\(\int \frac {3 x^2+x^3+3 x^2 \log (x)+e^{2 e^x} (3+x+3 \log (x))+e^{e^x} (6 x+2 x^2+6 x \log (x))+e^{e^x} (-3+3 e^x x) \log (x) \log (\frac {\log (x)}{2})+((6 x^2+2 x^3) \log (x)+3 x^2 \log ^2(x)+e^{2 e^x} ((6+2 x) \log (x)+3 \log ^2(x))+e^{e^x} ((12 x+4 x^2) \log (x)+6 x \log ^2(x))) \log (\frac {\log (x)}{2}) \log (\log (\frac {\log (x)}{2}))}{(3 e^{2 e^x} \log (x)+6 e^{e^x} x \log (x)+3 x^2 \log (x)) \log (\frac {\log (x)}{2})} \, dx\) [941]

Optimal result

Integrand size = 204, antiderivative size = 32 \[ \int \frac {3 x^2+x^3+3 x^2 \log (x)+e^{2 e^x} (3+x+3 \log (x))+e^{e^x} \left (6 x+2 x^2+6 x \log (x)\right )+e^{e^x} \left (-3+3 e^x x\right ) \log (x) \log \left (\frac {\log (x)}{2}\right )+\left (\left (6 x^2+2 x^3\right ) \log (x)+3 x^2 \log ^2(x)+e^{2 e^x} \left ((6+2 x) \log (x)+3 \log ^2(x)\right )+e^{e^x} \left (\left (12 x+4 x^2\right ) \log (x)+6 x \log ^2(x)\right )\right ) \log \left (\frac {\log (x)}{2}\right ) \log \left (\log \left (\frac {\log (x)}{2}\right )\right )}{\left (3 e^{2 e^x} \log (x)+6 e^{e^x} x \log (x)+3 x^2 \log (x)\right ) \log \left (\frac {\log (x)}{2}\right )} \, dx=-\frac {x}{e^{e^x}+x}+x \left (1+\frac {x}{3}+\log (x)\right ) \log \left (\log \left (\frac {\log (x)}{2}\right )\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {3 x^2+x^3+3 x^2 \log (x)+e^{2 e^x} (3+x+3 \log (x))+e^{e^x} \left (6 x+2 x^2+6 x \log (x)\right )+e^{e^x} \left (-3+3 e^x x\right ) \log (x) \log \left (\frac {\log (x)}{2}\right )+\left (\left (6 x^2+2 x^3\right ) \log (x)+3 x^2 \log ^2(x)+e^{2 e^x} \left ((6+2 x) \log (x)+3 \log ^2(x)\right )+e^{e^x} \left (\left (12 x+4 x^2\right ) \log (x)+6 x \log ^2(x)\right )\right ) \log \left (\frac {\log (x)}{2}\right ) \log \left (\log \left (\frac {\log (x)}{2}\right )\right )}{\left (3 e^{2 e^x} \log (x)+6 e^{e^x} x \log (x)+3 x^2 \log (x)\right ) \log \left (\frac {\log (x)}{2}\right )} \, dx=\frac {1}{3} \left (-\frac {3 x}{e^{e^x}+x}+x (3+x+3 \log (x)) \log \left (\log \left (\frac {\log (x)}{2}\right )\right )\right ) \] Input:

Integrate[(3*x^2 + x^3 + 3*x^2*Log[x] + E^(2*E^x)*(3 + x + 3*Log[x]) + E^E 
^x*(6*x + 2*x^2 + 6*x*Log[x]) + E^E^x*(-3 + 3*E^x*x)*Log[x]*Log[Log[x]/2] 
+ ((6*x^2 + 2*x^3)*Log[x] + 3*x^2*Log[x]^2 + E^(2*E^x)*((6 + 2*x)*Log[x] + 
 3*Log[x]^2) + E^E^x*((12*x + 4*x^2)*Log[x] + 6*x*Log[x]^2))*Log[Log[x]/2] 
*Log[Log[Log[x]/2]])/((3*E^(2*E^x)*Log[x] + 6*E^E^x*x*Log[x] + 3*x^2*Log[x 
])*Log[Log[x]/2]),x]
 

Output:

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

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.

\(\displaystyle \int \frac {x^3+3 x^2+3 x^2 \log (x)+e^{e^x} \left (2 x^2+6 x+6 x \log (x)\right )+\left (3 x^2 \log ^2(x)+e^{e^x} \left (\left (4 x^2+12 x\right ) \log (x)+6 x \log ^2(x)\right )+\left (2 x^3+6 x^2\right ) \log (x)+e^{2 e^x} \left (3 \log ^2(x)+(2 x+6) \log (x)\right )\right ) \log \left (\frac {\log (x)}{2}\right ) \log \left (\log \left (\frac {\log (x)}{2}\right )\right )+e^{2 e^x} (x+3 \log (x)+3)+e^{e^x} \left (3 e^x x-3\right ) \log (x) \log \left (\frac {\log (x)}{2}\right )}{\left (3 x^2 \log (x)+6 e^{e^x} x \log (x)+3 e^{2 e^x} \log (x)\right ) \log \left (\frac {\log (x)}{2}\right )} \, dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2009

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

Input:

Int[(3*x^2 + x^3 + 3*x^2*Log[x] + E^(2*E^x)*(3 + x + 3*Log[x]) + E^E^x*(6* 
x + 2*x^2 + 6*x*Log[x]) + E^E^x*(-3 + 3*E^x*x)*Log[x]*Log[Log[x]/2] + ((6* 
x^2 + 2*x^3)*Log[x] + 3*x^2*Log[x]^2 + E^(2*E^x)*((6 + 2*x)*Log[x] + 3*Log 
[x]^2) + E^E^x*((12*x + 4*x^2)*Log[x] + 6*x*Log[x]^2))*Log[Log[x]/2]*Log[L 
og[Log[x]/2]])/((3*E^(2*E^x)*Log[x] + 6*E^E^x*x*Log[x] + 3*x^2*Log[x])*Log 
[Log[x]/2]),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94

\[\left (x \ln \left (x \right )+\frac {x^{2}}{3}+x \right ) \ln \left (\ln \left (\frac {\ln \left (x \right )}{2}\right )\right )-\frac {x}{x +{\mathrm e}^{{\mathrm e}^{x}}}\]

Input:

int((((3*ln(x)^2+(2*x+6)*ln(x))*exp(exp(x))^2+(6*x*ln(x)^2+(4*x^2+12*x)*ln 
(x))*exp(exp(x))+3*x^2*ln(x)^2+(2*x^3+6*x^2)*ln(x))*ln(1/2*ln(x))*ln(ln(1/ 
2*ln(x)))+(3*exp(x)*x-3)*ln(x)*exp(exp(x))*ln(1/2*ln(x))+(3*ln(x)+3+x)*exp 
(exp(x))^2+(6*x*ln(x)+2*x^2+6*x)*exp(exp(x))+3*x^2*ln(x)+x^3+3*x^2)/(3*ln( 
x)*exp(exp(x))^2+6*x*ln(x)*exp(exp(x))+3*x^2*ln(x))/ln(1/2*ln(x)),x)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.62 \[ \int \frac {3 x^2+x^3+3 x^2 \log (x)+e^{2 e^x} (3+x+3 \log (x))+e^{e^x} \left (6 x+2 x^2+6 x \log (x)\right )+e^{e^x} \left (-3+3 e^x x\right ) \log (x) \log \left (\frac {\log (x)}{2}\right )+\left (\left (6 x^2+2 x^3\right ) \log (x)+3 x^2 \log ^2(x)+e^{2 e^x} \left ((6+2 x) \log (x)+3 \log ^2(x)\right )+e^{e^x} \left (\left (12 x+4 x^2\right ) \log (x)+6 x \log ^2(x)\right )\right ) \log \left (\frac {\log (x)}{2}\right ) \log \left (\log \left (\frac {\log (x)}{2}\right )\right )}{\left (3 e^{2 e^x} \log (x)+6 e^{e^x} x \log (x)+3 x^2 \log (x)\right ) \log \left (\frac {\log (x)}{2}\right )} \, dx=\frac {{\left (x^{3} + 3 \, x^{2} \log \left (x\right ) + 3 \, x^{2} + {\left (x^{2} + 3 \, x \log \left (x\right ) + 3 \, x\right )} e^{\left (e^{x}\right )}\right )} \log \left (\log \left (\frac {1}{2} \, \log \left (x\right )\right )\right ) - 3 \, x}{3 \, {\left (x + e^{\left (e^{x}\right )}\right )}} \] Input:

integrate((((3*log(x)^2+(2*x+6)*log(x))*exp(exp(x))^2+(6*x*log(x)^2+(4*x^2 
+12*x)*log(x))*exp(exp(x))+3*x^2*log(x)^2+(2*x^3+6*x^2)*log(x))*log(1/2*lo 
g(x))*log(log(1/2*log(x)))+(3*exp(x)*x-3)*log(x)*exp(exp(x))*log(1/2*log(x 
))+(3*log(x)+3+x)*exp(exp(x))^2+(6*x*log(x)+2*x^2+6*x)*exp(exp(x))+3*x^2*l 
og(x)+x^3+3*x^2)/(3*log(x)*exp(exp(x))^2+6*x*log(x)*exp(exp(x))+3*x^2*log( 
x))/log(1/2*log(x)),x, algorithm="fricas")
 

Output:

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

Sympy [A] (verification not implemented)

Time = 1.07 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84 \[ \int \frac {3 x^2+x^3+3 x^2 \log (x)+e^{2 e^x} (3+x+3 \log (x))+e^{e^x} \left (6 x+2 x^2+6 x \log (x)\right )+e^{e^x} \left (-3+3 e^x x\right ) \log (x) \log \left (\frac {\log (x)}{2}\right )+\left (\left (6 x^2+2 x^3\right ) \log (x)+3 x^2 \log ^2(x)+e^{2 e^x} \left ((6+2 x) \log (x)+3 \log ^2(x)\right )+e^{e^x} \left (\left (12 x+4 x^2\right ) \log (x)+6 x \log ^2(x)\right )\right ) \log \left (\frac {\log (x)}{2}\right ) \log \left (\log \left (\frac {\log (x)}{2}\right )\right )}{\left (3 e^{2 e^x} \log (x)+6 e^{e^x} x \log (x)+3 x^2 \log (x)\right ) \log \left (\frac {\log (x)}{2}\right )} \, dx=- \frac {x}{x + e^{e^{x}}} + \left (\frac {x^{2}}{3} + x \log {\left (x \right )} + x\right ) \log {\left (\log {\left (\frac {\log {\left (x \right )}}{2} \right )} \right )} \] Input:

integrate((((3*ln(x)**2+(2*x+6)*ln(x))*exp(exp(x))**2+(6*x*ln(x)**2+(4*x** 
2+12*x)*ln(x))*exp(exp(x))+3*x**2*ln(x)**2+(2*x**3+6*x**2)*ln(x))*ln(1/2*l 
n(x))*ln(ln(1/2*ln(x)))+(3*exp(x)*x-3)*ln(x)*exp(exp(x))*ln(1/2*ln(x))+(3* 
ln(x)+3+x)*exp(exp(x))**2+(6*x*ln(x)+2*x**2+6*x)*exp(exp(x))+3*x**2*ln(x)+ 
x**3+3*x**2)/(3*ln(x)*exp(exp(x))**2+6*x*ln(x)*exp(exp(x))+3*x**2*ln(x))/l 
n(1/2*ln(x)),x)
 

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (27) = 54\).

Time = 0.20 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.72 \[ \int \frac {3 x^2+x^3+3 x^2 \log (x)+e^{2 e^x} (3+x+3 \log (x))+e^{e^x} \left (6 x+2 x^2+6 x \log (x)\right )+e^{e^x} \left (-3+3 e^x x\right ) \log (x) \log \left (\frac {\log (x)}{2}\right )+\left (\left (6 x^2+2 x^3\right ) \log (x)+3 x^2 \log ^2(x)+e^{2 e^x} \left ((6+2 x) \log (x)+3 \log ^2(x)\right )+e^{e^x} \left (\left (12 x+4 x^2\right ) \log (x)+6 x \log ^2(x)\right )\right ) \log \left (\frac {\log (x)}{2}\right ) \log \left (\log \left (\frac {\log (x)}{2}\right )\right )}{\left (3 e^{2 e^x} \log (x)+6 e^{e^x} x \log (x)+3 x^2 \log (x)\right ) \log \left (\frac {\log (x)}{2}\right )} \, dx=\frac {{\left (x^{3} + 3 \, x^{2} \log \left (x\right ) + 3 \, x^{2} + {\left (x^{2} + 3 \, x \log \left (x\right ) + 3 \, x\right )} e^{\left (e^{x}\right )}\right )} \log \left (-\log \left (2\right ) + \log \left (\log \left (x\right )\right )\right ) - 3 \, x}{3 \, {\left (x + e^{\left (e^{x}\right )}\right )}} \] Input:

integrate((((3*log(x)^2+(2*x+6)*log(x))*exp(exp(x))^2+(6*x*log(x)^2+(4*x^2 
+12*x)*log(x))*exp(exp(x))+3*x^2*log(x)^2+(2*x^3+6*x^2)*log(x))*log(1/2*lo 
g(x))*log(log(1/2*log(x)))+(3*exp(x)*x-3)*log(x)*exp(exp(x))*log(1/2*log(x 
))+(3*log(x)+3+x)*exp(exp(x))^2+(6*x*log(x)+2*x^2+6*x)*exp(exp(x))+3*x^2*l 
og(x)+x^3+3*x^2)/(3*log(x)*exp(exp(x))^2+6*x*log(x)*exp(exp(x))+3*x^2*log( 
x))/log(1/2*log(x)),x, algorithm="maxima")
 

Output:

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

Giac [F(-1)]

Timed out. \[ \int \frac {3 x^2+x^3+3 x^2 \log (x)+e^{2 e^x} (3+x+3 \log (x))+e^{e^x} \left (6 x+2 x^2+6 x \log (x)\right )+e^{e^x} \left (-3+3 e^x x\right ) \log (x) \log \left (\frac {\log (x)}{2}\right )+\left (\left (6 x^2+2 x^3\right ) \log (x)+3 x^2 \log ^2(x)+e^{2 e^x} \left ((6+2 x) \log (x)+3 \log ^2(x)\right )+e^{e^x} \left (\left (12 x+4 x^2\right ) \log (x)+6 x \log ^2(x)\right )\right ) \log \left (\frac {\log (x)}{2}\right ) \log \left (\log \left (\frac {\log (x)}{2}\right )\right )}{\left (3 e^{2 e^x} \log (x)+6 e^{e^x} x \log (x)+3 x^2 \log (x)\right ) \log \left (\frac {\log (x)}{2}\right )} \, dx=\text {Timed out} \] Input:

integrate((((3*log(x)^2+(2*x+6)*log(x))*exp(exp(x))^2+(6*x*log(x)^2+(4*x^2 
+12*x)*log(x))*exp(exp(x))+3*x^2*log(x)^2+(2*x^3+6*x^2)*log(x))*log(1/2*lo 
g(x))*log(log(1/2*log(x)))+(3*exp(x)*x-3)*log(x)*exp(exp(x))*log(1/2*log(x 
))+(3*log(x)+3+x)*exp(exp(x))^2+(6*x*log(x)+2*x^2+6*x)*exp(exp(x))+3*x^2*l 
og(x)+x^3+3*x^2)/(3*log(x)*exp(exp(x))^2+6*x*log(x)*exp(exp(x))+3*x^2*log( 
x))/log(1/2*log(x)),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [B] (verification not implemented)

Time = 8.10 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.25 \[ \int \frac {3 x^2+x^3+3 x^2 \log (x)+e^{2 e^x} (3+x+3 \log (x))+e^{e^x} \left (6 x+2 x^2+6 x \log (x)\right )+e^{e^x} \left (-3+3 e^x x\right ) \log (x) \log \left (\frac {\log (x)}{2}\right )+\left (\left (6 x^2+2 x^3\right ) \log (x)+3 x^2 \log ^2(x)+e^{2 e^x} \left ((6+2 x) \log (x)+3 \log ^2(x)\right )+e^{e^x} \left (\left (12 x+4 x^2\right ) \log (x)+6 x \log ^2(x)\right )\right ) \log \left (\frac {\log (x)}{2}\right ) \log \left (\log \left (\frac {\log (x)}{2}\right )\right )}{\left (3 e^{2 e^x} \log (x)+6 e^{e^x} x \log (x)+3 x^2 \log (x)\right ) \log \left (\frac {\log (x)}{2}\right )} \, dx=\ln \left (\ln \left (\frac {\ln \left (x\right )}{2}\right )\right )\,\left (\frac {x^3+6\,x^2}{3\,x}-x+x\,\ln \left (x\right )\right )-\frac {x}{x+{\mathrm {e}}^{{\mathrm {e}}^x}} \] Input:

int((3*x^2*log(x) + exp(2*exp(x))*(x + 3*log(x) + 3) + exp(exp(x))*(6*x + 
6*x*log(x) + 2*x^2) + 3*x^2 + x^3 + log(log(log(x)/2))*log(log(x)/2)*(log( 
x)*(6*x^2 + 2*x^3) + 3*x^2*log(x)^2 + exp(2*exp(x))*(3*log(x)^2 + log(x)*( 
2*x + 6)) + exp(exp(x))*(6*x*log(x)^2 + log(x)*(12*x + 4*x^2))) + exp(exp( 
x))*log(log(x)/2)*log(x)*(3*x*exp(x) - 3))/(log(log(x)/2)*(3*x^2*log(x) + 
3*exp(2*exp(x))*log(x) + 6*x*exp(exp(x))*log(x))),x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 101, normalized size of antiderivative = 3.16 \[ \int \frac {3 x^2+x^3+3 x^2 \log (x)+e^{2 e^x} (3+x+3 \log (x))+e^{e^x} \left (6 x+2 x^2+6 x \log (x)\right )+e^{e^x} \left (-3+3 e^x x\right ) \log (x) \log \left (\frac {\log (x)}{2}\right )+\left (\left (6 x^2+2 x^3\right ) \log (x)+3 x^2 \log ^2(x)+e^{2 e^x} \left ((6+2 x) \log (x)+3 \log ^2(x)\right )+e^{e^x} \left (\left (12 x+4 x^2\right ) \log (x)+6 x \log ^2(x)\right )\right ) \log \left (\frac {\log (x)}{2}\right ) \log \left (\log \left (\frac {\log (x)}{2}\right )\right )}{\left (3 e^{2 e^x} \log (x)+6 e^{e^x} x \log (x)+3 x^2 \log (x)\right ) \log \left (\frac {\log (x)}{2}\right )} \, dx=\frac {3 e^{e^{x}} \mathrm {log}\left (\mathrm {log}\left (\frac {\mathrm {log}\left (x \right )}{2}\right )\right ) \mathrm {log}\left (x \right ) x +e^{e^{x}} \mathrm {log}\left (\mathrm {log}\left (\frac {\mathrm {log}\left (x \right )}{2}\right )\right ) x^{2}+3 e^{e^{x}} \mathrm {log}\left (\mathrm {log}\left (\frac {\mathrm {log}\left (x \right )}{2}\right )\right ) x +3 e^{e^{x}}+3 \,\mathrm {log}\left (\mathrm {log}\left (\frac {\mathrm {log}\left (x \right )}{2}\right )\right ) \mathrm {log}\left (x \right ) x^{2}+\mathrm {log}\left (\mathrm {log}\left (\frac {\mathrm {log}\left (x \right )}{2}\right )\right ) x^{3}+3 \,\mathrm {log}\left (\mathrm {log}\left (\frac {\mathrm {log}\left (x \right )}{2}\right )\right ) x^{2}}{3 e^{e^{x}}+3 x} \] Input:

int((((3*log(x)^2+(2*x+6)*log(x))*exp(exp(x))^2+(6*x*log(x)^2+(4*x^2+12*x) 
*log(x))*exp(exp(x))+3*x^2*log(x)^2+(2*x^3+6*x^2)*log(x))*log(1/2*log(x))* 
log(log(1/2*log(x)))+(3*exp(x)*x-3)*log(x)*exp(exp(x))*log(1/2*log(x))+(3* 
log(x)+3+x)*exp(exp(x))^2+(6*x*log(x)+2*x^2+6*x)*exp(exp(x))+3*x^2*log(x)+ 
x^3+3*x^2)/(3*log(x)*exp(exp(x))^2+6*x*log(x)*exp(exp(x))+3*x^2*log(x))/lo 
g(1/2*log(x)),x)
 

Output:

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