\(\int \frac {-12 e^{e^x}+(-6 x+6 x \log (3)) \log (x^2)-6 e^{e^x+x} x \log (x^2) \log (\log (x^2))+((3 x-3 x \log (3)) \log (x^2)+3 e^{e^x} \log (x^2) \log (\log (x^2))) \log (x-x \log (3)+e^{e^x} \log (\log (x^2)))}{((x-x \log (3)) \log (x^2)+e^{e^x} \log (x^2) \log (\log (x^2))) \log ^3(x-x \log (3)+e^{e^x} \log (\log (x^2)))} \, dx\) [17]

Optimal result

Integrand size = 144, antiderivative size = 24 \[ \int \frac {-12 e^{e^x}+(-6 x+6 x \log (3)) \log \left (x^2\right )-6 e^{e^x+x} x \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+\left ((3 x-3 x \log (3)) \log \left (x^2\right )+3 e^{e^x} \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right ) \log \left (x-x \log (3)+e^{e^x} \log \left (\log \left (x^2\right )\right )\right )}{\left ((x-x \log (3)) \log \left (x^2\right )+e^{e^x} \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right ) \log ^3\left (x-x \log (3)+e^{e^x} \log \left (\log \left (x^2\right )\right )\right )} \, dx=\frac {3 x}{\log ^2\left (x-x \log (3)+e^{e^x} \log \left (\log \left (x^2\right )\right )\right )} \] Output:

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {-12 e^{e^x}+(-6 x+6 x \log (3)) \log \left (x^2\right )-6 e^{e^x+x} x \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+\left ((3 x-3 x \log (3)) \log \left (x^2\right )+3 e^{e^x} \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right ) \log \left (x-x \log (3)+e^{e^x} \log \left (\log \left (x^2\right )\right )\right )}{\left ((x-x \log (3)) \log \left (x^2\right )+e^{e^x} \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right ) \log ^3\left (x-x \log (3)+e^{e^x} \log \left (\log \left (x^2\right )\right )\right )} \, dx=\frac {3 x}{\log ^2\left (x-x \log (3)+e^{e^x} \log \left (\log \left (x^2\right )\right )\right )} \] Input:

Integrate[(-12*E^E^x + (-6*x + 6*x*Log[3])*Log[x^2] - 6*E^(E^x + x)*x*Log[ 
x^2]*Log[Log[x^2]] + ((3*x - 3*x*Log[3])*Log[x^2] + 3*E^E^x*Log[x^2]*Log[L 
og[x^2]])*Log[x - x*Log[3] + E^E^x*Log[Log[x^2]]])/(((x - x*Log[3])*Log[x^ 
2] + E^E^x*Log[x^2]*Log[Log[x^2]])*Log[x - x*Log[3] + E^E^x*Log[Log[x^2]]] 
^3),x]
 

Output:

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

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

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {(6 x \log (3)-6 x) \log \left (x^2\right )-6 e^{x+e^x} x \log \left (\log \left (x^2\right )\right ) \log \left (x^2\right )+\left ((3 x-3 x \log (3)) \log \left (x^2\right )+3 e^{e^x} \log \left (\log \left (x^2\right )\right ) \log \left (x^2\right )\right ) \log \left (e^{e^x} \log \left (\log \left (x^2\right )\right )+x+x (-\log (3))\right )-12 e^{e^x}}{\log \left (x^2\right ) \left (e^{e^x} \log \left (\log \left (x^2\right )\right )+x (1-\log (3))\right ) \log ^3\left (e^{e^x} \log \left (\log \left (x^2\right )\right )+x (1-\log (3))\right )}dx\)

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle -12 \int \frac {1}{\log \left (x^2\right ) \log \left (\log \left (x^2\right )\right ) \log ^3\left ((1-\log (3)) x+e^{e^x} \log \left (\log \left (x^2\right )\right )\right )}dx+6 (1-\log (3)) \int \frac {x}{\left (-((1-\log (3)) x)-e^{e^x} \log \left (\log \left (x^2\right )\right )\right ) \log ^3\left ((1-\log (3)) x+e^{e^x} \log \left (\log \left (x^2\right )\right )\right )}dx+6 \int \frac {e^{x+e^x} x \log \left (\log \left (x^2\right )\right )}{\left (-((1-\log (3)) x)-e^{e^x} \log \left (\log \left (x^2\right )\right )\right ) \log ^3\left ((1-\log (3)) x+e^{e^x} \log \left (\log \left (x^2\right )\right )\right )}dx+12 (1-\log (3)) \int \frac {x}{\log \left (x^2\right ) \log \left (\log \left (x^2\right )\right ) \left ((1-\log (3)) x+e^{e^x} \log \left (\log \left (x^2\right )\right )\right ) \log ^3\left ((1-\log (3)) x+e^{e^x} \log \left (\log \left (x^2\right )\right )\right )}dx+3 \int \frac {1}{\log ^2\left ((1-\log (3)) x+e^{e^x} \log \left (\log \left (x^2\right )\right )\right )}dx\)

Input:

Int[(-12*E^E^x + (-6*x + 6*x*Log[3])*Log[x^2] - 6*E^(E^x + x)*x*Log[x^2]*L 
og[Log[x^2]] + ((3*x - 3*x*Log[3])*Log[x^2] + 3*E^E^x*Log[x^2]*Log[Log[x^2 
]])*Log[x - x*Log[3] + E^E^x*Log[Log[x^2]]])/(((x - x*Log[3])*Log[x^2] + E 
^E^x*Log[x^2]*Log[Log[x^2]])*Log[x - x*Log[3] + E^E^x*Log[Log[x^2]]]^3),x]
 

Output:

$Aborted
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.12 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.17

\[\frac {3 x}{\ln \left ({\mathrm e}^{{\mathrm e}^{x}} \ln \left (2 \ln \left (x \right )-\frac {i \pi \,\operatorname {csgn}\left (i x^{2}\right ) {\left (-\operatorname {csgn}\left (i x^{2}\right )+\operatorname {csgn}\left (i x \right )\right )}^{2}}{2}\right )-x \ln \left (3\right )+x \right )^{2}}\]

Input:

int(((3*ln(x^2)*exp(exp(x))*ln(ln(x^2))+(-3*x*ln(3)+3*x)*ln(x^2))*ln(exp(e 
xp(x))*ln(ln(x^2))-x*ln(3)+x)-6*x*exp(x)*ln(x^2)*exp(exp(x))*ln(ln(x^2))-1 
2*exp(exp(x))+(6*x*ln(3)-6*x)*ln(x^2))/(ln(x^2)*exp(exp(x))*ln(ln(x^2))+(- 
x*ln(3)+x)*ln(x^2))/ln(exp(exp(x))*ln(ln(x^2))-x*ln(3)+x)^3,x)
 

Output:

3*x/ln(exp(exp(x))*ln(2*ln(x)-1/2*I*Pi*csgn(I*x^2)*(-csgn(I*x^2)+csgn(I*x) 
)^2)-x*ln(3)+x)^2
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.50 \[ \int \frac {-12 e^{e^x}+(-6 x+6 x \log (3)) \log \left (x^2\right )-6 e^{e^x+x} x \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+\left ((3 x-3 x \log (3)) \log \left (x^2\right )+3 e^{e^x} \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right ) \log \left (x-x \log (3)+e^{e^x} \log \left (\log \left (x^2\right )\right )\right )}{\left ((x-x \log (3)) \log \left (x^2\right )+e^{e^x} \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right ) \log ^3\left (x-x \log (3)+e^{e^x} \log \left (\log \left (x^2\right )\right )\right )} \, dx=\frac {3 \, x}{\log \left (-{\left ({\left (x \log \left (3\right ) - x\right )} e^{x} - e^{\left (x + e^{x}\right )} \log \left (\log \left (x^{2}\right )\right )\right )} e^{\left (-x\right )}\right )^{2}} \] Input:

integrate(((3*log(x^2)*exp(exp(x))*log(log(x^2))+(-3*x*log(3)+3*x)*log(x^2 
))*log(exp(exp(x))*log(log(x^2))-x*log(3)+x)-6*x*exp(x)*log(x^2)*exp(exp(x 
))*log(log(x^2))-12*exp(exp(x))+(6*x*log(3)-6*x)*log(x^2))/(log(x^2)*exp(e 
xp(x))*log(log(x^2))+(-x*log(3)+x)*log(x^2))/log(exp(exp(x))*log(log(x^2)) 
-x*log(3)+x)^3,x, algorithm="fricas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int \frac {-12 e^{e^x}+(-6 x+6 x \log (3)) \log \left (x^2\right )-6 e^{e^x+x} x \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+\left ((3 x-3 x \log (3)) \log \left (x^2\right )+3 e^{e^x} \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right ) \log \left (x-x \log (3)+e^{e^x} \log \left (\log \left (x^2\right )\right )\right )}{\left ((x-x \log (3)) \log \left (x^2\right )+e^{e^x} \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right ) \log ^3\left (x-x \log (3)+e^{e^x} \log \left (\log \left (x^2\right )\right )\right )} \, dx=\text {Timed out} \] Input:

integrate(((3*ln(x**2)*exp(exp(x))*ln(ln(x**2))+(-3*x*ln(3)+3*x)*ln(x**2)) 
*ln(exp(exp(x))*ln(ln(x**2))-x*ln(3)+x)-6*x*exp(x)*ln(x**2)*exp(exp(x))*ln 
(ln(x**2))-12*exp(exp(x))+(6*x*ln(3)-6*x)*ln(x**2))/(ln(x**2)*exp(exp(x))* 
ln(ln(x**2))+(-x*ln(3)+x)*ln(x**2))/ln(exp(exp(x))*ln(ln(x**2))-x*ln(3)+x) 
**3,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.50 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12 \[ \int \frac {-12 e^{e^x}+(-6 x+6 x \log (3)) \log \left (x^2\right )-6 e^{e^x+x} x \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+\left ((3 x-3 x \log (3)) \log \left (x^2\right )+3 e^{e^x} \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right ) \log \left (x-x \log (3)+e^{e^x} \log \left (\log \left (x^2\right )\right )\right )}{\left ((x-x \log (3)) \log \left (x^2\right )+e^{e^x} \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right ) \log ^3\left (x-x \log (3)+e^{e^x} \log \left (\log \left (x^2\right )\right )\right )} \, dx=\frac {3 \, x}{\log \left (-x {\left (\log \left (3\right ) - 1\right )} + e^{\left (e^{x}\right )} \log \left (2\right ) + e^{\left (e^{x}\right )} \log \left (\log \left (x\right )\right )\right )^{2}} \] Input:

integrate(((3*log(x^2)*exp(exp(x))*log(log(x^2))+(-3*x*log(3)+3*x)*log(x^2 
))*log(exp(exp(x))*log(log(x^2))-x*log(3)+x)-6*x*exp(x)*log(x^2)*exp(exp(x 
))*log(log(x^2))-12*exp(exp(x))+(6*x*log(3)-6*x)*log(x^2))/(log(x^2)*exp(e 
xp(x))*log(log(x^2))+(-x*log(3)+x)*log(x^2))/log(exp(exp(x))*log(log(x^2)) 
-x*log(3)+x)^3,x, algorithm="maxima")
 

Output:

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

Giac [F(-1)]

Timed out. \[ \int \frac {-12 e^{e^x}+(-6 x+6 x \log (3)) \log \left (x^2\right )-6 e^{e^x+x} x \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+\left ((3 x-3 x \log (3)) \log \left (x^2\right )+3 e^{e^x} \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right ) \log \left (x-x \log (3)+e^{e^x} \log \left (\log \left (x^2\right )\right )\right )}{\left ((x-x \log (3)) \log \left (x^2\right )+e^{e^x} \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right ) \log ^3\left (x-x \log (3)+e^{e^x} \log \left (\log \left (x^2\right )\right )\right )} \, dx=\text {Timed out} \] Input:

integrate(((3*log(x^2)*exp(exp(x))*log(log(x^2))+(-3*x*log(3)+3*x)*log(x^2 
))*log(exp(exp(x))*log(log(x^2))-x*log(3)+x)-6*x*exp(x)*log(x^2)*exp(exp(x 
))*log(log(x^2))-12*exp(exp(x))+(6*x*log(3)-6*x)*log(x^2))/(log(x^2)*exp(e 
xp(x))*log(log(x^2))+(-x*log(3)+x)*log(x^2))/log(exp(exp(x))*log(log(x^2)) 
-x*log(3)+x)^3,x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

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

int(-(12*exp(exp(x)) + log(x^2)*(6*x - 6*x*log(3)) - log(x - x*log(3) + ex 
p(exp(x))*log(log(x^2)))*(log(x^2)*(3*x - 3*x*log(3)) + 3*log(x^2)*exp(exp 
(x))*log(log(x^2))) + 6*x*log(x^2)*exp(exp(x))*exp(x)*log(log(x^2)))/(log( 
x - x*log(3) + exp(exp(x))*log(log(x^2)))^3*(log(x^2)*(x - x*log(3)) + log 
(x^2)*exp(exp(x))*log(log(x^2)))),x)
 

Output:

int(-(12*exp(exp(x)) + log(x^2)*(6*x - 6*x*log(3)) - log(x - x*log(3) + ex 
p(exp(x))*log(log(x^2)))*(log(x^2)*(3*x - 3*x*log(3)) + 3*log(x^2)*exp(exp 
(x))*log(log(x^2))) + 6*x*log(x^2)*exp(exp(x))*exp(x)*log(log(x^2)))/(log( 
x - x*log(3) + exp(exp(x))*log(log(x^2)))^3*(log(x^2)*(x - x*log(3)) + log 
(x^2)*exp(exp(x))*log(log(x^2)))), x)
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {-12 e^{e^x}+(-6 x+6 x \log (3)) \log \left (x^2\right )-6 e^{e^x+x} x \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+\left ((3 x-3 x \log (3)) \log \left (x^2\right )+3 e^{e^x} \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right ) \log \left (x-x \log (3)+e^{e^x} \log \left (\log \left (x^2\right )\right )\right )}{\left ((x-x \log (3)) \log \left (x^2\right )+e^{e^x} \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right ) \log ^3\left (x-x \log (3)+e^{e^x} \log \left (\log \left (x^2\right )\right )\right )} \, dx=\frac {3 x}{\mathrm {log}\left (e^{e^{x}} \mathrm {log}\left (\mathrm {log}\left (x^{2}\right )\right )-\mathrm {log}\left (3\right ) x +x \right )^{2}} \] Input:

int(((3*log(x^2)*exp(exp(x))*log(log(x^2))+(-3*x*log(3)+3*x)*log(x^2))*log 
(exp(exp(x))*log(log(x^2))-x*log(3)+x)-6*x*exp(x)*log(x^2)*exp(exp(x))*log 
(log(x^2))-12*exp(exp(x))+(6*x*log(3)-6*x)*log(x^2))/(log(x^2)*exp(exp(x)) 
*log(log(x^2))+(-x*log(3)+x)*log(x^2))/log(exp(exp(x))*log(log(x^2))-x*log 
(3)+x)^3,x)
 

Output:

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