\(\int \frac {24-6 x+(12+24 x-6 x^2) \log (\frac {2}{x^2})+(12-3 x) \log (\frac {2}{x^2}) \log (-\frac {x}{(-4+x) \log (\frac {2}{x^2})})}{(-4 x^4+x^5) \log (\frac {2}{x^2})+(-8 x^3+2 x^4) \log (\frac {2}{x^2}) \log (-\frac {x}{(-4+x) \log (\frac {2}{x^2})})+(-4 x^2+x^3) \log (\frac {2}{x^2}) \log ^2(-\frac {x}{(-4+x) \log (\frac {2}{x^2})})} \, dx\) [2624]

Optimal result

Integrand size = 141, antiderivative size = 27 \[ \int \frac {24-6 x+\left (12+24 x-6 x^2\right ) \log \left (\frac {2}{x^2}\right )+(12-3 x) \log \left (\frac {2}{x^2}\right ) \log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )}{\left (-4 x^4+x^5\right ) \log \left (\frac {2}{x^2}\right )+\left (-8 x^3+2 x^4\right ) \log \left (\frac {2}{x^2}\right ) \log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )+\left (-4 x^2+x^3\right ) \log \left (\frac {2}{x^2}\right ) \log ^2\left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )} \, dx=\frac {3}{x \left (x+\log \left (\frac {x}{(4-x) \log \left (\frac {2}{x^2}\right )}\right )\right )} \] Output:

3/x/(x+ln(x/(4-x)/ln(2/x^2)))
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {24-6 x+\left (12+24 x-6 x^2\right ) \log \left (\frac {2}{x^2}\right )+(12-3 x) \log \left (\frac {2}{x^2}\right ) \log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )}{\left (-4 x^4+x^5\right ) \log \left (\frac {2}{x^2}\right )+\left (-8 x^3+2 x^4\right ) \log \left (\frac {2}{x^2}\right ) \log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )+\left (-4 x^2+x^3\right ) \log \left (\frac {2}{x^2}\right ) \log ^2\left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )} \, dx=\frac {3}{x \left (x+\log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )\right )} \] Input:

Integrate[(24 - 6*x + (12 + 24*x - 6*x^2)*Log[2/x^2] + (12 - 3*x)*Log[2/x^ 
2]*Log[-(x/((-4 + x)*Log[2/x^2]))])/((-4*x^4 + x^5)*Log[2/x^2] + (-8*x^3 + 
 2*x^4)*Log[2/x^2]*Log[-(x/((-4 + x)*Log[2/x^2]))] + (-4*x^2 + x^3)*Log[2/ 
x^2]*Log[-(x/((-4 + x)*Log[2/x^2]))]^2),x]
 

Output:

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

\(\displaystyle \int \frac {\left (-6 x^2+24 x+12\right ) \log \left (\frac {2}{x^2}\right )+(12-3 x) \log \left (\frac {2}{x^2}\right ) \log \left (-\frac {x}{(x-4) \log \left (\frac {2}{x^2}\right )}\right )-6 x+24}{\left (x^3-4 x^2\right ) \log \left (\frac {2}{x^2}\right ) \log ^2\left (-\frac {x}{(x-4) \log \left (\frac {2}{x^2}\right )}\right )+\left (x^5-4 x^4\right ) \log \left (\frac {2}{x^2}\right )+\left (2 x^4-8 x^3\right ) \log \left (\frac {2}{x^2}\right ) \log \left (-\frac {x}{(x-4) \log \left (\frac {2}{x^2}\right )}\right )} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {-\left (-6 x^2+24 x+12\right ) \log \left (\frac {2}{x^2}\right )-(12-3 x) \log \left (\frac {2}{x^2}\right ) \log \left (-\frac {x}{(x-4) \log \left (\frac {2}{x^2}\right )}\right )+6 x-24}{(4-x) x^2 \log \left (\frac {2}{x^2}\right ) \left (\log \left (-\frac {x}{(x-4) \log \left (\frac {2}{x^2}\right )}\right )+x\right )^2}dx\)

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3}{4} \int \frac {1}{(x-4) \left (x+\log \left (-\frac {x}{(x-4) \log \left (\frac {2}{x^2}\right )}\right )\right )^2}dx-3 \int \frac {1}{x^2 \left (x+\log \left (-\frac {x}{(x-4) \log \left (\frac {2}{x^2}\right )}\right )\right )^2}dx-\frac {15}{4} \int \frac {1}{x \left (x+\log \left (-\frac {x}{(x-4) \log \left (\frac {2}{x^2}\right )}\right )\right )^2}dx-6 \int \frac {1}{x^2 \log \left (\frac {2}{x^2}\right ) \left (x+\log \left (-\frac {x}{(x-4) \log \left (\frac {2}{x^2}\right )}\right )\right )^2}dx-3 \int \frac {1}{x^2 \left (x+\log \left (-\frac {x}{(x-4) \log \left (\frac {2}{x^2}\right )}\right )\right )}dx\)

Input:

Int[(24 - 6*x + (12 + 24*x - 6*x^2)*Log[2/x^2] + (12 - 3*x)*Log[2/x^2]*Log 
[-(x/((-4 + x)*Log[2/x^2]))])/((-4*x^4 + x^5)*Log[2/x^2] + (-8*x^3 + 2*x^4 
)*Log[2/x^2]*Log[-(x/((-4 + x)*Log[2/x^2]))] + (-4*x^2 + x^3)*Log[2/x^2]*L 
og[-(x/((-4 + x)*Log[2/x^2]))]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 1.33 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00

method	result	size
parallelrisch	\(\frac {3}{x \left (\ln \left (-\frac {x}{\left (x -4\right ) \ln \left (\frac {2}{x^{2}}\right )}\right )+x \right )}\)	\(27\)

Input:

int(((-3*x+12)*ln(2/x^2)*ln(-x/(x-4)/ln(2/x^2))+(-6*x^2+24*x+12)*ln(2/x^2) 
-6*x+24)/((x^3-4*x^2)*ln(2/x^2)*ln(-x/(x-4)/ln(2/x^2))^2+(2*x^4-8*x^3)*ln( 
2/x^2)*ln(-x/(x-4)/ln(2/x^2))+(x^5-4*x^4)*ln(2/x^2)),x,method=_RETURNVERBO 
SE)
 

Output:

3/x/(ln(-x/(x-4)/ln(2/x^2))+x)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {24-6 x+\left (12+24 x-6 x^2\right ) \log \left (\frac {2}{x^2}\right )+(12-3 x) \log \left (\frac {2}{x^2}\right ) \log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )}{\left (-4 x^4+x^5\right ) \log \left (\frac {2}{x^2}\right )+\left (-8 x^3+2 x^4\right ) \log \left (\frac {2}{x^2}\right ) \log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )+\left (-4 x^2+x^3\right ) \log \left (\frac {2}{x^2}\right ) \log ^2\left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )} \, dx=\frac {3}{x^{2} + x \log \left (-\frac {x}{{\left (x - 4\right )} \log \left (\frac {2}{x^{2}}\right )}\right )} \] Input:

integrate(((-3*x+12)*log(2/x^2)*log(-x/(-4+x)/log(2/x^2))+(-6*x^2+24*x+12) 
*log(2/x^2)-6*x+24)/((x^3-4*x^2)*log(2/x^2)*log(-x/(-4+x)/log(2/x^2))^2+(2 
*x^4-8*x^3)*log(2/x^2)*log(-x/(-4+x)/log(2/x^2))+(x^5-4*x^4)*log(2/x^2)),x 
, algorithm="fricas")
 

Output:

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

Sympy [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \frac {24-6 x+\left (12+24 x-6 x^2\right ) \log \left (\frac {2}{x^2}\right )+(12-3 x) \log \left (\frac {2}{x^2}\right ) \log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )}{\left (-4 x^4+x^5\right ) \log \left (\frac {2}{x^2}\right )+\left (-8 x^3+2 x^4\right ) \log \left (\frac {2}{x^2}\right ) \log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )+\left (-4 x^2+x^3\right ) \log \left (\frac {2}{x^2}\right ) \log ^2\left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )} \, dx=\frac {3}{x^{2} + x \log {\left (- \frac {x}{\left (x - 4\right ) \log {\left (\frac {2}{x^{2}} \right )}} \right )}} \] Input:

integrate(((-3*x+12)*ln(2/x**2)*ln(-x/(-4+x)/ln(2/x**2))+(-6*x**2+24*x+12) 
*ln(2/x**2)-6*x+24)/((x**3-4*x**2)*ln(2/x**2)*ln(-x/(-4+x)/ln(2/x**2))**2+ 
(2*x**4-8*x**3)*ln(2/x**2)*ln(-x/(-4+x)/ln(2/x**2))+(x**5-4*x**4)*ln(2/x** 
2)),x)
 

Output:

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

Maxima [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19 \[ \int \frac {24-6 x+\left (12+24 x-6 x^2\right ) \log \left (\frac {2}{x^2}\right )+(12-3 x) \log \left (\frac {2}{x^2}\right ) \log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )}{\left (-4 x^4+x^5\right ) \log \left (\frac {2}{x^2}\right )+\left (-8 x^3+2 x^4\right ) \log \left (\frac {2}{x^2}\right ) \log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )+\left (-4 x^2+x^3\right ) \log \left (\frac {2}{x^2}\right ) \log ^2\left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )} \, dx=\frac {3}{x^{2} - x \log \left (x - 4\right ) + x \log \left (x\right ) - x \log \left (-\log \left (2\right ) + 2 \, \log \left (x\right )\right )} \] Input:

integrate(((-3*x+12)*log(2/x^2)*log(-x/(-4+x)/log(2/x^2))+(-6*x^2+24*x+12) 
*log(2/x^2)-6*x+24)/((x^3-4*x^2)*log(2/x^2)*log(-x/(-4+x)/log(2/x^2))^2+(2 
*x^4-8*x^3)*log(2/x^2)*log(-x/(-4+x)/log(2/x^2))+(x^5-4*x^4)*log(2/x^2)),x 
, algorithm="maxima")
 

Output:

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

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.39 (sec) , antiderivative size = 699, normalized size of antiderivative = 25.89 \[ \int \frac {24-6 x+\left (12+24 x-6 x^2\right ) \log \left (\frac {2}{x^2}\right )+(12-3 x) \log \left (\frac {2}{x^2}\right ) \log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )}{\left (-4 x^4+x^5\right ) \log \left (\frac {2}{x^2}\right )+\left (-8 x^3+2 x^4\right ) \log \left (\frac {2}{x^2}\right ) \log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )+\left (-4 x^2+x^3\right ) \log \left (\frac {2}{x^2}\right ) \log ^2\left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )} \, dx=\text {Too large to display} \] Input:

integrate(((-3*x+12)*log(2/x^2)*log(-x/(-4+x)/log(2/x^2))+(-6*x^2+24*x+12) 
*log(2/x^2)-6*x+24)/((x^3-4*x^2)*log(2/x^2)*log(-x/(-4+x)/log(2/x^2))^2+(2 
*x^4-8*x^3)*log(2/x^2)*log(-x/(-4+x)/log(2/x^2))+(x^5-4*x^4)*log(2/x^2)),x 
, algorithm="giac")
 

Output:

-3*(x^2*log(2)*log(2/x^2) - 2*x^2*log(x)*log(2/x^2) - 4*x*log(2)*log(2/x^2 
) + 8*x*log(x)*log(2/x^2) + 2*x*log(2/x^2) - 4*log(2)*log(2/x^2) + 8*log(x 
)*log(2/x^2) - 8*log(2/x^2))/(-I*pi*x^3*log(2)*log(2/x^2) - x^4*log(2)*log 
(2/x^2) + x^3*log(2)*log(x*log(2/x^2) - 4*log(2/x^2))*log(2/x^2) + 2*I*pi* 
x^3*log(x)*log(2/x^2) + 2*x^4*log(x)*log(2/x^2) - x^3*log(2)*log(x)*log(2/ 
x^2) - 2*x^3*log(x*log(2/x^2) - 4*log(2/x^2))*log(x)*log(2/x^2) + 2*x^3*lo 
g(x)^2*log(2/x^2) + 4*I*pi*x^2*log(2)*log(2/x^2) + 4*x^3*log(2)*log(2/x^2) 
 - 4*x^2*log(2)*log(x*log(2/x^2) - 4*log(2/x^2))*log(2/x^2) - 8*I*pi*x^2*l 
og(x)*log(2/x^2) - 8*x^3*log(x)*log(2/x^2) + 4*x^2*log(2)*log(x)*log(2/x^2 
) + 8*x^2*log(x*log(2/x^2) - 4*log(2/x^2))*log(x)*log(2/x^2) - 8*x^2*log(x 
)^2*log(2/x^2) - 2*I*pi*x^2*log(2) - 2*x^3*log(2) + 2*x^2*log(2)*log(x*log 
(2/x^2) - 4*log(2/x^2)) + 4*I*pi*x^2*log(x) + 4*x^3*log(x) - 2*x^2*log(2)* 
log(x) - 4*x^2*log(x*log(2/x^2) - 4*log(2/x^2))*log(x) + 4*x^2*log(x)^2 + 
4*I*pi*x*log(2)*log(2/x^2) + 4*x^2*log(2)*log(2/x^2) - 4*x*log(2)*log(x*lo 
g(2/x^2) - 4*log(2/x^2))*log(2/x^2) - 8*I*pi*x*log(x)*log(2/x^2) - 8*x^2*l 
og(x)*log(2/x^2) + 4*x*log(2)*log(x)*log(2/x^2) + 8*x*log(x*log(2/x^2) - 4 
*log(2/x^2))*log(x)*log(2/x^2) - 8*x*log(x)^2*log(2/x^2) + 8*I*pi*x*log(2) 
 + 8*x^2*log(2) - 8*x*log(2)*log(x*log(2/x^2) - 4*log(2/x^2)) - 16*I*pi*x* 
log(x) - 16*x^2*log(x) + 8*x*log(2)*log(x) + 16*x*log(x*log(2/x^2) - 4*log 
(2/x^2))*log(x) - 16*x*log(x)^2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {24-6 x+\left (12+24 x-6 x^2\right ) \log \left (\frac {2}{x^2}\right )+(12-3 x) \log \left (\frac {2}{x^2}\right ) \log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )}{\left (-4 x^4+x^5\right ) \log \left (\frac {2}{x^2}\right )+\left (-8 x^3+2 x^4\right ) \log \left (\frac {2}{x^2}\right ) \log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )+\left (-4 x^2+x^3\right ) \log \left (\frac {2}{x^2}\right ) \log ^2\left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )} \, dx=\int \frac {6\,x-\ln \left (\frac {2}{x^2}\right )\,\left (-6\,x^2+24\,x+12\right )+\ln \left (-\frac {x}{\ln \left (\frac {2}{x^2}\right )\,\left (x-4\right )}\right )\,\ln \left (\frac {2}{x^2}\right )\,\left (3\,x-12\right )-24}{\ln \left (\frac {2}{x^2}\right )\,\left (4\,x^2-x^3\right )\,{\ln \left (-\frac {x}{\ln \left (\frac {2}{x^2}\right )\,\left (x-4\right )}\right )}^2+\ln \left (\frac {2}{x^2}\right )\,\left (8\,x^3-2\,x^4\right )\,\ln \left (-\frac {x}{\ln \left (\frac {2}{x^2}\right )\,\left (x-4\right )}\right )+\ln \left (\frac {2}{x^2}\right )\,\left (4\,x^4-x^5\right )} \,d x \] Input:

int((6*x - log(2/x^2)*(24*x - 6*x^2 + 12) + log(-x/(log(2/x^2)*(x - 4)))*l 
og(2/x^2)*(3*x - 12) - 24)/(log(2/x^2)*(4*x^4 - x^5) + log(-x/(log(2/x^2)* 
(x - 4)))*log(2/x^2)*(8*x^3 - 2*x^4) + log(-x/(log(2/x^2)*(x - 4)))^2*log( 
2/x^2)*(4*x^2 - x^3)),x)
 

Output:

int((6*x - log(2/x^2)*(24*x - 6*x^2 + 12) + log(-x/(log(2/x^2)*(x - 4)))*l 
og(2/x^2)*(3*x - 12) - 24)/(log(2/x^2)*(4*x^4 - x^5) + log(-x/(log(2/x^2)* 
(x - 4)))*log(2/x^2)*(8*x^3 - 2*x^4) + log(-x/(log(2/x^2)*(x - 4)))^2*log( 
2/x^2)*(4*x^2 - x^3)), x)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19 \[ \int \frac {24-6 x+\left (12+24 x-6 x^2\right ) \log \left (\frac {2}{x^2}\right )+(12-3 x) \log \left (\frac {2}{x^2}\right ) \log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )}{\left (-4 x^4+x^5\right ) \log \left (\frac {2}{x^2}\right )+\left (-8 x^3+2 x^4\right ) \log \left (\frac {2}{x^2}\right ) \log \left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )+\left (-4 x^2+x^3\right ) \log \left (\frac {2}{x^2}\right ) \log ^2\left (-\frac {x}{(-4+x) \log \left (\frac {2}{x^2}\right )}\right )} \, dx=\frac {3}{x \left (\mathrm {log}\left (-\frac {x}{\mathrm {log}\left (\frac {2}{x^{2}}\right ) x -4 \,\mathrm {log}\left (\frac {2}{x^{2}}\right )}\right )+x \right )} \] Input:

int(((-3*x+12)*log(2/x^2)*log(-x/(-4+x)/log(2/x^2))+(-6*x^2+24*x+12)*log(2 
/x^2)-6*x+24)/((x^3-4*x^2)*log(2/x^2)*log(-x/(-4+x)/log(2/x^2))^2+(2*x^4-8 
*x^3)*log(2/x^2)*log(-x/(-4+x)/log(2/x^2))+(x^5-4*x^4)*log(2/x^2)),x)
 

Output:

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