\(\int \frac {(-6 x+3 \log (4)) \log (\frac {x}{2})+(-2 x^2+x \log (4)) \log ^2(\frac {x}{2})+(3 x-3 \log (4)+(-x^2+x \log (4)) \log ^2(\frac {x}{2})) \log (-x^2+x \log (4))}{(-x^2+x \log (4)) \log ^2(\frac {x}{2})} \, dx\) [2168]

Optimal result

Integrand size = 95, antiderivative size = 22 \[ \int \frac {(-6 x+3 \log (4)) \log \left (\frac {x}{2}\right )+\left (-2 x^2+x \log (4)\right ) \log ^2\left (\frac {x}{2}\right )+\left (3 x-3 \log (4)+\left (-x^2+x \log (4)\right ) \log ^2\left (\frac {x}{2}\right )\right ) \log \left (-x^2+x \log (4)\right )}{\left (-x^2+x \log (4)\right ) \log ^2\left (\frac {x}{2}\right )} \, dx=\left (x+\frac {3}{\log \left (\frac {x}{2}\right )}\right ) \log (x (-x+\log (4))) \] Output:

ln(x*(2*ln(2)-x))*(3/ln(1/2*x)+x)
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.27 \[ \int \frac {(-6 x+3 \log (4)) \log \left (\frac {x}{2}\right )+\left (-2 x^2+x \log (4)\right ) \log ^2\left (\frac {x}{2}\right )+\left (3 x-3 \log (4)+\left (-x^2+x \log (4)\right ) \log ^2\left (\frac {x}{2}\right )\right ) \log \left (-x^2+x \log (4)\right )}{\left (-x^2+x \log (4)\right ) \log ^2\left (\frac {x}{2}\right )} \, dx=\frac {\left (3+x \log \left (\frac {x}{2}\right )\right ) \log (x (-x+\log (4)))}{\log \left (\frac {x}{2}\right )} \] Input:

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

Output:

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

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

\(\Big \downarrow \) 2026

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

$Aborted
 

Maple [A] (verified)

Time = 55.58 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.77

method	result	size
parallelrisch	\(\frac {x \ln \left (\frac {x}{2}\right ) \ln \left (x \left (2 \ln \left (2\right )-x \right )\right )+3 \ln \left (x \left (2 \ln \left (2\right )-x \right )\right )}{\ln \left (\frac {x}{2}\right )}\)	\(39\)
risch	\(\frac {\left (-6+2 x \ln \left (2\right )-2 x \ln \left (x \right )\right ) \ln \left (\ln \left (2\right )-\frac {x}{2}\right )}{2 \ln \left (2\right )-2 \ln \left (x \right )}+\frac {-2 i \ln \left (2\right ) \pi x \,\operatorname {csgn}\left (i \left (\ln \left (2\right )-\frac {x}{2}\right )\right ) \operatorname {csgn}\left (i x \left (\ln \left (2\right )-\frac {x}{2}\right )\right )^{2}-2 i \ln \left (2\right ) \pi x \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x \left (\ln \left (2\right )-\frac {x}{2}\right )\right )^{2}+2 i \pi x \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x \left (\ln \left (2\right )-\frac {x}{2}\right )\right )^{2} \ln \left (x \right )+4 x \ln \left (x \right )^{2}+12 \ln \left (2\right )-4 x \ln \left (2\right ) \ln \left (x \right )+2 i \ln \left (2\right ) \pi x \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i \left (\ln \left (2\right )-\frac {x}{2}\right )\right ) \operatorname {csgn}\left (i x \left (\ln \left (2\right )-\frac {x}{2}\right )\right )-6 i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i \left (\ln \left (2\right )-\frac {x}{2}\right )\right ) \operatorname {csgn}\left (i x \left (\ln \left (2\right )-\frac {x}{2}\right )\right )+2 i \pi x \,\operatorname {csgn}\left (i \left (\ln \left (2\right )-\frac {x}{2}\right )\right ) \operatorname {csgn}\left (i x \left (\ln \left (2\right )-\frac {x}{2}\right )\right )^{2} \ln \left (x \right )+6 i \pi \,\operatorname {csgn}\left (i \left (\ln \left (2\right )-\frac {x}{2}\right )\right ) \operatorname {csgn}\left (i x \left (\ln \left (2\right )-\frac {x}{2}\right )\right )^{2}-2 i \pi x \operatorname {csgn}\left (i x \left (\ln \left (2\right )-\frac {x}{2}\right )\right )^{3} \ln \left (x \right )-6 i \pi \operatorname {csgn}\left (i x \left (\ln \left (2\right )-\frac {x}{2}\right )\right )^{3}+2 i \ln \left (2\right ) \pi x \operatorname {csgn}\left (i x \left (\ln \left (2\right )-\frac {x}{2}\right )\right )^{3}-2 i \pi x \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i \left (\ln \left (2\right )-\frac {x}{2}\right )\right ) \operatorname {csgn}\left (i x \left (\ln \left (2\right )-\frac {x}{2}\right )\right ) \ln \left (x \right )+6 i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x \left (\ln \left (2\right )-\frac {x}{2}\right )\right )^{2}}{-4 \ln \left (2\right )+4 \ln \left (x \right )}\)	\(377\)

Input:

int((((2*x*ln(2)-x^2)*ln(1/2*x)^2-6*ln(2)+3*x)*ln(2*x*ln(2)-x^2)+(2*x*ln(2 
)-2*x^2)*ln(1/2*x)^2+(6*ln(2)-6*x)*ln(1/2*x))/(2*x*ln(2)-x^2)/ln(1/2*x)^2, 
x,method=_RETURNVERBOSE)
 

Output:

(x*ln(1/2*x)*ln(x*(2*ln(2)-x))+3*ln(x*(2*ln(2)-x)))/ln(1/2*x)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.23 \[ \int \frac {(-6 x+3 \log (4)) \log \left (\frac {x}{2}\right )+\left (-2 x^2+x \log (4)\right ) \log ^2\left (\frac {x}{2}\right )+\left (3 x-3 \log (4)+\left (-x^2+x \log (4)\right ) \log ^2\left (\frac {x}{2}\right )\right ) \log \left (-x^2+x \log (4)\right )}{\left (-x^2+x \log (4)\right ) \log ^2\left (\frac {x}{2}\right )} \, dx=\frac {{\left (x \log \left (\frac {1}{2} \, x\right ) + 3\right )} \log \left (-x^{2} + 2 \, x \log \left (2\right )\right )}{\log \left (\frac {1}{2} \, x\right )} \] Input:

integrate((((2*x*log(2)-x^2)*log(1/2*x)^2-6*log(2)+3*x)*log(2*x*log(2)-x^2 
)+(2*x*log(2)-2*x^2)*log(1/2*x)^2+(6*log(2)-6*x)*log(1/2*x))/(2*x*log(2)-x 
^2)/log(1/2*x)^2,x, algorithm="fricas")
 

Output:

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

Sympy [F(-2)]

Exception generated. \[ \int \frac {(-6 x+3 \log (4)) \log \left (\frac {x}{2}\right )+\left (-2 x^2+x \log (4)\right ) \log ^2\left (\frac {x}{2}\right )+\left (3 x-3 \log (4)+\left (-x^2+x \log (4)\right ) \log ^2\left (\frac {x}{2}\right )\right ) \log \left (-x^2+x \log (4)\right )}{\left (-x^2+x \log (4)\right ) \log ^2\left (\frac {x}{2}\right )} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((((2*x*ln(2)-x**2)*ln(1/2*x)**2-6*ln(2)+3*x)*ln(2*x*ln(2)-x**2)+ 
(2*x*ln(2)-2*x**2)*ln(1/2*x)**2+(6*ln(2)-6*x)*ln(1/2*x))/(2*x*ln(2)-x**2)/ 
ln(1/2*x)**2,x)
 

Output:

Exception raised: TypeError >> '>' not supported between instances of 'Pol 
y' and 'int'
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (21) = 42\).

Time = 0.16 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.23 \[ \int \frac {(-6 x+3 \log (4)) \log \left (\frac {x}{2}\right )+\left (-2 x^2+x \log (4)\right ) \log ^2\left (\frac {x}{2}\right )+\left (3 x-3 \log (4)+\left (-x^2+x \log (4)\right ) \log ^2\left (\frac {x}{2}\right )\right ) \log \left (-x^2+x \log (4)\right )}{\left (-x^2+x \log (4)\right ) \log ^2\left (\frac {x}{2}\right )} \, dx=\frac {x \log \left (2\right ) \log \left (x\right ) - x \log \left (x\right )^{2} + {\left (x \log \left (2\right ) - x \log \left (x\right ) - 3\right )} \log \left (-x + 2 \, \log \left (2\right )\right ) - 3 \, \log \left (2\right )}{\log \left (2\right ) - \log \left (x\right )} \] Input:

integrate((((2*x*log(2)-x^2)*log(1/2*x)^2-6*log(2)+3*x)*log(2*x*log(2)-x^2 
)+(2*x*log(2)-2*x^2)*log(1/2*x)^2+(6*log(2)-6*x)*log(1/2*x))/(2*x*log(2)-x 
^2)/log(1/2*x)^2,x, algorithm="maxima")
 

Output:

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

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.86 \[ \int \frac {(-6 x+3 \log (4)) \log \left (\frac {x}{2}\right )+\left (-2 x^2+x \log (4)\right ) \log ^2\left (\frac {x}{2}\right )+\left (3 x-3 \log (4)+\left (-x^2+x \log (4)\right ) \log ^2\left (\frac {x}{2}\right )\right ) \log \left (-x^2+x \log (4)\right )}{\left (-x^2+x \log (4)\right ) \log ^2\left (\frac {x}{2}\right )} \, dx=x \log \left (x\right ) + {\left (x - \frac {3}{\log \left (2\right ) - \log \left (x\right )}\right )} \log \left (-x + 2 \, \log \left (2\right )\right ) - \frac {3 \, \log \left (2\right )}{\log \left (2\right ) - \log \left (x\right )} \] Input:

integrate((((2*x*log(2)-x^2)*log(1/2*x)^2-6*log(2)+3*x)*log(2*x*log(2)-x^2 
)+(2*x*log(2)-2*x^2)*log(1/2*x)^2+(6*log(2)-6*x)*log(1/2*x))/(2*x*log(2)-x 
^2)/log(1/2*x)^2,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int((log(2*x*log(2) - x^2)*(3*x - 6*log(2) + log(x/2)^2*(2*x*log(2) - x^2) 
) - log(x/2)*(6*x - 6*log(2)) + log(x/2)^2*(2*x*log(2) - 2*x^2))/(log(x/2) 
^2*(2*x*log(2) - x^2)),x)
 

Output:

int((log(2*x*log(2) - x^2)*(3*x - 6*log(2) + log(x/2)^2*(2*x*log(2) - x^2) 
) - log(x/2)*(6*x - 6*log(2)) + log(x/2)^2*(2*x*log(2) - 2*x^2))/(log(x/2) 
^2*(2*x*log(2) - x^2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.95 \[ \int \frac {(-6 x+3 \log (4)) \log \left (\frac {x}{2}\right )+\left (-2 x^2+x \log (4)\right ) \log ^2\left (\frac {x}{2}\right )+\left (3 x-3 \log (4)+\left (-x^2+x \log (4)\right ) \log ^2\left (\frac {x}{2}\right )\right ) \log \left (-x^2+x \log (4)\right )}{\left (-x^2+x \log (4)\right ) \log ^2\left (\frac {x}{2}\right )} \, dx=\frac {-2 \,\mathrm {log}\left (2 \,\mathrm {log}\left (2\right ) x -x^{2}\right ) \mathrm {log}\left (\frac {x}{2}\right ) \mathrm {log}\left (2\right )+\mathrm {log}\left (2 \,\mathrm {log}\left (2\right ) x -x^{2}\right ) \mathrm {log}\left (\frac {x}{2}\right ) x +3 \,\mathrm {log}\left (2 \,\mathrm {log}\left (2\right ) x -x^{2}\right )+2 \,\mathrm {log}\left (2 \,\mathrm {log}\left (2\right )-x \right ) \mathrm {log}\left (\frac {x}{2}\right ) \mathrm {log}\left (2\right )+2 \,\mathrm {log}\left (\frac {x}{2}\right ) \mathrm {log}\left (x \right ) \mathrm {log}\left (2\right )}{\mathrm {log}\left (\frac {x}{2}\right )} \] Input:

int((((2*x*log(2)-x^2)*log(1/2*x)^2-6*log(2)+3*x)*log(2*x*log(2)-x^2)+(2*x 
*log(2)-2*x^2)*log(1/2*x)^2+(6*log(2)-6*x)*log(1/2*x))/(2*x*log(2)-x^2)/lo 
g(1/2*x)^2,x)
 

Output:

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