\(\int \frac {-4 x-4 e^x x^2-8 x^3+(4+4 e^x x+8 x^2) \log (x)+(e^x (2-2 x)+2 x^2-2 x^3+(2-2 x) \log (x)) \log (e^x+x^2+\log (x)) \log (\log (e^x+x^2+\log (x)))}{(-e^x x^2-x^4+(e^x x-x^2+x^3) \log (x)+x \log ^2(x)) \log (e^x+x^2+\log (x)) \log (\log (e^x+x^2+\log (x)))} \, dx\) [683]

Optimal result

Integrand size = 148, antiderivative size = 23 \[ \int \frac {-4 x-4 e^x x^2-8 x^3+\left (4+4 e^x x+8 x^2\right ) \log (x)+\left (e^x (2-2 x)+2 x^2-2 x^3+(2-2 x) \log (x)\right ) \log \left (e^x+x^2+\log (x)\right ) \log \left (\log \left (e^x+x^2+\log (x)\right )\right )}{\left (-e^x x^2-x^4+\left (e^x x-x^2+x^3\right ) \log (x)+x \log ^2(x)\right ) \log \left (e^x+x^2+\log (x)\right ) \log \left (\log \left (e^x+x^2+\log (x)\right )\right )} \, dx=\log \left ((-x+\log (x))^2 \log ^4\left (\log \left (e^x+x^2+\log (x)\right )\right )\right ) \] Output:

ln((ln(x)-x)^2*ln(ln(ln(x)+x^2+exp(x)))^4)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {-4 x-4 e^x x^2-8 x^3+\left (4+4 e^x x+8 x^2\right ) \log (x)+\left (e^x (2-2 x)+2 x^2-2 x^3+(2-2 x) \log (x)\right ) \log \left (e^x+x^2+\log (x)\right ) \log \left (\log \left (e^x+x^2+\log (x)\right )\right )}{\left (-e^x x^2-x^4+\left (e^x x-x^2+x^3\right ) \log (x)+x \log ^2(x)\right ) \log \left (e^x+x^2+\log (x)\right ) \log \left (\log \left (e^x+x^2+\log (x)\right )\right )} \, dx=2 \log (x-\log (x))+4 \log \left (\log \left (\log \left (e^x+x^2+\log (x)\right )\right )\right ) \] Input:

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

Output:

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

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.

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle 4 \int \frac {1}{\log \left (x^2+e^x+\log (x)\right ) \log \left (\log \left (x^2+e^x+\log (x)\right )\right )}dx+4 \int \frac {1}{x \left (x^2+e^x+\log (x)\right ) \log \left (x^2+e^x+\log (x)\right ) \log \left (\log \left (x^2+e^x+\log (x)\right )\right )}dx+8 \int \frac {x}{\left (x^2+e^x+\log (x)\right ) \log \left (x^2+e^x+\log (x)\right ) \log \left (\log \left (x^2+e^x+\log (x)\right )\right )}dx-4 \int \frac {x^2}{\left (x^2+e^x+\log (x)\right ) \log \left (x^2+e^x+\log (x)\right ) \log \left (\log \left (x^2+e^x+\log (x)\right )\right )}dx-4 \int \frac {\log (x)}{\left (x^2+e^x+\log (x)\right ) \log \left (x^2+e^x+\log (x)\right ) \log \left (\log \left (x^2+e^x+\log (x)\right )\right )}dx+2 \log (x-\log (x))\)

Input:

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

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04

\[2 \ln \left (\ln \left (x \right )-x \right )+4 \ln \left (\ln \left (\ln \left (\ln \left (x \right )+x^{2}+{\mathrm e}^{x}\right )\right )\right )\]

Input:

int((((2-2*x)*ln(x)+(2-2*x)*exp(x)-2*x^3+2*x^2)*ln(ln(x)+x^2+exp(x))*ln(ln 
(ln(x)+x^2+exp(x)))+(4*exp(x)*x+8*x^2+4)*ln(x)-4*exp(x)*x^2-8*x^3-4*x)/(x* 
ln(x)^2+(exp(x)*x+x^3-x^2)*ln(x)-exp(x)*x^2-x^4)/ln(ln(x)+x^2+exp(x))/ln(l 
n(ln(x)+x^2+exp(x))),x)
 

Output:

2*ln(ln(x)-x)+4*ln(ln(ln(ln(x)+x^2+exp(x))))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {-4 x-4 e^x x^2-8 x^3+\left (4+4 e^x x+8 x^2\right ) \log (x)+\left (e^x (2-2 x)+2 x^2-2 x^3+(2-2 x) \log (x)\right ) \log \left (e^x+x^2+\log (x)\right ) \log \left (\log \left (e^x+x^2+\log (x)\right )\right )}{\left (-e^x x^2-x^4+\left (e^x x-x^2+x^3\right ) \log (x)+x \log ^2(x)\right ) \log \left (e^x+x^2+\log (x)\right ) \log \left (\log \left (e^x+x^2+\log (x)\right )\right )} \, dx=2 \, \log \left (-x + \log \left (x\right )\right ) + 4 \, \log \left (\log \left (\log \left (x^{2} + e^{x} + \log \left (x\right )\right )\right )\right ) \] Input:

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

Output:

2*log(-x + log(x)) + 4*log(log(log(x^2 + e^x + log(x))))
 

Sympy [A] (verification not implemented)

Time = 54.45 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {-4 x-4 e^x x^2-8 x^3+\left (4+4 e^x x+8 x^2\right ) \log (x)+\left (e^x (2-2 x)+2 x^2-2 x^3+(2-2 x) \log (x)\right ) \log \left (e^x+x^2+\log (x)\right ) \log \left (\log \left (e^x+x^2+\log (x)\right )\right )}{\left (-e^x x^2-x^4+\left (e^x x-x^2+x^3\right ) \log (x)+x \log ^2(x)\right ) \log \left (e^x+x^2+\log (x)\right ) \log \left (\log \left (e^x+x^2+\log (x)\right )\right )} \, dx=2 \log {\left (- x + \log {\left (x \right )} \right )} + 4 \log {\left (\log {\left (\log {\left (x^{2} + e^{x} + \log {\left (x \right )} \right )} \right )} \right )} \] Input:

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

Output:

2*log(-x + log(x)) + 4*log(log(log(x**2 + exp(x) + log(x))))
 

Maxima [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {-4 x-4 e^x x^2-8 x^3+\left (4+4 e^x x+8 x^2\right ) \log (x)+\left (e^x (2-2 x)+2 x^2-2 x^3+(2-2 x) \log (x)\right ) \log \left (e^x+x^2+\log (x)\right ) \log \left (\log \left (e^x+x^2+\log (x)\right )\right )}{\left (-e^x x^2-x^4+\left (e^x x-x^2+x^3\right ) \log (x)+x \log ^2(x)\right ) \log \left (e^x+x^2+\log (x)\right ) \log \left (\log \left (e^x+x^2+\log (x)\right )\right )} \, dx=2 \, \log \left (-x + \log \left (x\right )\right ) + 4 \, \log \left (\log \left (\log \left (x^{2} + e^{x} + \log \left (x\right )\right )\right )\right ) \] Input:

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

Output:

2*log(-x + log(x)) + 4*log(log(log(x^2 + e^x + log(x))))
 

Giac [F(-2)]

Exception generated. \[ \int \frac {-4 x-4 e^x x^2-8 x^3+\left (4+4 e^x x+8 x^2\right ) \log (x)+\left (e^x (2-2 x)+2 x^2-2 x^3+(2-2 x) \log (x)\right ) \log \left (e^x+x^2+\log (x)\right ) \log \left (\log \left (e^x+x^2+\log (x)\right )\right )}{\left (-e^x x^2-x^4+\left (e^x x-x^2+x^3\right ) \log (x)+x \log ^2(x)\right ) \log \left (e^x+x^2+\log (x)\right ) \log \left (\log \left (e^x+x^2+\log (x)\right )\right )} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Sign error %%%{ln(w),0%%%}Sign erro 
r %%%{ln(w),0%%%}Done
 

Mupad [B] (verification not implemented)

Time = 3.21 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {-4 x-4 e^x x^2-8 x^3+\left (4+4 e^x x+8 x^2\right ) \log (x)+\left (e^x (2-2 x)+2 x^2-2 x^3+(2-2 x) \log (x)\right ) \log \left (e^x+x^2+\log (x)\right ) \log \left (\log \left (e^x+x^2+\log (x)\right )\right )}{\left (-e^x x^2-x^4+\left (e^x x-x^2+x^3\right ) \log (x)+x \log ^2(x)\right ) \log \left (e^x+x^2+\log (x)\right ) \log \left (\log \left (e^x+x^2+\log (x)\right )\right )} \, dx=4\,\ln \left (\ln \left (\ln \left ({\mathrm {e}}^x+\ln \left (x\right )+x^2\right )\right )\right )+2\,\ln \left (\ln \left (x\right )-x\right ) \] Input:

int((4*x + 4*x^2*exp(x) - log(x)*(4*x*exp(x) + 8*x^2 + 4) + 8*x^3 + log(ex 
p(x) + log(x) + x^2)*log(log(exp(x) + log(x) + x^2))*(exp(x)*(2*x - 2) + l 
og(x)*(2*x - 2) - 2*x^2 + 2*x^3))/(log(exp(x) + log(x) + x^2)*log(log(exp( 
x) + log(x) + x^2))*(x^2*exp(x) - x*log(x)^2 - log(x)*(x*exp(x) - x^2 + x^ 
3) + x^4)),x)
 

Output:

4*log(log(log(exp(x) + log(x) + x^2))) + 2*log(log(x) - x)
 

Reduce [F]

\[ \int \frac {-4 x-4 e^x x^2-8 x^3+\left (4+4 e^x x+8 x^2\right ) \log (x)+\left (e^x (2-2 x)+2 x^2-2 x^3+(2-2 x) \log (x)\right ) \log \left (e^x+x^2+\log (x)\right ) \log \left (\log \left (e^x+x^2+\log (x)\right )\right )}{\left (-e^x x^2-x^4+\left (e^x x-x^2+x^3\right ) \log (x)+x \log ^2(x)\right ) \log \left (e^x+x^2+\log (x)\right ) \log \left (\log \left (e^x+x^2+\log (x)\right )\right )} \, dx=\int \frac {\left (\left (-2 x +2\right ) \mathrm {log}\left (x \right )+\left (-2 x +2\right ) {\mathrm e}^{x}-2 x^{3}+2 x^{2}\right ) \mathrm {log}\left (\mathrm {log}\left (x \right )+x^{2}+{\mathrm e}^{x}\right ) \mathrm {log}\left (\mathrm {log}\left (\mathrm {log}\left (x \right )+x^{2}+{\mathrm e}^{x}\right )\right )+\left (4 \,{\mathrm e}^{x} x +8 x^{2}+4\right ) \mathrm {log}\left (x \right )-4 \,{\mathrm e}^{x} x^{2}-8 x^{3}-4 x}{\left (\mathrm {log}\left (x \right )^{2} x +\left ({\mathrm e}^{x} x +x^{3}-x^{2}\right ) \mathrm {log}\left (x \right )-{\mathrm e}^{x} x^{2}-x^{4}\right ) \mathrm {log}\left (\mathrm {log}\left (x \right )+x^{2}+{\mathrm e}^{x}\right ) \mathrm {log}\left (\mathrm {log}\left (\mathrm {log}\left (x \right )+x^{2}+{\mathrm e}^{x}\right )\right )}d x \] Input:

int((((2-2*x)*log(x)+(2-2*x)*exp(x)-2*x^3+2*x^2)*log(log(x)+x^2+exp(x))*lo 
g(log(log(x)+x^2+exp(x)))+(4*exp(x)*x+8*x^2+4)*log(x)-4*exp(x)*x^2-8*x^3-4 
*x)/(x*log(x)^2+(exp(x)*x+x^3-x^2)*log(x)-exp(x)*x^2-x^4)/log(log(x)+x^2+e 
xp(x))/log(log(log(x)+x^2+exp(x))),x)
 

Output:

int((((2-2*x)*log(x)+(2-2*x)*exp(x)-2*x^3+2*x^2)*log(log(x)+x^2+exp(x))*lo 
g(log(log(x)+x^2+exp(x)))+(4*exp(x)*x+8*x^2+4)*log(x)-4*exp(x)*x^2-8*x^3-4 
*x)/(x*log(x)^2+(exp(x)*x+x^3-x^2)*log(x)-exp(x)*x^2-x^4)/log(log(x)+x^2+e 
xp(x))/log(log(log(x)+x^2+exp(x))),x)