\(\int \frac {-2+2 x+2 x^2-6 x^3+4 x^2 \log (4)+(x-3 x^3+2 x^2 \log (4)) \log (x^2)+(4 x^2+2 x^2 \log (x^2)) \log (2+\log (x^2))}{2 x^2-2 x^4+2 x^3 \log (4)+(x^2-x^4+x^3 \log (4)) \log (x^2)+(-2 x+2 x^3+(-x+x^3) \log (x^2)) \log (2+\log (x^2))} \, dx\) [1345]

Optimal result

Integrand size = 134, antiderivative size = 30 \[ \int \frac {-2+2 x+2 x^2-6 x^3+4 x^2 \log (4)+\left (x-3 x^3+2 x^2 \log (4)\right ) \log \left (x^2\right )+\left (4 x^2+2 x^2 \log \left (x^2\right )\right ) \log \left (2+\log \left (x^2\right )\right )}{2 x^2-2 x^4+2 x^3 \log (4)+\left (x^2-x^4+x^3 \log (4)\right ) \log \left (x^2\right )+\left (-2 x+2 x^3+\left (-x+x^3\right ) \log \left (x^2\right )\right ) \log \left (2+\log \left (x^2\right )\right )} \, dx=\log \left (x^2 \log (4)+\frac {\left (x-x^3\right ) \left (x-\log \left (2+\log \left (x^2\right )\right )\right )}{x}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13 \[ \int \frac {-2+2 x+2 x^2-6 x^3+4 x^2 \log (4)+\left (x-3 x^3+2 x^2 \log (4)\right ) \log \left (x^2\right )+\left (4 x^2+2 x^2 \log \left (x^2\right )\right ) \log \left (2+\log \left (x^2\right )\right )}{2 x^2-2 x^4+2 x^3 \log (4)+\left (x^2-x^4+x^3 \log (4)\right ) \log \left (x^2\right )+\left (-2 x+2 x^3+\left (-x+x^3\right ) \log \left (x^2\right )\right ) \log \left (2+\log \left (x^2\right )\right )} \, dx=\log \left (-x+x^3-x^2 \log (4)+\log \left (2+\log \left (x^2\right )\right )-x^2 \log \left (2+\log \left (x^2\right )\right )\right ) \] Input:

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

Output:

Log[-x + x^3 - x^2*Log[4] + Log[2 + Log[x^2]] - x^2*Log[2 + Log[x^2]]]
 

Rubi [A] (verified)

Time = 0.93 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {6, 7292, 7235}

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.

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

\(\Big \downarrow \) 6

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 7235

\(\displaystyle \log \left (-x^3+x^2 \log \left (\log \left (x^2\right )+2\right )+x^2 \log (4)-\log \left (\log \left (x^2\right )+2\right )+x\right )\)

Input:

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

Output:

Log[x - x^3 + x^2*Log[4] - Log[2 + Log[x^2]] + x^2*Log[2 + Log[x^2]]]
 

Defintions of rubi rules used

rule 6

Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v 
+ (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] &&  !FreeQ[Fx, x]
 

rule 7235

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*L 
og[RemoveContent[y, x]], x] /;  !FalseQ[q]]
 

rule 7292

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

Maple [A] (verified)

Time = 1.42 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17

method	result	size
parallelrisch	\(\ln \left (-2 x^{2} \ln \left (2\right )+x^{3}-\ln \left (2+\ln \left (x^{2}\right )\right ) x^{2}-x +\ln \left (2+\ln \left (x^{2}\right )\right )\right )\)	\(35\)

Input:

int(((2*x^2*ln(x^2)+4*x^2)*ln(2+ln(x^2))+(4*x^2*ln(2)-3*x^3+x)*ln(x^2)+8*x 
^2*ln(2)-6*x^3+2*x^2+2*x-2)/(((x^3-x)*ln(x^2)+2*x^3-2*x)*ln(2+ln(x^2))+(2* 
x^3*ln(2)-x^4+x^2)*ln(x^2)+4*x^3*ln(2)-2*x^4+2*x^2),x,method=_RETURNVERBOS 
E)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.50 \[ \int \frac {-2+2 x+2 x^2-6 x^3+4 x^2 \log (4)+\left (x-3 x^3+2 x^2 \log (4)\right ) \log \left (x^2\right )+\left (4 x^2+2 x^2 \log \left (x^2\right )\right ) \log \left (2+\log \left (x^2\right )\right )}{2 x^2-2 x^4+2 x^3 \log (4)+\left (x^2-x^4+x^3 \log (4)\right ) \log \left (x^2\right )+\left (-2 x+2 x^3+\left (-x+x^3\right ) \log \left (x^2\right )\right ) \log \left (2+\log \left (x^2\right )\right )} \, dx=\log \left (x^{2} - 1\right ) + \log \left (-\frac {x^{3} - 2 \, x^{2} \log \left (2\right ) - {\left (x^{2} - 1\right )} \log \left (\log \left (x^{2}\right ) + 2\right ) - x}{x^{2} - 1}\right ) \] Input:

integrate(((2*x^2*log(x^2)+4*x^2)*log(2+log(x^2))+(4*x^2*log(2)-3*x^3+x)*l 
og(x^2)+8*x^2*log(2)-6*x^3+2*x^2+2*x-2)/(((x^3-x)*log(x^2)+2*x^3-2*x)*log( 
2+log(x^2))+(2*x^3*log(2)-x^4+x^2)*log(x^2)+4*x^3*log(2)-2*x^4+2*x^2),x, a 
lgorithm="fricas")
 

Output:

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

Sympy [A] (verification not implemented)

Time = 0.60 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13 \[ \int \frac {-2+2 x+2 x^2-6 x^3+4 x^2 \log (4)+\left (x-3 x^3+2 x^2 \log (4)\right ) \log \left (x^2\right )+\left (4 x^2+2 x^2 \log \left (x^2\right )\right ) \log \left (2+\log \left (x^2\right )\right )}{2 x^2-2 x^4+2 x^3 \log (4)+\left (x^2-x^4+x^3 \log (4)\right ) \log \left (x^2\right )+\left (-2 x+2 x^3+\left (-x+x^3\right ) \log \left (x^2\right )\right ) \log \left (2+\log \left (x^2\right )\right )} \, dx=\log {\left (x^{2} - 1 \right )} + \log {\left (\log {\left (\log {\left (x^{2} \right )} + 2 \right )} + \frac {- x^{3} + 2 x^{2} \log {\left (2 \right )} + x}{x^{2} - 1} \right )} \] Input:

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.57 \[ \int \frac {-2+2 x+2 x^2-6 x^3+4 x^2 \log (4)+\left (x-3 x^3+2 x^2 \log (4)\right ) \log \left (x^2\right )+\left (4 x^2+2 x^2 \log \left (x^2\right )\right ) \log \left (2+\log \left (x^2\right )\right )}{2 x^2-2 x^4+2 x^3 \log (4)+\left (x^2-x^4+x^3 \log (4)\right ) \log \left (x^2\right )+\left (-2 x+2 x^3+\left (-x+x^3\right ) \log \left (x^2\right )\right ) \log \left (2+\log \left (x^2\right )\right )} \, dx=\log \left (x + 1\right ) + \log \left (x - 1\right ) + \log \left (-\frac {x^{3} - 3 \, x^{2} \log \left (2\right ) - {\left (x^{2} - 1\right )} \log \left (\log \left (x\right ) + 1\right ) - x + \log \left (2\right )}{x^{2} - 1}\right ) \] Input:

integrate(((2*x^2*log(x^2)+4*x^2)*log(2+log(x^2))+(4*x^2*log(2)-3*x^3+x)*l 
og(x^2)+8*x^2*log(2)-6*x^3+2*x^2+2*x-2)/(((x^3-x)*log(x^2)+2*x^3-2*x)*log( 
2+log(x^2))+(2*x^3*log(2)-x^4+x^2)*log(x^2)+4*x^3*log(2)-2*x^4+2*x^2),x, a 
lgorithm="maxima")
 

Output:

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

Giac [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17 \[ \int \frac {-2+2 x+2 x^2-6 x^3+4 x^2 \log (4)+\left (x-3 x^3+2 x^2 \log (4)\right ) \log \left (x^2\right )+\left (4 x^2+2 x^2 \log \left (x^2\right )\right ) \log \left (2+\log \left (x^2\right )\right )}{2 x^2-2 x^4+2 x^3 \log (4)+\left (x^2-x^4+x^3 \log (4)\right ) \log \left (x^2\right )+\left (-2 x+2 x^3+\left (-x+x^3\right ) \log \left (x^2\right )\right ) \log \left (2+\log \left (x^2\right )\right )} \, dx=\log \left (-x^{3} + 2 \, x^{2} \log \left (2\right ) + x^{2} \log \left (\log \left (x^{2}\right ) + 2\right ) + x - \log \left (\log \left (x^{2}\right ) + 2\right )\right ) \] Input:

integrate(((2*x^2*log(x^2)+4*x^2)*log(2+log(x^2))+(4*x^2*log(2)-3*x^3+x)*l 
og(x^2)+8*x^2*log(2)-6*x^3+2*x^2+2*x-2)/(((x^3-x)*log(x^2)+2*x^3-2*x)*log( 
2+log(x^2))+(2*x^3*log(2)-x^4+x^2)*log(x^2)+4*x^3*log(2)-2*x^4+2*x^2),x, a 
lgorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17 \[ \int \frac {-2+2 x+2 x^2-6 x^3+4 x^2 \log (4)+\left (x-3 x^3+2 x^2 \log (4)\right ) \log \left (x^2\right )+\left (4 x^2+2 x^2 \log \left (x^2\right )\right ) \log \left (2+\log \left (x^2\right )\right )}{2 x^2-2 x^4+2 x^3 \log (4)+\left (x^2-x^4+x^3 \log (4)\right ) \log \left (x^2\right )+\left (-2 x+2 x^3+\left (-x+x^3\right ) \log \left (x^2\right )\right ) \log \left (2+\log \left (x^2\right )\right )} \, dx=\mathrm {log}\left (\mathrm {log}\left (\mathrm {log}\left (x^{2}\right )+2\right ) x^{2}-\mathrm {log}\left (\mathrm {log}\left (x^{2}\right )+2\right )+2 \,\mathrm {log}\left (2\right ) x^{2}-x^{3}+x \right ) \] Input:

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

Output:

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