\(\int \frac {(-8 x+4 x^2) \log (-2+x)+(2 x^2-x^3+(2-x) \log (3)) \log (-2+x) \log ^2(x^2)+(-2 x^2 \log (x^2)+(-x^3+x \log (3)) \log ^2(x^2)) \log (\frac {-2 x+(-x^2+\log (3)) \log (x^2)}{x \log (x^2)})}{(4 x^2-2 x^3) \log (x^2)+(2 x^3-x^4+(-2 x+x^2) \log (3)) \log ^2(x^2)} \, dx\) [1337]

Optimal result

Integrand size = 149, antiderivative size = 24 \[ \int \frac {\left (-8 x+4 x^2\right ) \log (-2+x)+\left (2 x^2-x^3+(2-x) \log (3)\right ) \log (-2+x) \log ^2\left (x^2\right )+\left (-2 x^2 \log \left (x^2\right )+\left (-x^3+x \log (3)\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {-2 x+\left (-x^2+\log (3)\right ) \log \left (x^2\right )}{x \log \left (x^2\right )}\right )}{\left (4 x^2-2 x^3\right ) \log \left (x^2\right )+\left (2 x^3-x^4+\left (-2 x+x^2\right ) \log (3)\right ) \log ^2\left (x^2\right )} \, dx=\log (-2+x) \log \left (-x+\frac {\log (3)}{x}-\frac {2}{\log \left (x^2\right )}\right ) \] Output:

ln(-2+x)*ln(ln(3)/x-x-2/ln(x^2))
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-8 x+4 x^2\right ) \log (-2+x)+\left (2 x^2-x^3+(2-x) \log (3)\right ) \log (-2+x) \log ^2\left (x^2\right )+\left (-2 x^2 \log \left (x^2\right )+\left (-x^3+x \log (3)\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {-2 x+\left (-x^2+\log (3)\right ) \log \left (x^2\right )}{x \log \left (x^2\right )}\right )}{\left (4 x^2-2 x^3\right ) \log \left (x^2\right )+\left (2 x^3-x^4+\left (-2 x+x^2\right ) \log (3)\right ) \log ^2\left (x^2\right )} \, dx=\log (-2+x) \log \left (-x+\frac {\log (3)}{x}-\frac {2}{\log \left (x^2\right )}\right ) \] Input:

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

Output:

Log[-2 + x]*Log[-x + Log[3]/x - 2/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.

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle -2 \int \frac {\log (x-2)}{x \log \left (x^2\right )}dx-2 \int \frac {\log (x-2)}{\log \left (x^2\right ) x^2+2 x-\log (3) \log \left (x^2\right )}dx-2 \log (3) \int \frac {\log (x-2)}{x \left (\log \left (x^2\right ) x^2+2 x-\log (3) \log \left (x^2\right )\right )}dx+2 \int \frac {x \log (x-2)}{\log \left (x^2\right ) x^2+2 x-\log (3) \log \left (x^2\right )}dx+\frac {\log (9) \int \frac {\log (x-2)}{\left (\sqrt {\log (3)}-x\right ) \left (\log \left (x^2\right ) x^2+2 x-\log (3) \log \left (x^2\right )\right )}dx}{\sqrt {\log (3)}}+\frac {\log (9) \int \frac {\log (x-2)}{\left (x+\sqrt {\log (3)}\right ) \left (\log \left (x^2\right ) x^2+2 x-\log (3) \log \left (x^2\right )\right )}dx}{\sqrt {\log (3)}}+\int \frac {\log \left (-x-\frac {2}{\log \left (x^2\right )}+\frac {\log (3)}{x}\right )}{x-2}dx-\operatorname {PolyLog}\left (2,1-\frac {x}{2}\right )+\operatorname {PolyLog}\left (2,\frac {2-x}{2-\sqrt {\log (3)}}\right )+\operatorname {PolyLog}\left (2,\frac {2-x}{2+\sqrt {\log (3)}}\right )-\log (x-2) \log \left (\frac {x}{2}\right )+\log (x-2) \log \left (\frac {x-\sqrt {\log (3)}}{2-\sqrt {\log (3)}}\right )+\log (x-2) \log \left (\frac {x+\sqrt {\log (3)}}{2+\sqrt {\log (3)}}\right )\)

Input:

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

Output:

$Aborted
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 154.82 (sec) , antiderivative size = 2484, normalized size of antiderivative = 103.50

method	result	size
risch	\(\text {Expression too large to display}\)	\(2484\)

Input:

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

Output:

ln(-2+x)*ln(-4*I*x-Pi*x^2*csgn(I*x)^2*csgn(I*x^2)+Pi*ln(3)*csgn(I*x)^2*csg 
n(I*x^2)+2*Pi*x^2*csgn(I*x)*csgn(I*x^2)^2-2*Pi*ln(3)*csgn(I*x)*csgn(I*x^2) 
^2-4*I*x^2*ln(x)+4*I*ln(3)*ln(x)-Pi*x^2*csgn(I*x^2)^3+Pi*ln(3)*csgn(I*x^2) 
^3)-ln(-2+x)*ln(Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi 
*csgn(I*x)*csgn(I*x^2)^2)-ln(x)*ln(-2+x)-1/2*I*Pi*ln(-2+x)*csgn(I/x)*csgn( 
I/(Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*cs 
gn(I*x^2)^2)*(-4*I*x-Pi*x^2*csgn(I*x)^2*csgn(I*x^2)+Pi*ln(3)*csgn(I*x)^2*c 
sgn(I*x^2)+2*Pi*x^2*csgn(I*x)*csgn(I*x^2)^2-2*Pi*ln(3)*csgn(I*x)*csgn(I*x^ 
2)^2-4*I*x^2*ln(x)+4*I*ln(3)*ln(x)-Pi*x^2*csgn(I*x^2)^3+Pi*ln(3)*csgn(I*x^ 
2)^3))*csgn(I/x*(-4*I*x-Pi*x^2*csgn(I*x)^2*csgn(I*x^2)+Pi*ln(3)*csgn(I*x)^ 
2*csgn(I*x^2)+2*Pi*x^2*csgn(I*x)*csgn(I*x^2)^2-2*Pi*ln(3)*csgn(I*x)*csgn(I 
*x^2)^2-4*I*x^2*ln(x)+4*I*ln(3)*ln(x)-Pi*x^2*csgn(I*x^2)^3+Pi*ln(3)*csgn(I 
*x^2)^3)/(Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn( 
I*x)*csgn(I*x^2)^2))+1/2*I*Pi*ln(-2+x)*csgn(I/x)*csgn(I/x*(-4*I*x-Pi*x^2*c 
sgn(I*x)^2*csgn(I*x^2)+Pi*ln(3)*csgn(I*x)^2*csgn(I*x^2)+2*Pi*x^2*csgn(I*x) 
*csgn(I*x^2)^2-2*Pi*ln(3)*csgn(I*x)*csgn(I*x^2)^2-4*I*x^2*ln(x)+4*I*ln(3)* 
ln(x)-Pi*x^2*csgn(I*x^2)^3+Pi*ln(3)*csgn(I*x^2)^3)/(Pi*csgn(I*x^2)^3+4*I*l 
n(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2))^2+1/2*I*Pi* 
ln(-2+x)*csgn(I/(Pi*csgn(I*x^2)^3+4*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*P 
i*csgn(I*x)*csgn(I*x^2)^2))*csgn(-4*x+I*Pi*x^2*csgn(I*x)^2*csgn(I*x^2)-...
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.42 \[ \int \frac {\left (-8 x+4 x^2\right ) \log (-2+x)+\left (2 x^2-x^3+(2-x) \log (3)\right ) \log (-2+x) \log ^2\left (x^2\right )+\left (-2 x^2 \log \left (x^2\right )+\left (-x^3+x \log (3)\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {-2 x+\left (-x^2+\log (3)\right ) \log \left (x^2\right )}{x \log \left (x^2\right )}\right )}{\left (4 x^2-2 x^3\right ) \log \left (x^2\right )+\left (2 x^3-x^4+\left (-2 x+x^2\right ) \log (3)\right ) \log ^2\left (x^2\right )} \, dx=\log \left (x - 2\right ) \log \left (-\frac {{\left (x^{2} - \log \left (3\right )\right )} \log \left (x^{2}\right ) + 2 \, x}{x \log \left (x^{2}\right )}\right ) \] Input:

integrate((((x*log(3)-x^3)*log(x^2)^2-2*x^2*log(x^2))*log(((log(3)-x^2)*lo 
g(x^2)-2*x)/x/log(x^2))+((2-x)*log(3)-x^3+2*x^2)*log(-2+x)*log(x^2)^2+(4*x 
^2-8*x)*log(-2+x))/(((x^2-2*x)*log(3)-x^4+2*x^3)*log(x^2)^2+(-2*x^3+4*x^2) 
*log(x^2)),x, algorithm="fricas")
 

Output:

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

Sympy [A] (verification not implemented)

Time = 0.63 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12 \[ \int \frac {\left (-8 x+4 x^2\right ) \log (-2+x)+\left (2 x^2-x^3+(2-x) \log (3)\right ) \log (-2+x) \log ^2\left (x^2\right )+\left (-2 x^2 \log \left (x^2\right )+\left (-x^3+x \log (3)\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {-2 x+\left (-x^2+\log (3)\right ) \log \left (x^2\right )}{x \log \left (x^2\right )}\right )}{\left (4 x^2-2 x^3\right ) \log \left (x^2\right )+\left (2 x^3-x^4+\left (-2 x+x^2\right ) \log (3)\right ) \log ^2\left (x^2\right )} \, dx=\log {\left (\frac {- 2 x + \left (- x^{2} + \log {\left (3 \right )}\right ) \log {\left (x^{2} \right )}}{x \log {\left (x^{2} \right )}} \right )} \log {\left (x - 2 \right )} \] Input:

integrate((((x*ln(3)-x**3)*ln(x**2)**2-2*x**2*ln(x**2))*ln(((ln(3)-x**2)*l 
n(x**2)-2*x)/x/ln(x**2))+((2-x)*ln(3)-x**3+2*x**2)*ln(-2+x)*ln(x**2)**2+(4 
*x**2-8*x)*ln(-2+x))/(((x**2-2*x)*ln(3)-x**4+2*x**3)*ln(x**2)**2+(-2*x**3+ 
4*x**2)*ln(x**2)),x)
 

Output:

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

Maxima [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.46 \[ \int \frac {\left (-8 x+4 x^2\right ) \log (-2+x)+\left (2 x^2-x^3+(2-x) \log (3)\right ) \log (-2+x) \log ^2\left (x^2\right )+\left (-2 x^2 \log \left (x^2\right )+\left (-x^3+x \log (3)\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {-2 x+\left (-x^2+\log (3)\right ) \log \left (x^2\right )}{x \log \left (x^2\right )}\right )}{\left (4 x^2-2 x^3\right ) \log \left (x^2\right )+\left (2 x^3-x^4+\left (-2 x+x^2\right ) \log (3)\right ) \log ^2\left (x^2\right )} \, dx=-{\left (\log \left (x\right ) + \log \left (\log \left (x\right )\right )\right )} \log \left (x - 2\right ) + \log \left (-{\left (x^{2} - \log \left (3\right )\right )} \log \left (x\right ) - x\right ) \log \left (x - 2\right ) \] Input:

integrate((((x*log(3)-x^3)*log(x^2)^2-2*x^2*log(x^2))*log(((log(3)-x^2)*lo 
g(x^2)-2*x)/x/log(x^2))+((2-x)*log(3)-x^3+2*x^2)*log(-2+x)*log(x^2)^2+(4*x 
^2-8*x)*log(-2+x))/(((x^2-2*x)*log(3)-x^4+2*x^3)*log(x^2)^2+(-2*x^3+4*x^2) 
*log(x^2)),x, algorithm="maxima")
 

Output:

-(log(x) + log(log(x)))*log(x - 2) + log(-(x^2 - log(3))*log(x) - x)*log(x 
 - 2)
 

Giac [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.92 \[ \int \frac {\left (-8 x+4 x^2\right ) \log (-2+x)+\left (2 x^2-x^3+(2-x) \log (3)\right ) \log (-2+x) \log ^2\left (x^2\right )+\left (-2 x^2 \log \left (x^2\right )+\left (-x^3+x \log (3)\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {-2 x+\left (-x^2+\log (3)\right ) \log \left (x^2\right )}{x \log \left (x^2\right )}\right )}{\left (4 x^2-2 x^3\right ) \log \left (x^2\right )+\left (2 x^3-x^4+\left (-2 x+x^2\right ) \log (3)\right ) \log ^2\left (x^2\right )} \, dx=\log \left (-x^{2} \log \left (x^{2}\right ) + \log \left (3\right ) \log \left (x^{2}\right ) - 2 \, x\right ) \log \left (x - 2\right ) - \log \left (x - 2\right ) \log \left (x\right ) - \log \left (x - 2\right ) \log \left (\log \left (x^{2}\right )\right ) \] Input:

integrate((((x*log(3)-x^3)*log(x^2)^2-2*x^2*log(x^2))*log(((log(3)-x^2)*lo 
g(x^2)-2*x)/x/log(x^2))+((2-x)*log(3)-x^3+2*x^2)*log(-2+x)*log(x^2)^2+(4*x 
^2-8*x)*log(-2+x))/(((x^2-2*x)*log(3)-x^4+2*x^3)*log(x^2)^2+(-2*x^3+4*x^2) 
*log(x^2)),x, algorithm="giac")
 

Output:

log(-x^2*log(x^2) + log(3)*log(x^2) - 2*x)*log(x - 2) - log(x - 2)*log(x) 
- log(x - 2)*log(log(x^2))
 

Mupad [B] (verification not implemented)

Time = 4.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.46 \[ \int \frac {\left (-8 x+4 x^2\right ) \log (-2+x)+\left (2 x^2-x^3+(2-x) \log (3)\right ) \log (-2+x) \log ^2\left (x^2\right )+\left (-2 x^2 \log \left (x^2\right )+\left (-x^3+x \log (3)\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {-2 x+\left (-x^2+\log (3)\right ) \log \left (x^2\right )}{x \log \left (x^2\right )}\right )}{\left (4 x^2-2 x^3\right ) \log \left (x^2\right )+\left (2 x^3-x^4+\left (-2 x+x^2\right ) \log (3)\right ) \log ^2\left (x^2\right )} \, dx=\ln \left (-\frac {2\,x-\ln \left (x^2\right )\,\left (\ln \left (3\right )-x^2\right )}{x\,\ln \left (x^2\right )}\right )\,\ln \left (x-2\right ) \] Input:

int(-(log(x - 2)*(8*x - 4*x^2) + log(-(2*x - log(x^2)*(log(3) - x^2))/(x*l 
og(x^2)))*(2*x^2*log(x^2) - log(x^2)^2*(x*log(3) - x^3)) + log(x - 2)*log( 
x^2)^2*(log(3)*(x - 2) - 2*x^2 + x^3))/(log(x^2)*(4*x^2 - 2*x^3) - log(x^2 
)^2*(log(3)*(2*x - x^2) - 2*x^3 + x^4)),x)
 

Output:

log(-(2*x - log(x^2)*(log(3) - x^2))/(x*log(x^2)))*log(x - 2)
 

Reduce [F]

\[ \int \frac {\left (-8 x+4 x^2\right ) \log (-2+x)+\left (2 x^2-x^3+(2-x) \log (3)\right ) \log (-2+x) \log ^2\left (x^2\right )+\left (-2 x^2 \log \left (x^2\right )+\left (-x^3+x \log (3)\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {-2 x+\left (-x^2+\log (3)\right ) \log \left (x^2\right )}{x \log \left (x^2\right )}\right )}{\left (4 x^2-2 x^3\right ) \log \left (x^2\right )+\left (2 x^3-x^4+\left (-2 x+x^2\right ) \log (3)\right ) \log ^2\left (x^2\right )} \, dx=\int \frac {\left (\left (\mathrm {log}\left (3\right ) x -x^{3}\right ) \mathrm {log}\left (x^{2}\right )^{2}-2 \,\mathrm {log}\left (x^{2}\right ) x^{2}\right ) \mathrm {log}\left (\frac {\left (\mathrm {log}\left (3\right )-x^{2}\right ) \mathrm {log}\left (x^{2}\right )-2 x}{x \,\mathrm {log}\left (x^{2}\right )}\right )+\left (\left (-x +2\right ) \mathrm {log}\left (3\right )-x^{3}+2 x^{2}\right ) \mathrm {log}\left (x -2\right ) \mathrm {log}\left (x^{2}\right )^{2}+\left (4 x^{2}-8 x \right ) \mathrm {log}\left (x -2\right )}{\left (\left (x^{2}-2 x \right ) \mathrm {log}\left (3\right )-x^{4}+2 x^{3}\right ) \mathrm {log}\left (x^{2}\right )^{2}+\left (-2 x^{3}+4 x^{2}\right ) \mathrm {log}\left (x^{2}\right )}d x \] Input:

int((((x*log(3)-x^3)*log(x^2)^2-2*x^2*log(x^2))*log(((log(3)-x^2)*log(x^2) 
-2*x)/x/log(x^2))+((2-x)*log(3)-x^3+2*x^2)*log(-2+x)*log(x^2)^2+(4*x^2-8*x 
)*log(-2+x))/(((x^2-2*x)*log(3)-x^4+2*x^3)*log(x^2)^2+(-2*x^3+4*x^2)*log(x 
^2)),x)
 

Output:

int((((x*log(3)-x^3)*log(x^2)^2-2*x^2*log(x^2))*log(((log(3)-x^2)*log(x^2) 
-2*x)/x/log(x^2))+((2-x)*log(3)-x^3+2*x^2)*log(-2+x)*log(x^2)^2+(4*x^2-8*x 
)*log(-2+x))/(((x^2-2*x)*log(3)-x^4+2*x^3)*log(x^2)^2+(-2*x^3+4*x^2)*log(x 
^2)),x)