\(\int \frac {4+2 \log (x^2)+2 \log (x^2) \log (x \log (x^2))-2 \log (x^2) \log (x \log (x^2)) \log (3 \log (x \log (x^2)))}{(-2 x-3 x^2) \log (x^2) \log (x \log (x^2))+2 x \log (x^2) \log (x \log (x^2)) \log (3 \log (x \log (x^2)))} \, dx\) [1024]

Optimal result

Integrand size = 93, antiderivative size = 23 \[ \int \frac {4+2 \log \left (x^2\right )+2 \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right )-2 \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right ) \log \left (3 \log \left (x \log \left (x^2\right )\right )\right )}{\left (-2 x-3 x^2\right ) \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right )+2 x \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right ) \log \left (3 \log \left (x \log \left (x^2\right )\right )\right )} \, dx=\log \left (-3-\frac {2}{x}+\frac {2 \log \left (3 \log \left (x \log \left (x^2\right )\right )\right )}{x}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {4+2 \log \left (x^2\right )+2 \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right )-2 \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right ) \log \left (3 \log \left (x \log \left (x^2\right )\right )\right )}{\left (-2 x-3 x^2\right ) \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right )+2 x \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right ) \log \left (3 \log \left (x \log \left (x^2\right )\right )\right )} \, dx=-\log (x)+\log \left (2+3 x-2 \log \left (3 \log \left (x \log \left (x^2\right )\right )\right )\right ) \] Input:

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

Output:

-Log[x] + Log[2 + 3*x - 2*Log[3*Log[x*Log[x^2]]]]
 

Rubi [A] (verified)

Time = 1.06 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.35, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {7292, 27, 25, 7293, 2009}

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.

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7292

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =! 
= u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [A] (verified)

Time = 2.32 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04

method	result	size
parallelrisch	\(-\frac {\ln \left (x^{2}\right )}{2}+\ln \left (-\frac {2 \ln \left (3 \ln \left (x \ln \left (x^{2}\right )\right )\right )}{3}+x +\frac {2}{3}\right )\)	\(24\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {4+2 \log \left (x^2\right )+2 \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right )-2 \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right ) \log \left (3 \log \left (x \log \left (x^2\right )\right )\right )}{\left (-2 x-3 x^2\right ) \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right )+2 x \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right ) \log \left (3 \log \left (x \log \left (x^2\right )\right )\right )} \, dx=-\frac {1}{2} \, \log \left (x^{2}\right ) + \log \left (-3 \, x + 2 \, \log \left (3 \, \log \left (x \log \left (x^{2}\right )\right )\right ) - 2\right ) \] Input:

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

Output:

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

Sympy [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {4+2 \log \left (x^2\right )+2 \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right )-2 \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right ) \log \left (3 \log \left (x \log \left (x^2\right )\right )\right )}{\left (-2 x-3 x^2\right ) \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right )+2 x \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right ) \log \left (3 \log \left (x \log \left (x^2\right )\right )\right )} \, dx=- \log {\left (x \right )} + \log {\left (- \frac {3 x}{2} + \log {\left (3 \log {\left (x \log {\left (x^{2} \right )} \right )} \right )} - 1 \right )} \] Input:

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {4+2 \log \left (x^2\right )+2 \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right )-2 \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right ) \log \left (3 \log \left (x \log \left (x^2\right )\right )\right )}{\left (-2 x-3 x^2\right ) \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right )+2 x \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right ) \log \left (3 \log \left (x \log \left (x^2\right )\right )\right )} \, dx=-\log \left (x\right ) + \log \left (-\frac {3}{2} \, x + \log \left (3\right ) + \log \left (\log \left (2\right ) + \log \left (x\right ) + \log \left (\log \left (x\right )\right )\right ) - 1\right ) \] Input:

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

Output:

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

Giac [F]

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

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

Output:

integrate(-2*(log(x^2)*log(x*log(x^2))*log(3*log(x*log(x^2))) - log(x^2)*l 
og(x*log(x^2)) - log(x^2) - 2)/(2*x*log(x^2)*log(x*log(x^2))*log(3*log(x*l 
og(x^2))) - (3*x^2 + 2*x)*log(x^2)*log(x*log(x^2))), x)
 

Mupad [B] (verification not implemented)

Time = 3.45 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {4+2 \log \left (x^2\right )+2 \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right )-2 \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right ) \log \left (3 \log \left (x \log \left (x^2\right )\right )\right )}{\left (-2 x-3 x^2\right ) \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right )+2 x \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right ) \log \left (3 \log \left (x \log \left (x^2\right )\right )\right )} \, dx=\ln \left (\ln \left (3\right )-\frac {3\,x}{2}+\ln \left (\ln \left (x\,\ln \left (x^2\right )\right )\right )-1\right )-\ln \left (x\right ) \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {4+2 \log \left (x^2\right )+2 \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right )-2 \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right ) \log \left (3 \log \left (x \log \left (x^2\right )\right )\right )}{\left (-2 x-3 x^2\right ) \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right )+2 x \log \left (x^2\right ) \log \left (x \log \left (x^2\right )\right ) \log \left (3 \log \left (x \log \left (x^2\right )\right )\right )} \, dx=\mathrm {log}\left (2 \,\mathrm {log}\left (3 \,\mathrm {log}\left (\mathrm {log}\left (x^{2}\right ) x \right )\right )-3 x -2\right )-\mathrm {log}\left (x \right ) \] Input:

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

Output:

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