log ^2((2-x) \log (x))+\left (256 x^7-512 x^6-1600 x^5+3200 x^4+2500 x^3-5000 x^2\right ) \log (x)} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {75 \left (\log (x) \left (24 x^3-64 x^2-25 x+8 (x-2) \log ^2(-((x-2) \log (x)))+16 (5-2 x) x \log (-((x-2) \log (x)))+50\right )-16 (x-2) (x-\log (-((x-2) \log (x))))\right )}{4 (2-x) x^2 \log (x) \left (-8 x^2-8 \log ^2(-((x-2) \log (x)))+16 x \log (-((x-2) \log (x)))+25\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {75}{4} \int \frac {16 (2-x) (x-\log ((2-x) \log (x)))+\log (x) \left (24 x^3-64 x^2+16 (5-2 x) \log ((2-x) \log (x)) x-25 x-8 (2-x) \log ^2((2-x) \log (x))+50\right )}{(2-x) x^2 \log (x) \left (-8 x^2+16 \log ((2-x) \log (x)) x-8 \log ^2((2-x) \log (x))+25\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {75}{4} \int \left (-\frac {16 \left (\log (x) x^2-3 \log (x) x-x+2\right ) (x-\log (-((x-2) \log (x))))}{(x-2) x^2 \log (x) \left (8 x^2-16 \log (-((x-2) \log (x))) x+8 \log ^2(-((x-2) \log (x)))-25\right )^2}-\frac {1}{x^2 \left (8 x^2-16 \log (-((x-2) \log (x))) x+8 \log ^2(-((x-2) \log (x)))-25\right )}\right )dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \frac {75}{4} \int \left (-\frac {16 \left (\log (x) x^2-3 \log (x) x-x+2\right ) (x-\log (-((x-2) \log (x))))}{(x-2) x^2 \log (x) \left (8 x^2-16 \log (-((x-2) \log (x))) x+8 \log ^2(-((x-2) \log (x)))-25\right )^2}-\frac {1}{x^2 \left (8 x^2-16 \log (-((x-2) \log (x))) x+8 \log ^2(-((x-2) \log (x)))-25\right )}\right )dx\)

Input:

Int[(-2400*x + 1200*x^2 + (-3750 + 1875*x + 4800*x^2 - 1800*x^3)*Log[x] + 
(2400 - 1200*x + (-6000*x + 2400*x^2)*Log[x])*Log[(2 - x)*Log[x]] + (1200 
- 600*x)*Log[x]*Log[(2 - x)*Log[x]]^2)/((-5000*x^2 + 2500*x^3 + 3200*x^4 - 
 1600*x^5 - 512*x^6 + 256*x^7)*Log[x] + (-6400*x^3 + 3200*x^4 + 2048*x^5 - 
 1024*x^6)*Log[x]*Log[(2 - x)*Log[x]] + (3200*x^2 - 1600*x^3 - 3072*x^4 + 
1536*x^5)*Log[x]*Log[(2 - x)*Log[x]]^2 + (2048*x^3 - 1024*x^4)*Log[x]*Log[ 
(2 - x)*Log[x]]^3 + (-512*x^2 + 256*x^3)*Log[x]*Log[(2 - x)*Log[x]]^4),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 21.34 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.33

method	result	size
parallelrisch	\(\frac {75}{4 x \left (8 x^{2}-16 x \ln \left (\left (2-x \right ) \ln \left (x \right )\right )+8 \ln \left (\left (2-x \right ) \ln \left (x \right )\right )^{2}-25\right )}\)	\(40\)
risch	\(\text {Expression too large to display}\)	\(829\)
default	\(\text {Expression too large to display}\)	\(1699\)
parts	\(\text {Expression too large to display}\)	\(1699\)

Input:

int(((-600*x+1200)*ln(x)*ln((2-x)*ln(x))^2+((2400*x^2-6000*x)*ln(x)-1200*x 
+2400)*ln((2-x)*ln(x))+(-1800*x^3+4800*x^2+1875*x-3750)*ln(x)+1200*x^2-240 
0*x)/((256*x^3-512*x^2)*ln(x)*ln((2-x)*ln(x))^4+(-1024*x^4+2048*x^3)*ln(x) 
*ln((2-x)*ln(x))^3+(1536*x^5-3072*x^4-1600*x^3+3200*x^2)*ln(x)*ln((2-x)*ln 
(x))^2+(-1024*x^6+2048*x^5+3200*x^4-6400*x^3)*ln(x)*ln((2-x)*ln(x))+(256*x 
^7-512*x^6-1600*x^5+3200*x^4+2500*x^3-5000*x^2)*ln(x)),x,method=_RETURNVER 
BOSE)
 

Output:

75/4/x/(8*x^2-16*x*ln((2-x)*ln(x))+8*ln((2-x)*ln(x))^2-25)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.30 \[ \int \frac {-2400 x+1200 x^2+\left (-3750+1875 x+4800 x^2-1800 x^3\right ) \log (x)+\left (2400-1200 x+\left (-6000 x+2400 x^2\right ) \log (x)\right ) \log ((2-x) \log (x))+(1200-600 x) \log (x) \log ^2((2-x) \log (x))}{\left (-5000 x^2+2500 x^3+3200 x^4-1600 x^5-512 x^6+256 x^7\right ) \log (x)+\left (-6400 x^3+3200 x^4+2048 x^5-1024 x^6\right ) \log (x) \log ((2-x) \log (x))+\left (3200 x^2-1600 x^3-3072 x^4+1536 x^5\right ) \log (x) \log ^2((2-x) \log (x))+\left (2048 x^3-1024 x^4\right ) \log (x) \log ^3((2-x) \log (x))+\left (-512 x^2+256 x^3\right ) \log (x) \log ^4((2-x) \log (x))} \, dx=\frac {75}{4 \, {\left (8 \, x^{3} - 16 \, x^{2} \log \left (-{\left (x - 2\right )} \log \left (x\right )\right ) + 8 \, x \log \left (-{\left (x - 2\right )} \log \left (x\right )\right )^{2} - 25 \, x\right )}} \] Input:

integrate(((-600*x+1200)*log(x)*log((2-x)*log(x))^2+((2400*x^2-6000*x)*log 
(x)-1200*x+2400)*log((2-x)*log(x))+(-1800*x^3+4800*x^2+1875*x-3750)*log(x) 
+1200*x^2-2400*x)/((256*x^3-512*x^2)*log(x)*log((2-x)*log(x))^4+(-1024*x^4 
+2048*x^3)*log(x)*log((2-x)*log(x))^3+(1536*x^5-3072*x^4-1600*x^3+3200*x^2 
)*log(x)*log((2-x)*log(x))^2+(-1024*x^6+2048*x^5+3200*x^4-6400*x^3)*log(x) 
*log((2-x)*log(x))+(256*x^7-512*x^6-1600*x^5+3200*x^4+2500*x^3-5000*x^2)*l 
og(x)),x, algorithm="fricas")
 

Output:

75/4/(8*x^3 - 16*x^2*log(-(x - 2)*log(x)) + 8*x*log(-(x - 2)*log(x))^2 - 2 
5*x)
 

Sympy [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20 \[ \int \frac {-2400 x+1200 x^2+\left (-3750+1875 x+4800 x^2-1800 x^3\right ) \log (x)+\left (2400-1200 x+\left (-6000 x+2400 x^2\right ) \log (x)\right ) \log ((2-x) \log (x))+(1200-600 x) \log (x) \log ^2((2-x) \log (x))}{\left (-5000 x^2+2500 x^3+3200 x^4-1600 x^5-512 x^6+256 x^7\right ) \log (x)+\left (-6400 x^3+3200 x^4+2048 x^5-1024 x^6\right ) \log (x) \log ((2-x) \log (x))+\left (3200 x^2-1600 x^3-3072 x^4+1536 x^5\right ) \log (x) \log ^2((2-x) \log (x))+\left (2048 x^3-1024 x^4\right ) \log (x) \log ^3((2-x) \log (x))+\left (-512 x^2+256 x^3\right ) \log (x) \log ^4((2-x) \log (x))} \, dx=\frac {75}{32 x^{3} - 64 x^{2} \log {\left (\left (2 - x\right ) \log {\left (x \right )} \right )} + 32 x \log {\left (\left (2 - x\right ) \log {\left (x \right )} \right )}^{2} - 100 x} \] Input:

integrate(((-600*x+1200)*ln(x)*ln((2-x)*ln(x))**2+((2400*x**2-6000*x)*ln(x 
)-1200*x+2400)*ln((2-x)*ln(x))+(-1800*x**3+4800*x**2+1875*x-3750)*ln(x)+12 
00*x**2-2400*x)/((256*x**3-512*x**2)*ln(x)*ln((2-x)*ln(x))**4+(-1024*x**4+ 
2048*x**3)*ln(x)*ln((2-x)*ln(x))**3+(1536*x**5-3072*x**4-1600*x**3+3200*x* 
*2)*ln(x)*ln((2-x)*ln(x))**2+(-1024*x**6+2048*x**5+3200*x**4-6400*x**3)*ln 
(x)*ln((2-x)*ln(x))+(256*x**7-512*x**6-1600*x**5+3200*x**4+2500*x**3-5000* 
x**2)*ln(x)),x)
 

Output:

75/(32*x**3 - 64*x**2*log((2 - x)*log(x)) + 32*x*log((2 - x)*log(x))**2 - 
100*x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (25) = 50\).

Time = 0.11 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.93 \[ \int \frac {-2400 x+1200 x^2+\left (-3750+1875 x+4800 x^2-1800 x^3\right ) \log (x)+\left (2400-1200 x+\left (-6000 x+2400 x^2\right ) \log (x)\right ) \log ((2-x) \log (x))+(1200-600 x) \log (x) \log ^2((2-x) \log (x))}{\left (-5000 x^2+2500 x^3+3200 x^4-1600 x^5-512 x^6+256 x^7\right ) \log (x)+\left (-6400 x^3+3200 x^4+2048 x^5-1024 x^6\right ) \log (x) \log ((2-x) \log (x))+\left (3200 x^2-1600 x^3-3072 x^4+1536 x^5\right ) \log (x) \log ^2((2-x) \log (x))+\left (2048 x^3-1024 x^4\right ) \log (x) \log ^3((2-x) \log (x))+\left (-512 x^2+256 x^3\right ) \log (x) \log ^4((2-x) \log (x))} \, dx=\frac {75}{4 \, {\left (8 \, x^{3} + 8 \, x \log \left (-x + 2\right )^{2} - 16 \, x^{2} \log \left (\log \left (x\right )\right ) + 8 \, x \log \left (\log \left (x\right )\right )^{2} - 16 \, {\left (x^{2} - x \log \left (\log \left (x\right )\right )\right )} \log \left (-x + 2\right ) - 25 \, x\right )}} \] Input:

integrate(((-600*x+1200)*log(x)*log((2-x)*log(x))^2+((2400*x^2-6000*x)*log 
(x)-1200*x+2400)*log((2-x)*log(x))+(-1800*x^3+4800*x^2+1875*x-3750)*log(x) 
+1200*x^2-2400*x)/((256*x^3-512*x^2)*log(x)*log((2-x)*log(x))^4+(-1024*x^4 
+2048*x^3)*log(x)*log((2-x)*log(x))^3+(1536*x^5-3072*x^4-1600*x^3+3200*x^2 
)*log(x)*log((2-x)*log(x))^2+(-1024*x^6+2048*x^5+3200*x^4-6400*x^3)*log(x) 
*log((2-x)*log(x))+(256*x^7-512*x^6-1600*x^5+3200*x^4+2500*x^3-5000*x^2)*l 
og(x)),x, algorithm="maxima")
 

Output:

75/4/(8*x^3 + 8*x*log(-x + 2)^2 - 16*x^2*log(log(x)) + 8*x*log(log(x))^2 - 
 16*(x^2 - x*log(log(x)))*log(-x + 2) - 25*x)
 

Giac [A] (verification not implemented)

Time = 5.73 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.50 \[ \int \frac {-2400 x+1200 x^2+\left (-3750+1875 x+4800 x^2-1800 x^3\right ) \log (x)+\left (2400-1200 x+\left (-6000 x+2400 x^2\right ) \log (x)\right ) \log ((2-x) \log (x))+(1200-600 x) \log (x) \log ^2((2-x) \log (x))}{\left (-5000 x^2+2500 x^3+3200 x^4-1600 x^5-512 x^6+256 x^7\right ) \log (x)+\left (-6400 x^3+3200 x^4+2048 x^5-1024 x^6\right ) \log (x) \log ((2-x) \log (x))+\left (3200 x^2-1600 x^3-3072 x^4+1536 x^5\right ) \log (x) \log ^2((2-x) \log (x))+\left (2048 x^3-1024 x^4\right ) \log (x) \log ^3((2-x) \log (x))+\left (-512 x^2+256 x^3\right ) \log (x) \log ^4((2-x) \log (x))} \, dx=\frac {75}{4 \, {\left (8 \, x^{3} - 16 \, x^{2} \log \left (-x \log \left (x\right ) + 2 \, \log \left (x\right )\right ) + 8 \, x \log \left (-x \log \left (x\right ) + 2 \, \log \left (x\right )\right )^{2} - 25 \, x\right )}} \] Input:

integrate(((-600*x+1200)*log(x)*log((2-x)*log(x))^2+((2400*x^2-6000*x)*log 
(x)-1200*x+2400)*log((2-x)*log(x))+(-1800*x^3+4800*x^2+1875*x-3750)*log(x) 
+1200*x^2-2400*x)/((256*x^3-512*x^2)*log(x)*log((2-x)*log(x))^4+(-1024*x^4 
+2048*x^3)*log(x)*log((2-x)*log(x))^3+(1536*x^5-3072*x^4-1600*x^3+3200*x^2 
)*log(x)*log((2-x)*log(x))^2+(-1024*x^6+2048*x^5+3200*x^4-6400*x^3)*log(x) 
*log((2-x)*log(x))+(256*x^7-512*x^6-1600*x^5+3200*x^4+2500*x^3-5000*x^2)*l 
og(x)),x, algorithm="giac")
 

Output:

75/4/(8*x^3 - 16*x^2*log(-x*log(x) + 2*log(x)) + 8*x*log(-x*log(x) + 2*log 
(x))^2 - 25*x)
 

Mupad [B] (verification not implemented)

Time = 4.04 (sec) , antiderivative size = 270, normalized size of antiderivative = 9.00 \[ \int \frac {-2400 x+1200 x^2+\left (-3750+1875 x+4800 x^2-1800 x^3\right ) \log (x)+\left (2400-1200 x+\left (-6000 x+2400 x^2\right ) \log (x)\right ) \log ((2-x) \log (x))+(1200-600 x) \log (x) \log ^2((2-x) \log (x))}{\left (-5000 x^2+2500 x^3+3200 x^4-1600 x^5-512 x^6+256 x^7\right ) \log (x)+\left (-6400 x^3+3200 x^4+2048 x^5-1024 x^6\right ) \log (x) \log ((2-x) \log (x))+\left (3200 x^2-1600 x^3-3072 x^4+1536 x^5\right ) \log (x) \log ^2((2-x) \log (x))+\left (2048 x^3-1024 x^4\right ) \log (x) \log ^3((2-x) \log (x))+\left (-512 x^2+256 x^3\right ) \log (x) \log ^4((2-x) \log (x))} \, dx=-\frac {300\,x^2\,{\ln \left (x\right )}^2+\frac {75\,x^8\,{\ln \left (x\right )}^4}{4}-x^5\,\left (1125\,{\ln \left (x\right )}^4+1125\,{\ln \left (x\right )}^3+150\,{\ln \left (x\right )}^2\right )+x^4\,\left (675\,{\ln \left (x\right )}^4+1650\,{\ln \left (x\right )}^3+450\,{\ln \left (x\right )}^2\right )+x^6\,\left (\frac {2775\,{\ln \left (x\right )}^4}{4}+\frac {675\,{\ln \left (x\right )}^3}{2}+\frac {75\,{\ln \left (x\right )}^2}{4}\right )-x^7\,\left (\frac {375\,{\ln \left (x\right )}^4}{2}+\frac {75\,{\ln \left (x\right )}^3}{2}\right )-x^3\,\left (900\,{\ln \left (x\right )}^3+600\,{\ln \left (x\right )}^2\right )}{x^2\,\ln \left (x\right )\,\left (x-2\right )\,\left (-8\,x^2+16\,x\,\ln \left (-\ln \left (x\right )\,\left (x-2\right )\right )-8\,{\ln \left (-\ln \left (x\right )\,\left (x-2\right )\right )}^2+25\right )\,\left (x^6\,{\ln \left (x\right )}^3-8\,x^5\,{\ln \left (x\right )}^3-2\,x^5\,{\ln \left (x\right )}^2+21\,x^4\,{\ln \left (x\right )}^3+14\,x^4\,{\ln \left (x\right )}^2+x^4\,\ln \left (x\right )-18\,x^3\,{\ln \left (x\right )}^3-32\,x^3\,{\ln \left (x\right )}^2-6\,x^3\,\ln \left (x\right )+24\,x^2\,{\ln \left (x\right )}^2+12\,x^2\,\ln \left (x\right )-8\,x\,\ln \left (x\right )\right )} \] Input:

int((2400*x + log(-log(x)*(x - 2))*(1200*x + log(x)*(6000*x - 2400*x^2) - 
2400) - 1200*x^2 - log(x)*(1875*x + 4800*x^2 - 1800*x^3 - 3750) + log(-log 
(x)*(x - 2))^2*log(x)*(600*x - 1200))/(log(x)*(5000*x^2 - 2500*x^3 - 3200* 
x^4 + 1600*x^5 + 512*x^6 - 256*x^7) + log(-log(x)*(x - 2))*log(x)*(6400*x^ 
3 - 3200*x^4 - 2048*x^5 + 1024*x^6) - log(-log(x)*(x - 2))^2*log(x)*(3200* 
x^2 - 1600*x^3 - 3072*x^4 + 1536*x^5) + log(-log(x)*(x - 2))^4*log(x)*(512 
*x^2 - 256*x^3) - log(-log(x)*(x - 2))^3*log(x)*(2048*x^3 - 1024*x^4)),x)
 

Output:

-(300*x^2*log(x)^2 + (75*x^8*log(x)^4)/4 - x^5*(150*log(x)^2 + 1125*log(x) 
^3 + 1125*log(x)^4) + x^4*(450*log(x)^2 + 1650*log(x)^3 + 675*log(x)^4) + 
x^6*((75*log(x)^2)/4 + (675*log(x)^3)/2 + (2775*log(x)^4)/4) - x^7*((75*lo 
g(x)^3)/2 + (375*log(x)^4)/2) - x^3*(600*log(x)^2 + 900*log(x)^3))/(x^2*lo 
g(x)*(x - 2)*(16*x*log(-log(x)*(x - 2)) - 8*x^2 - 8*log(-log(x)*(x - 2))^2 
 + 25)*(12*x^2*log(x) - 6*x^3*log(x) + x^4*log(x) + 24*x^2*log(x)^2 - 32*x 
^3*log(x)^2 - 18*x^3*log(x)^3 + 14*x^4*log(x)^2 + 21*x^4*log(x)^3 - 2*x^5* 
log(x)^2 - 8*x^5*log(x)^3 + x^6*log(x)^3 - 8*x*log(x)))
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.43 \[ \int \frac {-2400 x+1200 x^2+\left (-3750+1875 x+4800 x^2-1800 x^3\right ) \log (x)+\left (2400-1200 x+\left (-6000 x+2400 x^2\right ) \log (x)\right ) \log ((2-x) \log (x))+(1200-600 x) \log (x) \log ^2((2-x) \log (x))}{\left (-5000 x^2+2500 x^3+3200 x^4-1600 x^5-512 x^6+256 x^7\right ) \log (x)+\left (-6400 x^3+3200 x^4+2048 x^5-1024 x^6\right ) \log (x) \log ((2-x) \log (x))+\left (3200 x^2-1600 x^3-3072 x^4+1536 x^5\right ) \log (x) \log ^2((2-x) \log (x))+\left (2048 x^3-1024 x^4\right ) \log (x) \log ^3((2-x) \log (x))+\left (-512 x^2+256 x^3\right ) \log (x) \log ^4((2-x) \log (x))} \, dx=\frac {75}{4 x \left (8 \mathrm {log}\left (-\mathrm {log}\left (x \right ) x +2 \,\mathrm {log}\left (x \right )\right )^{2}-16 \,\mathrm {log}\left (-\mathrm {log}\left (x \right ) x +2 \,\mathrm {log}\left (x \right )\right ) x +8 x^{2}-25\right )} \] Input:

int(((-600*x+1200)*log(x)*log((2-x)*log(x))^2+((2400*x^2-6000*x)*log(x)-12 
00*x+2400)*log((2-x)*log(x))+(-1800*x^3+4800*x^2+1875*x-3750)*log(x)+1200* 
x^2-2400*x)/((256*x^3-512*x^2)*log(x)*log((2-x)*log(x))^4+(-1024*x^4+2048* 
x^3)*log(x)*log((2-x)*log(x))^3+(1536*x^5-3072*x^4-1600*x^3+3200*x^2)*log( 
x)*log((2-x)*log(x))^2+(-1024*x^6+2048*x^5+3200*x^4-6400*x^3)*log(x)*log(( 
2-x)*log(x))+(256*x^7-512*x^6-1600*x^5+3200*x^4+2500*x^3-5000*x^2)*log(x)) 
,x)
 

Output:

75/(4*x*(8*log( - log(x)*x + 2*log(x))**2 - 16*log( - log(x)*x + 2*log(x)) 
*x + 8*x**2 - 25))