\(\int \frac {-e^4-x+(-e^4-x) \log (x)+e^4 \log (x) \log (2 x \log (x))+((-e^8-2 e^4 x-x^2) \log (x)+x^x ((e^8+2 e^4 x+x^2) \log (x)+(e^8+2 e^4 x+x^2) \log ^2(x))) \log ^2(2 x \log (x))}{(e^8+2 e^4 x+x^2) \log (x) \log ^2(2 x \log (x))} \, dx\) [2662]

Optimal result

Integrand size = 130, antiderivative size = 25 \[ \int \frac {-e^4-x+\left (-e^4-x\right ) \log (x)+e^4 \log (x) \log (2 x \log (x))+\left (\left (-e^8-2 e^4 x-x^2\right ) \log (x)+x^x \left (\left (e^8+2 e^4 x+x^2\right ) \log (x)+\left (e^8+2 e^4 x+x^2\right ) \log ^2(x)\right )\right ) \log ^2(2 x \log (x))}{\left (e^8+2 e^4 x+x^2\right ) \log (x) \log ^2(2 x \log (x))} \, dx=-4-x+x^x+\frac {x}{\left (e^4+x\right ) \log (2 x \log (x))} \] Output:

exp(x*ln(x))-4+1/(x+exp(4))*x/ln(2*x*ln(x))-x
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {-e^4-x+\left (-e^4-x\right ) \log (x)+e^4 \log (x) \log (2 x \log (x))+\left (\left (-e^8-2 e^4 x-x^2\right ) \log (x)+x^x \left (\left (e^8+2 e^4 x+x^2\right ) \log (x)+\left (e^8+2 e^4 x+x^2\right ) \log ^2(x)\right )\right ) \log ^2(2 x \log (x))}{\left (e^8+2 e^4 x+x^2\right ) \log (x) \log ^2(2 x \log (x))} \, dx=-x+x^x+\frac {x}{\left (e^4+x\right ) \log (2 x \log (x))} \] Input:

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

Output:

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

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

\(\Big \downarrow \) 2007

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

$Aborted
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(66\) vs. \(2(26)=52\).

Time = 39.14 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.68

method	result	size
parallelrisch	\(\frac {{\mathrm e}^{8} \ln \left (2 x \ln \left (x \right )\right )+{\mathrm e}^{4} \ln \left (2 x \ln \left (x \right )\right ) {\mathrm e}^{x \ln \left (x \right )}-\ln \left (2 x \ln \left (x \right )\right ) x^{2}+\ln \left (2 x \ln \left (x \right )\right ) {\mathrm e}^{x \ln \left (x \right )} x +x}{\ln \left (2 x \ln \left (x \right )\right ) \left (x +{\mathrm e}^{4}\right )}\)	\(67\)
risch	\(x^{x}-x +\frac {2 i x}{\left (x +{\mathrm e}^{4}\right ) \left (\pi \,\operatorname {csgn}\left (i \ln \left (x \right )\right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x \ln \left (x \right )\right )-\pi \,\operatorname {csgn}\left (i \ln \left (x \right )\right ) \operatorname {csgn}\left (i x \ln \left (x \right )\right )^{2}-\pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x \ln \left (x \right )\right )^{2}+\pi \operatorname {csgn}\left (i x \ln \left (x \right )\right )^{3}+2 i \ln \left (2\right )+2 i \ln \left (\ln \left (x \right )\right )+2 i \ln \left (x \right )\right )}\)	\(103\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.72 \[ \int \frac {-e^4-x+\left (-e^4-x\right ) \log (x)+e^4 \log (x) \log (2 x \log (x))+\left (\left (-e^8-2 e^4 x-x^2\right ) \log (x)+x^x \left (\left (e^8+2 e^4 x+x^2\right ) \log (x)+\left (e^8+2 e^4 x+x^2\right ) \log ^2(x)\right )\right ) \log ^2(2 x \log (x))}{\left (e^8+2 e^4 x+x^2\right ) \log (x) \log ^2(2 x \log (x))} \, dx=\frac {{\left ({\left (x + e^{4}\right )} x^{x} - x^{2} - x e^{4}\right )} \log \left (2 \, x \log \left (x\right )\right ) + x}{{\left (x + e^{4}\right )} \log \left (2 \, x \log \left (x\right )\right )} \] Input:

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

Output:

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

Sympy [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {-e^4-x+\left (-e^4-x\right ) \log (x)+e^4 \log (x) \log (2 x \log (x))+\left (\left (-e^8-2 e^4 x-x^2\right ) \log (x)+x^x \left (\left (e^8+2 e^4 x+x^2\right ) \log (x)+\left (e^8+2 e^4 x+x^2\right ) \log ^2(x)\right )\right ) \log ^2(2 x \log (x))}{\left (e^8+2 e^4 x+x^2\right ) \log (x) \log ^2(2 x \log (x))} \, dx=- x + \frac {x}{\left (x + e^{4}\right ) \log {\left (2 x \log {\left (x \right )} \right )}} + e^{x \log {\left (x \right )}} \] Input:

integrate(((((exp(4)**2+2*x*exp(4)+x**2)*ln(x)**2+(exp(4)**2+2*x*exp(4)+x* 
*2)*ln(x))*exp(x*ln(x))+(-exp(4)**2-2*x*exp(4)-x**2)*ln(x))*ln(2*x*ln(x))* 
*2+exp(4)*ln(x)*ln(2*x*ln(x))+(-exp(4)-x)*ln(x)-exp(4)-x)/(exp(4)**2+2*x*e 
xp(4)+x**2)/ln(x)/ln(2*x*ln(x))**2,x)
 

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (24) = 48\).

Time = 0.16 (sec) , antiderivative size = 98, normalized size of antiderivative = 3.92 \[ \int \frac {-e^4-x+\left (-e^4-x\right ) \log (x)+e^4 \log (x) \log (2 x \log (x))+\left (\left (-e^8-2 e^4 x-x^2\right ) \log (x)+x^x \left (\left (e^8+2 e^4 x+x^2\right ) \log (x)+\left (e^8+2 e^4 x+x^2\right ) \log ^2(x)\right )\right ) \log ^2(2 x \log (x))}{\left (e^8+2 e^4 x+x^2\right ) \log (x) \log ^2(2 x \log (x))} \, dx=-\frac {x^{2} \log \left (2\right ) - {\left (x \log \left (2\right ) + e^{4} \log \left (2\right ) + {\left (x + e^{4}\right )} \log \left (x\right ) + {\left (x + e^{4}\right )} \log \left (\log \left (x\right )\right )\right )} x^{x} + {\left (e^{4} \log \left (2\right ) - 1\right )} x + {\left (x^{2} + x e^{4}\right )} \log \left (x\right ) + {\left (x^{2} + x e^{4}\right )} \log \left (\log \left (x\right )\right )}{x \log \left (2\right ) + e^{4} \log \left (2\right ) + {\left (x + e^{4}\right )} \log \left (x\right ) + {\left (x + e^{4}\right )} \log \left (\log \left (x\right )\right )} \] Input:

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

Output:

-(x^2*log(2) - (x*log(2) + e^4*log(2) + (x + e^4)*log(x) + (x + e^4)*log(l 
og(x)))*x^x + (e^4*log(2) - 1)*x + (x^2 + x*e^4)*log(x) + (x^2 + x*e^4)*lo 
g(log(x)))/(x*log(2) + e^4*log(2) + (x + e^4)*log(x) + (x + e^4)*log(log(x 
)))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1320 vs. \(2 (24) = 48\).

Time = 3.06 (sec) , antiderivative size = 1320, normalized size of antiderivative = 52.80 \[ \int \frac {-e^4-x+\left (-e^4-x\right ) \log (x)+e^4 \log (x) \log (2 x \log (x))+\left (\left (-e^8-2 e^4 x-x^2\right ) \log (x)+x^x \left (\left (e^8+2 e^4 x+x^2\right ) \log (x)+\left (e^8+2 e^4 x+x^2\right ) \log ^2(x)\right )\right ) \log ^2(2 x \log (x))}{\left (e^8+2 e^4 x+x^2\right ) \log (x) \log ^2(2 x \log (x))} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 2.94 (sec) , antiderivative size = 118, normalized size of antiderivative = 4.72 \[ \int \frac {-e^4-x+\left (-e^4-x\right ) \log (x)+e^4 \log (x) \log (2 x \log (x))+\left (\left (-e^8-2 e^4 x-x^2\right ) \log (x)+x^x \left (\left (e^8+2 e^4 x+x^2\right ) \log (x)+\left (e^8+2 e^4 x+x^2\right ) \log ^2(x)\right )\right ) \log ^2(2 x \log (x))}{\left (e^8+2 e^4 x+x^2\right ) \log (x) \log ^2(2 x \log (x))} \, dx=\frac {\frac {x}{x+{\mathrm {e}}^4}-\frac {x\,\ln \left (2\,x\,\ln \left (x\right )\right )\,{\mathrm {e}}^4\,\ln \left (x\right )}{{\left (x+{\mathrm {e}}^4\right )}^2\,\left (\ln \left (x\right )+1\right )}}{\ln \left (2\,x\,\ln \left (x\right )\right )}-x+x^x-\frac {\frac {2\,x^2\,{\mathrm {e}}^4}{{\left (x+{\mathrm {e}}^4\right )}^3}-\frac {x\,\ln \left (x\right )\,\left ({\mathrm {e}}^8-x\,{\mathrm {e}}^4\right )}{{\left (x+{\mathrm {e}}^4\right )}^3}}{\ln \left (x\right )+1}+\frac {2\,x^2\,{\mathrm {e}}^4}{x^3+3\,{\mathrm {e}}^4\,x^2+3\,{\mathrm {e}}^8\,x+{\mathrm {e}}^{12}} \] Input:

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

Output:

(x/(x + exp(4)) - (x*log(2*x*log(x))*exp(4)*log(x))/((x + exp(4))^2*(log(x 
) + 1)))/log(2*x*log(x)) - x + x^x - ((2*x^2*exp(4))/(x + exp(4))^3 - (x*l 
og(x)*(exp(8) - x*exp(4)))/(x + exp(4))^3)/(log(x) + 1) + (2*x^2*exp(4))/( 
exp(12) + 3*x*exp(8) + 3*x^2*exp(4) + x^3)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.60 \[ \int \frac {-e^4-x+\left (-e^4-x\right ) \log (x)+e^4 \log (x) \log (2 x \log (x))+\left (\left (-e^8-2 e^4 x-x^2\right ) \log (x)+x^x \left (\left (e^8+2 e^4 x+x^2\right ) \log (x)+\left (e^8+2 e^4 x+x^2\right ) \log ^2(x)\right )\right ) \log ^2(2 x \log (x))}{\left (e^8+2 e^4 x+x^2\right ) \log (x) \log ^2(2 x \log (x))} \, dx=\frac {x^{x} \mathrm {log}\left (2 \,\mathrm {log}\left (x \right ) x \right ) e^{4}+x^{x} \mathrm {log}\left (2 \,\mathrm {log}\left (x \right ) x \right ) x -\mathrm {log}\left (2 \,\mathrm {log}\left (x \right ) x \right ) e^{4} x -\mathrm {log}\left (2 \,\mathrm {log}\left (x \right ) x \right ) x^{2}+x}{\mathrm {log}\left (2 \,\mathrm {log}\left (x \right ) x \right ) \left (e^{4}+x \right )} \] Input:

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

Output:

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