\(\int \frac {(-4-2 x) \log ^2(8 x+4 x^2)+(4+4 x) \log (x) \log (8 x+4 x^2) \log (\log (x))+((2 x+x^2) \log (x)+(2 x+x^2) \log ^2(x)) \log ^3(\log (x))}{(2 x+x^2) \log (x) \log ^3(\log (x))} \, dx\) [2821]

Optimal result

Integrand size = 88, antiderivative size = 26 \[ \int \frac {(-4-2 x) \log ^2\left (8 x+4 x^2\right )+(4+4 x) \log (x) \log \left (8 x+4 x^2\right ) \log (\log (x))+\left (\left (2 x+x^2\right ) \log (x)+\left (2 x+x^2\right ) \log ^2(x)\right ) \log ^3(\log (x))}{\left (2 x+x^2\right ) \log (x) \log ^3(\log (x))} \, dx=4+\log \left (3 e^{\frac {\log ^2(4 x (2+x))}{\log ^2(\log (x))}}\right )+x \log (x) \] Output:

x*ln(x)+4+ln(3*exp(ln(4*x*(2+x))^2/ln(ln(x))^2))
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \[ \int \frac {(-4-2 x) \log ^2\left (8 x+4 x^2\right )+(4+4 x) \log (x) \log \left (8 x+4 x^2\right ) \log (\log (x))+\left (\left (2 x+x^2\right ) \log (x)+\left (2 x+x^2\right ) \log ^2(x)\right ) \log ^3(\log (x))}{\left (2 x+x^2\right ) \log (x) \log ^3(\log (x))} \, dx=x \log (x)+\frac {\log ^2(4 x (2+x))}{\log ^2(\log (x))} \] Input:

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

Output:

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

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

\(\Big \downarrow \) 2026

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

$Aborted
 

Maple [A] (verified)

Time = 2.95 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.27

method	result	size
parallelrisch	\(-\frac {-4 x \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{2}-4 \ln \left (4 x^{2}+8 x \right )^{2}}{4 \ln \left (\ln \left (x \right )\right )^{2}}\)	\(33\)
risch	\(x \ln \left (x \right )+\frac {-\pi ^{2} \operatorname {csgn}\left (i x \left (2+x \right )\right )^{6}+16 \ln \left (2\right ) \ln \left (x \right )+4 \ln \left (2+x \right )^{2}+16 \ln \left (2\right )^{2}+4 \ln \left (x \right )^{2}+8 \ln \left (x \right ) \ln \left (2+x \right )+16 \ln \left (2\right ) \ln \left (2+x \right )-\pi ^{2} \operatorname {csgn}\left (i \left (2+x \right )\right )^{2} \operatorname {csgn}\left (i x \left (2+x \right )\right )^{4}+2 \pi ^{2} \operatorname {csgn}\left (i \left (2+x \right )\right ) \operatorname {csgn}\left (i x \left (2+x \right )\right )^{5}+2 \pi ^{2} \operatorname {csgn}\left (i x \left (2+x \right )\right )^{5} \operatorname {csgn}\left (i x \right )-\pi ^{2} \operatorname {csgn}\left (i x \left (2+x \right )\right )^{4} \operatorname {csgn}\left (i x \right )^{2}+2 \pi ^{2} \operatorname {csgn}\left (i \left (2+x \right )\right ) \operatorname {csgn}\left (i x \left (2+x \right )\right )^{3} \operatorname {csgn}\left (i x \right )^{2}-8 i \ln \left (2\right ) \pi \operatorname {csgn}\left (i x \left (2+x \right )\right )^{3}-4 i \ln \left (x \right ) \pi \operatorname {csgn}\left (i x \left (2+x \right )\right )^{3}-4 i \ln \left (2+x \right ) \pi \operatorname {csgn}\left (i x \left (2+x \right )\right )^{3}+2 \pi ^{2} \operatorname {csgn}\left (i \left (2+x \right )\right )^{2} \operatorname {csgn}\left (i x \left (2+x \right )\right )^{3} \operatorname {csgn}\left (i x \right )-4 \pi ^{2} \operatorname {csgn}\left (i \left (2+x \right )\right ) \operatorname {csgn}\left (i x \left (2+x \right )\right )^{4} \operatorname {csgn}\left (i x \right )-\pi ^{2} \operatorname {csgn}\left (i \left (2+x \right )\right )^{2} \operatorname {csgn}\left (i x \left (2+x \right )\right )^{2} \operatorname {csgn}\left (i x \right )^{2}+8 i \ln \left (2\right ) \pi \operatorname {csgn}\left (i x \left (2+x \right )\right )^{2} \operatorname {csgn}\left (i x \right )+4 i \ln \left (x \right ) \pi \,\operatorname {csgn}\left (i \left (2+x \right )\right ) \operatorname {csgn}\left (i x \left (2+x \right )\right )^{2}+4 i \ln \left (x \right ) \pi \operatorname {csgn}\left (i x \left (2+x \right )\right )^{2} \operatorname {csgn}\left (i x \right )+4 i \ln \left (2+x \right ) \pi \,\operatorname {csgn}\left (i \left (2+x \right )\right ) \operatorname {csgn}\left (i x \left (2+x \right )\right )^{2}+4 i \ln \left (2+x \right ) \pi \operatorname {csgn}\left (i x \left (2+x \right )\right )^{2} \operatorname {csgn}\left (i x \right )+8 i \ln \left (2\right ) \pi \,\operatorname {csgn}\left (i \left (2+x \right )\right ) \operatorname {csgn}\left (i x \left (2+x \right )\right )^{2}-8 i \ln \left (2\right ) \pi \,\operatorname {csgn}\left (i \left (2+x \right )\right ) \operatorname {csgn}\left (i x \left (2+x \right )\right ) \operatorname {csgn}\left (i x \right )-4 i \ln \left (x \right ) \pi \,\operatorname {csgn}\left (i \left (2+x \right )\right ) \operatorname {csgn}\left (i x \left (2+x \right )\right ) \operatorname {csgn}\left (i x \right )-4 i \ln \left (2+x \right ) \pi \,\operatorname {csgn}\left (i \left (2+x \right )\right ) \operatorname {csgn}\left (i x \left (2+x \right )\right ) \operatorname {csgn}\left (i x \right )}{4 \ln \left (\ln \left (x \right )\right )^{2}}\)	\(541\)

Input:

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

Output:

-1/4*(-4*x*ln(x)*ln(ln(x))^2-4*ln(4*x^2+8*x)^2)/ln(ln(x))^2
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {(-4-2 x) \log ^2\left (8 x+4 x^2\right )+(4+4 x) \log (x) \log \left (8 x+4 x^2\right ) \log (\log (x))+\left (\left (2 x+x^2\right ) \log (x)+\left (2 x+x^2\right ) \log ^2(x)\right ) \log ^3(\log (x))}{\left (2 x+x^2\right ) \log (x) \log ^3(\log (x))} \, dx=\frac {x \log \left (x\right ) \log \left (\log \left (x\right )\right )^{2} + \log \left (4 \, x^{2} + 8 \, x\right )^{2}}{\log \left (\log \left (x\right )\right )^{2}} \] Input:

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

Output:

(x*log(x)*log(log(x))^2 + log(4*x^2 + 8*x)^2)/log(log(x))^2
 

Sympy [A] (verification not implemented)

Time = 0.37 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {(-4-2 x) \log ^2\left (8 x+4 x^2\right )+(4+4 x) \log (x) \log \left (8 x+4 x^2\right ) \log (\log (x))+\left (\left (2 x+x^2\right ) \log (x)+\left (2 x+x^2\right ) \log ^2(x)\right ) \log ^3(\log (x))}{\left (2 x+x^2\right ) \log (x) \log ^3(\log (x))} \, dx=x \log {\left (x \right )} + \frac {\log {\left (4 x^{2} + 8 x \right )}^{2}}{\log {\left (\log {\left (x \right )} \right )}^{2}} \] Input:

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

Output:

x*log(x) + log(4*x**2 + 8*x)**2/log(log(x))**2
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (25) = 50\).

Time = 0.19 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.96 \[ \int \frac {(-4-2 x) \log ^2\left (8 x+4 x^2\right )+(4+4 x) \log (x) \log \left (8 x+4 x^2\right ) \log (\log (x))+\left (\left (2 x+x^2\right ) \log (x)+\left (2 x+x^2\right ) \log ^2(x)\right ) \log ^3(\log (x))}{\left (2 x+x^2\right ) \log (x) \log ^3(\log (x))} \, dx=\frac {x \log \left (x\right ) \log \left (\log \left (x\right )\right )^{2} + 4 \, \log \left (2\right )^{2} + 2 \, {\left (2 \, \log \left (2\right ) + \log \left (x\right )\right )} \log \left (x + 2\right ) + \log \left (x + 2\right )^{2} + 4 \, \log \left (2\right ) \log \left (x\right ) + \log \left (x\right )^{2}}{\log \left (\log \left (x\right )\right )^{2}} \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.69 \[ \int \frac {(-4-2 x) \log ^2\left (8 x+4 x^2\right )+(4+4 x) \log (x) \log \left (8 x+4 x^2\right ) \log (\log (x))+\left (\left (2 x+x^2\right ) \log (x)+\left (2 x+x^2\right ) \log ^2(x)\right ) \log ^3(\log (x))}{\left (2 x+x^2\right ) \log (x) \log ^3(\log (x))} \, dx=x \log \left (x\right ) + \frac {\log \left (4 \, x + 8\right )^{2}}{\log \left (\log \left (x\right )\right )^{2}} + \frac {2 \, \log \left (4 \, x + 8\right ) \log \left (x\right )}{\log \left (\log \left (x\right )\right )^{2}} + \frac {\log \left (x\right )^{2}}{\log \left (\log \left (x\right )\right )^{2}} \] Input:

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

Output:

x*log(x) + log(4*x + 8)^2/log(log(x))^2 + 2*log(4*x + 8)*log(x)/log(log(x) 
)^2 + log(x)^2/log(log(x))^2
 

Mupad [B] (verification not implemented)

Time = 3.05 (sec) , antiderivative size = 190, normalized size of antiderivative = 7.31 \[ \int \frac {(-4-2 x) \log ^2\left (8 x+4 x^2\right )+(4+4 x) \log (x) \log \left (8 x+4 x^2\right ) \log (\log (x))+\left (\left (2 x+x^2\right ) \log (x)+\left (2 x+x^2\right ) \log ^2(x)\right ) \log ^3(\log (x))}{\left (2 x+x^2\right ) \log (x) \log ^3(\log (x))} \, dx=2\,\ln \left (4\,x^2+8\,x\right )\,\ln \left (x\right )-\frac {16\,{\ln \left (x\right )}^2}{x^2+4\,x+4}+4\,{\ln \left (x\right )}^2+\frac {{\ln \left (4\,x^2+8\,x\right )}^2}{{\ln \left (\ln \left (x\right )\right )}^2}+x\,\ln \left (x\right )-\frac {2\,\ln \left (4\,x^2+8\,x\right )\,\ln \left (x\right )}{x+2}-\frac {16\,x\,{\ln \left (x\right )}^2}{x^2+4\,x+4}-\frac {4\,\ln \left (4\,x^2+8\,x\right )\,\ln \left (x\right )}{x^2+4\,x+4}-\frac {4\,x^2\,{\ln \left (x\right )}^2}{x^2+4\,x+4}-\frac {6\,x\,\ln \left (4\,x^2+8\,x\right )\,\ln \left (x\right )}{x^2+4\,x+4}-\frac {2\,x^2\,\ln \left (4\,x^2+8\,x\right )\,\ln \left (x\right )}{x^2+4\,x+4} \] Input:

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

Output:

2*log(8*x + 4*x^2)*log(x) - (16*log(x)^2)/(4*x + x^2 + 4) + 4*log(x)^2 + l 
og(8*x + 4*x^2)^2/log(log(x))^2 + x*log(x) - (2*log(8*x + 4*x^2)*log(x))/( 
x + 2) - (16*x*log(x)^2)/(4*x + x^2 + 4) - (4*log(8*x + 4*x^2)*log(x))/(4* 
x + x^2 + 4) - (4*x^2*log(x)^2)/(4*x + x^2 + 4) - (6*x*log(8*x + 4*x^2)*lo 
g(x))/(4*x + x^2 + 4) - (2*x^2*log(8*x + 4*x^2)*log(x))/(4*x + x^2 + 4)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {(-4-2 x) \log ^2\left (8 x+4 x^2\right )+(4+4 x) \log (x) \log \left (8 x+4 x^2\right ) \log (\log (x))+\left (\left (2 x+x^2\right ) \log (x)+\left (2 x+x^2\right ) \log ^2(x)\right ) \log ^3(\log (x))}{\left (2 x+x^2\right ) \log (x) \log ^3(\log (x))} \, dx=\frac {\mathrm {log}\left (\mathrm {log}\left (x \right )\right )^{2} \mathrm {log}\left (x \right ) x +\mathrm {log}\left (4 x^{2}+8 x \right )^{2}}{\mathrm {log}\left (\mathrm {log}\left (x \right )\right )^{2}} \] Input:

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

Output:

(log(log(x))**2*log(x)*x + log(4*x**2 + 8*x)**2)/log(log(x))**2