\(\int \frac {-108 x^2+e^{\frac {e^x}{x}} (e^x (162-162 x)-162 x) \log (x)+(108 x^2-486 x^3) \log (x)+(36 x+(-36 x+216 x^2) \log (x)) \log (\frac {x}{2 \log (x)})-18 x \log (x) \log ^2(\frac {x}{2 \log (x)})}{81 e^{\frac {2 e^x}{x}} x^3 \log (x)+162 e^{\frac {e^x}{x}} x^5 \log (x)+81 x^7 \log (x)+(-108 e^{\frac {e^x}{x}} x^4 \log (x)-108 x^6 \log (x)) \log (\frac {x}{2 \log (x)})+(18 e^{\frac {e^x}{x}} x^3 \log (x)+54 x^5 \log (x)) \log ^2(\frac {x}{2 \log (x)})-12 x^4 \log (x) \log ^3(\frac {x}{2 \log (x)})+x^3 \log (x) \log ^4(\frac {x}{2 \log (x)})} \, dx\) [635]

Optimal result

Integrand size = 242, antiderivative size = 35 \[ \int \frac {-108 x^2+e^{\frac {e^x}{x}} \left (e^x (162-162 x)-162 x\right ) \log (x)+\left (108 x^2-486 x^3\right ) \log (x)+\left (36 x+\left (-36 x+216 x^2\right ) \log (x)\right ) \log \left (\frac {x}{2 \log (x)}\right )-18 x \log (x) \log ^2\left (\frac {x}{2 \log (x)}\right )}{81 e^{\frac {2 e^x}{x}} x^3 \log (x)+162 e^{\frac {e^x}{x}} x^5 \log (x)+81 x^7 \log (x)+\left (-108 e^{\frac {e^x}{x}} x^4 \log (x)-108 x^6 \log (x)\right ) \log \left (\frac {x}{2 \log (x)}\right )+\left (18 e^{\frac {e^x}{x}} x^3 \log (x)+54 x^5 \log (x)\right ) \log ^2\left (\frac {x}{2 \log (x)}\right )-12 x^4 \log (x) \log ^3\left (\frac {x}{2 \log (x)}\right )+x^3 \log (x) \log ^4\left (\frac {x}{2 \log (x)}\right )} \, dx=\frac {2}{x \left (e^{\frac {e^x}{x}}+\left (x-\frac {1}{3} \log \left (\frac {x}{2 \log (x)}\right )\right )^2\right )} \] Output:

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

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.40 \[ \int \frac {-108 x^2+e^{\frac {e^x}{x}} \left (e^x (162-162 x)-162 x\right ) \log (x)+\left (108 x^2-486 x^3\right ) \log (x)+\left (36 x+\left (-36 x+216 x^2\right ) \log (x)\right ) \log \left (\frac {x}{2 \log (x)}\right )-18 x \log (x) \log ^2\left (\frac {x}{2 \log (x)}\right )}{81 e^{\frac {2 e^x}{x}} x^3 \log (x)+162 e^{\frac {e^x}{x}} x^5 \log (x)+81 x^7 \log (x)+\left (-108 e^{\frac {e^x}{x}} x^4 \log (x)-108 x^6 \log (x)\right ) \log \left (\frac {x}{2 \log (x)}\right )+\left (18 e^{\frac {e^x}{x}} x^3 \log (x)+54 x^5 \log (x)\right ) \log ^2\left (\frac {x}{2 \log (x)}\right )-12 x^4 \log (x) \log ^3\left (\frac {x}{2 \log (x)}\right )+x^3 \log (x) \log ^4\left (\frac {x}{2 \log (x)}\right )} \, dx=\frac {18}{x \left (9 e^{\frac {e^x}{x}}+9 x^2-6 x \log \left (\frac {x}{2 \log (x)}\right )+\log ^2\left (\frac {x}{2 \log (x)}\right )\right )} \] Input:

Integrate[(-108*x^2 + E^(E^x/x)*(E^x*(162 - 162*x) - 162*x)*Log[x] + (108* 
x^2 - 486*x^3)*Log[x] + (36*x + (-36*x + 216*x^2)*Log[x])*Log[x/(2*Log[x]) 
] - 18*x*Log[x]*Log[x/(2*Log[x])]^2)/(81*E^((2*E^x)/x)*x^3*Log[x] + 162*E^ 
(E^x/x)*x^5*Log[x] + 81*x^7*Log[x] + (-108*E^(E^x/x)*x^4*Log[x] - 108*x^6* 
Log[x])*Log[x/(2*Log[x])] + (18*E^(E^x/x)*x^3*Log[x] + 54*x^5*Log[x])*Log[ 
x/(2*Log[x])]^2 - 12*x^4*Log[x]*Log[x/(2*Log[x])]^3 + x^3*Log[x]*Log[x/(2* 
Log[x])]^4),x]
 

Output:

18/(x*(9*E^(E^x/x) + 9*x^2 - 6*x*Log[x/(2*Log[x])] + Log[x/(2*Log[x])]^2))
 

Rubi [A] (verified)

Time = 3.32 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.40, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.008, Rules used = {7292, 7238}

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.

\(\displaystyle \int \frac {-108 x^2+\left (\left (216 x^2-36 x\right ) \log (x)+36 x\right ) \log \left (\frac {x}{2 \log (x)}\right )+\left (108 x^2-486 x^3\right ) \log (x)-18 x \log (x) \log ^2\left (\frac {x}{2 \log (x)}\right )+e^{\frac {e^x}{x}} \left (e^x (162-162 x)-162 x\right ) \log (x)}{81 x^7 \log (x)+162 e^{\frac {e^x}{x}} x^5 \log (x)-12 x^4 \log (x) \log ^3\left (\frac {x}{2 \log (x)}\right )+x^3 \log (x) \log ^4\left (\frac {x}{2 \log (x)}\right )+81 e^{\frac {2 e^x}{x}} x^3 \log (x)+\left (-108 x^6 \log (x)-108 e^{\frac {e^x}{x}} x^4 \log (x)\right ) \log \left (\frac {x}{2 \log (x)}\right )+\left (54 x^5 \log (x)+18 e^{\frac {e^x}{x}} x^3 \log (x)\right ) \log ^2\left (\frac {x}{2 \log (x)}\right )} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {-108 x^2+\left (\left (216 x^2-36 x\right ) \log (x)+36 x\right ) \log \left (\frac {x}{2 \log (x)}\right )+\left (108 x^2-486 x^3\right ) \log (x)-18 x \log (x) \log ^2\left (\frac {x}{2 \log (x)}\right )+e^{\frac {e^x}{x}} \left (e^x (162-162 x)-162 x\right ) \log (x)}{x^3 \log (x) \left (9 x^2+9 e^{\frac {e^x}{x}}+\log ^2\left (\frac {x}{2 \log (x)}\right )-6 x \log \left (\frac {x}{2 \log (x)}\right )\right )^2}dx\)

\(\Big \downarrow \) 7238

\(\displaystyle \frac {18}{x \left (9 x^2+9 e^{\frac {e^x}{x}}+\log ^2\left (\frac {x}{2 \log (x)}\right )-6 x \log \left (\frac {x}{2 \log (x)}\right )\right )}\)

Input:

Int[(-108*x^2 + E^(E^x/x)*(E^x*(162 - 162*x) - 162*x)*Log[x] + (108*x^2 - 
486*x^3)*Log[x] + (36*x + (-36*x + 216*x^2)*Log[x])*Log[x/(2*Log[x])] - 18 
*x*Log[x]*Log[x/(2*Log[x])]^2)/(81*E^((2*E^x)/x)*x^3*Log[x] + 162*E^(E^x/x 
)*x^5*Log[x] + 81*x^7*Log[x] + (-108*E^(E^x/x)*x^4*Log[x] - 108*x^6*Log[x] 
)*Log[x/(2*Log[x])] + (18*E^(E^x/x)*x^3*Log[x] + 54*x^5*Log[x])*Log[x/(2*L 
og[x])]^2 - 12*x^4*Log[x]*Log[x/(2*Log[x])]^3 + x^3*Log[x]*Log[x/(2*Log[x] 
)]^4),x]
 

Output:

18/(x*(9*E^(E^x/x) + 9*x^2 - 6*x*Log[x/(2*Log[x])] + Log[x/(2*Log[x])]^2))
 

Defintions of rubi rules used

rule 7238

Int[(u_)*(y_)^(m_.)*(z_)^(n_.), x_Symbol] :> With[{q = DerivativeDivides[y* 
z, u*z^(n - m), x]}, Simp[q*y^(m + 1)*(z^(m + 1)/(m + 1)), x] /;  !FalseQ[q 
]] /; FreeQ[{m, n}, x] && NeQ[m, -1]
 

rule 7292

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

Maple [C] (warning: unable to verify)

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

Time = 0.09 (sec) , antiderivative size = 680, normalized size of antiderivative = 19.43

\[\text {Expression too large to display}\]

Input:

int((-18*x*ln(x)*ln(1/2*x/ln(x))^2+((216*x^2-36*x)*ln(x)+36*x)*ln(1/2*x/ln 
(x))+((-162*x+162)*exp(x)-162*x)*ln(x)*exp(exp(x)/x)+(-486*x^3+108*x^2)*ln 
(x)-108*x^2)/(x^3*ln(x)*ln(1/2*x/ln(x))^4-12*x^4*ln(x)*ln(1/2*x/ln(x))^3+( 
18*x^3*ln(x)*exp(exp(x)/x)+54*x^5*ln(x))*ln(1/2*x/ln(x))^2+(-108*x^4*ln(x) 
*exp(exp(x)/x)-108*x^6*ln(x))*ln(1/2*x/ln(x))+81*x^3*ln(x)*exp(exp(x)/x)^2 
+162*x^5*ln(x)*exp(exp(x)/x)+81*x^7*ln(x)),x)
 

Output:

72/x/(8*ln(2)*ln(ln(x))-8*ln(2)*ln(x)+24*x*ln(ln(x))-8*ln(x)*ln(ln(x))+24* 
x*ln(2)+36*exp(exp(x)/x)+4*ln(ln(x))^2-24*x*ln(x)+4*ln(x)^2+36*x^2+4*ln(2) 
^2+2*Pi^2*csgn(I*x/ln(x))^3*csgn(I*x)^2*csgn(I/ln(x))+2*Pi^2*csgn(I*x/ln(x 
))^5*csgn(I*x)+4*I*ln(2)*Pi*csgn(I*x/ln(x))^3+12*I*Pi*x*csgn(I*x/ln(x))^3- 
4*I*ln(x)*Pi*csgn(I*x/ln(x))^3+4*I*ln(ln(x))*Pi*csgn(I*x/ln(x))^3-Pi^2*csg 
n(I*x/ln(x))^6+2*Pi^2*csgn(I*x/ln(x))^3*csgn(I/ln(x))^2*csgn(I*x)+2*Pi^2*c 
sgn(I*x/ln(x))^5*csgn(I/ln(x))-Pi^2*csgn(I*x/ln(x))^2*csgn(I*x)^2*csgn(I/l 
n(x))^2-4*Pi^2*csgn(I*x/ln(x))^4*csgn(I*x)*csgn(I/ln(x))-Pi^2*csgn(I*x/ln( 
x))^4*csgn(I*x)^2-Pi^2*csgn(I*x/ln(x))^4*csgn(I/ln(x))^2-12*I*Pi*x*csgn(I* 
x/ln(x))^2*csgn(I*x)-12*I*Pi*x*csgn(I*x/ln(x))^2*csgn(I/ln(x))-4*I*ln(2)*P 
i*csgn(I*x/ln(x))^2*csgn(I*x)-4*I*ln(2)*Pi*csgn(I*x/ln(x))^2*csgn(I/ln(x)) 
+4*I*ln(x)*Pi*csgn(I*x/ln(x))^2*csgn(I*x)+4*I*ln(x)*Pi*csgn(I*x/ln(x))^2*c 
sgn(I/ln(x))-4*I*ln(ln(x))*Pi*csgn(I*x/ln(x))^2*csgn(I*x)-4*I*ln(ln(x))*Pi 
*csgn(I*x/ln(x))^2*csgn(I/ln(x))+4*I*ln(2)*Pi*csgn(I*x/ln(x))*csgn(I*x)*cs 
gn(I/ln(x))+12*I*Pi*x*csgn(I*x/ln(x))*csgn(I*x)*csgn(I/ln(x))-4*I*ln(x)*Pi 
*csgn(I*x/ln(x))*csgn(I*x)*csgn(I/ln(x))+4*I*ln(ln(x))*Pi*csgn(I*x/ln(x))* 
csgn(I*x)*csgn(I/ln(x)))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.29 \[ \int \frac {-108 x^2+e^{\frac {e^x}{x}} \left (e^x (162-162 x)-162 x\right ) \log (x)+\left (108 x^2-486 x^3\right ) \log (x)+\left (36 x+\left (-36 x+216 x^2\right ) \log (x)\right ) \log \left (\frac {x}{2 \log (x)}\right )-18 x \log (x) \log ^2\left (\frac {x}{2 \log (x)}\right )}{81 e^{\frac {2 e^x}{x}} x^3 \log (x)+162 e^{\frac {e^x}{x}} x^5 \log (x)+81 x^7 \log (x)+\left (-108 e^{\frac {e^x}{x}} x^4 \log (x)-108 x^6 \log (x)\right ) \log \left (\frac {x}{2 \log (x)}\right )+\left (18 e^{\frac {e^x}{x}} x^3 \log (x)+54 x^5 \log (x)\right ) \log ^2\left (\frac {x}{2 \log (x)}\right )-12 x^4 \log (x) \log ^3\left (\frac {x}{2 \log (x)}\right )+x^3 \log (x) \log ^4\left (\frac {x}{2 \log (x)}\right )} \, dx=\frac {18}{9 \, x^{3} - 6 \, x^{2} \log \left (\frac {x}{2 \, \log \left (x\right )}\right ) + x \log \left (\frac {x}{2 \, \log \left (x\right )}\right )^{2} + 9 \, x e^{\left (\frac {e^{x}}{x}\right )}} \] Input:

integrate((-18*x*log(x)*log(1/2*x/log(x))^2+((216*x^2-36*x)*log(x)+36*x)*l 
og(1/2*x/log(x))+((-162*x+162)*exp(x)-162*x)*log(x)*exp(exp(x)/x)+(-486*x^ 
3+108*x^2)*log(x)-108*x^2)/(x^3*log(x)*log(1/2*x/log(x))^4-12*x^4*log(x)*l 
og(1/2*x/log(x))^3+(18*x^3*log(x)*exp(exp(x)/x)+54*x^5*log(x))*log(1/2*x/l 
og(x))^2+(-108*x^4*log(x)*exp(exp(x)/x)-108*x^6*log(x))*log(1/2*x/log(x))+ 
81*x^3*log(x)*exp(exp(x)/x)^2+162*x^5*log(x)*exp(exp(x)/x)+81*x^7*log(x)), 
x, algorithm="fricas")
 

Output:

18/(9*x^3 - 6*x^2*log(1/2*x/log(x)) + x*log(1/2*x/log(x))^2 + 9*x*e^(e^x/x 
))
 

Sympy [A] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.17 \[ \int \frac {-108 x^2+e^{\frac {e^x}{x}} \left (e^x (162-162 x)-162 x\right ) \log (x)+\left (108 x^2-486 x^3\right ) \log (x)+\left (36 x+\left (-36 x+216 x^2\right ) \log (x)\right ) \log \left (\frac {x}{2 \log (x)}\right )-18 x \log (x) \log ^2\left (\frac {x}{2 \log (x)}\right )}{81 e^{\frac {2 e^x}{x}} x^3 \log (x)+162 e^{\frac {e^x}{x}} x^5 \log (x)+81 x^7 \log (x)+\left (-108 e^{\frac {e^x}{x}} x^4 \log (x)-108 x^6 \log (x)\right ) \log \left (\frac {x}{2 \log (x)}\right )+\left (18 e^{\frac {e^x}{x}} x^3 \log (x)+54 x^5 \log (x)\right ) \log ^2\left (\frac {x}{2 \log (x)}\right )-12 x^4 \log (x) \log ^3\left (\frac {x}{2 \log (x)}\right )+x^3 \log (x) \log ^4\left (\frac {x}{2 \log (x)}\right )} \, dx=\frac {18}{9 x^{3} - 6 x^{2} \log {\left (\frac {x}{2 \log {\left (x \right )}} \right )} + 9 x e^{\frac {e^{x}}{x}} + x \log {\left (\frac {x}{2 \log {\left (x \right )}} \right )}^{2}} \] Input:

integrate((-18*x*ln(x)*ln(1/2*x/ln(x))**2+((216*x**2-36*x)*ln(x)+36*x)*ln( 
1/2*x/ln(x))+((-162*x+162)*exp(x)-162*x)*ln(x)*exp(exp(x)/x)+(-486*x**3+10 
8*x**2)*ln(x)-108*x**2)/(x**3*ln(x)*ln(1/2*x/ln(x))**4-12*x**4*ln(x)*ln(1/ 
2*x/ln(x))**3+(18*x**3*ln(x)*exp(exp(x)/x)+54*x**5*ln(x))*ln(1/2*x/ln(x))* 
*2+(-108*x**4*ln(x)*exp(exp(x)/x)-108*x**6*ln(x))*ln(1/2*x/ln(x))+81*x**3* 
ln(x)*exp(exp(x)/x)**2+162*x**5*ln(x)*exp(exp(x)/x)+81*x**7*ln(x)),x)
 

Output:

18/(9*x**3 - 6*x**2*log(x/(2*log(x))) + 9*x*exp(exp(x)/x) + x*log(x/(2*log 
(x)))**2)
                                                                                    
                                                                                    
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (33) = 66\).

Time = 0.39 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.29 \[ \int \frac {-108 x^2+e^{\frac {e^x}{x}} \left (e^x (162-162 x)-162 x\right ) \log (x)+\left (108 x^2-486 x^3\right ) \log (x)+\left (36 x+\left (-36 x+216 x^2\right ) \log (x)\right ) \log \left (\frac {x}{2 \log (x)}\right )-18 x \log (x) \log ^2\left (\frac {x}{2 \log (x)}\right )}{81 e^{\frac {2 e^x}{x}} x^3 \log (x)+162 e^{\frac {e^x}{x}} x^5 \log (x)+81 x^7 \log (x)+\left (-108 e^{\frac {e^x}{x}} x^4 \log (x)-108 x^6 \log (x)\right ) \log \left (\frac {x}{2 \log (x)}\right )+\left (18 e^{\frac {e^x}{x}} x^3 \log (x)+54 x^5 \log (x)\right ) \log ^2\left (\frac {x}{2 \log (x)}\right )-12 x^4 \log (x) \log ^3\left (\frac {x}{2 \log (x)}\right )+x^3 \log (x) \log ^4\left (\frac {x}{2 \log (x)}\right )} \, dx=\frac {18}{9 \, x^{3} + 6 \, x^{2} \log \left (2\right ) + x \log \left (2\right )^{2} + x \log \left (x\right )^{2} + x \log \left (\log \left (x\right )\right )^{2} + 9 \, x e^{\left (\frac {e^{x}}{x}\right )} - 2 \, {\left (3 \, x^{2} + x \log \left (2\right )\right )} \log \left (x\right ) + 2 \, {\left (3 \, x^{2} + x \log \left (2\right ) - x \log \left (x\right )\right )} \log \left (\log \left (x\right )\right )} \] Input:

integrate((-18*x*log(x)*log(1/2*x/log(x))^2+((216*x^2-36*x)*log(x)+36*x)*l 
og(1/2*x/log(x))+((-162*x+162)*exp(x)-162*x)*log(x)*exp(exp(x)/x)+(-486*x^ 
3+108*x^2)*log(x)-108*x^2)/(x^3*log(x)*log(1/2*x/log(x))^4-12*x^4*log(x)*l 
og(1/2*x/log(x))^3+(18*x^3*log(x)*exp(exp(x)/x)+54*x^5*log(x))*log(1/2*x/l 
og(x))^2+(-108*x^4*log(x)*exp(exp(x)/x)-108*x^6*log(x))*log(1/2*x/log(x))+ 
81*x^3*log(x)*exp(exp(x)/x)^2+162*x^5*log(x)*exp(exp(x)/x)+81*x^7*log(x)), 
x, algorithm="maxima")
 

Output:

18/(9*x^3 + 6*x^2*log(2) + x*log(2)^2 + x*log(x)^2 + x*log(log(x))^2 + 9*x 
*e^(e^x/x) - 2*(3*x^2 + x*log(2))*log(x) + 2*(3*x^2 + x*log(2) - x*log(x)) 
*log(log(x)))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 7627 vs. \(2 (33) = 66\).

Time = 0.66 (sec) , antiderivative size = 7627, normalized size of antiderivative = 217.91 \[ \int \frac {-108 x^2+e^{\frac {e^x}{x}} \left (e^x (162-162 x)-162 x\right ) \log (x)+\left (108 x^2-486 x^3\right ) \log (x)+\left (36 x+\left (-36 x+216 x^2\right ) \log (x)\right ) \log \left (\frac {x}{2 \log (x)}\right )-18 x \log (x) \log ^2\left (\frac {x}{2 \log (x)}\right )}{81 e^{\frac {2 e^x}{x}} x^3 \log (x)+162 e^{\frac {e^x}{x}} x^5 \log (x)+81 x^7 \log (x)+\left (-108 e^{\frac {e^x}{x}} x^4 \log (x)-108 x^6 \log (x)\right ) \log \left (\frac {x}{2 \log (x)}\right )+\left (18 e^{\frac {e^x}{x}} x^3 \log (x)+54 x^5 \log (x)\right ) \log ^2\left (\frac {x}{2 \log (x)}\right )-12 x^4 \log (x) \log ^3\left (\frac {x}{2 \log (x)}\right )+x^3 \log (x) \log ^4\left (\frac {x}{2 \log (x)}\right )} \, dx=\text {Too large to display} \] Input:

integrate((-18*x*log(x)*log(1/2*x/log(x))^2+((216*x^2-36*x)*log(x)+36*x)*l 
og(1/2*x/log(x))+((-162*x+162)*exp(x)-162*x)*log(x)*exp(exp(x)/x)+(-486*x^ 
3+108*x^2)*log(x)-108*x^2)/(x^3*log(x)*log(1/2*x/log(x))^4-12*x^4*log(x)*l 
og(1/2*x/log(x))^3+(18*x^3*log(x)*exp(exp(x)/x)+54*x^5*log(x))*log(1/2*x/l 
og(x))^2+(-108*x^4*log(x)*exp(exp(x)/x)-108*x^6*log(x))*log(1/2*x/log(x))+ 
81*x^3*log(x)*exp(exp(x)/x)^2+162*x^5*log(x)*exp(exp(x)/x)+81*x^7*log(x)), 
x, algorithm="giac")
 

Output:

18*(27*x^4*e^(2*x)*log(x) - 54*x^4*e^x*log(x) + 27*x^3*e^(2*x)*log(2)*log( 
x) - 36*x^3*e^x*log(2)*log(x) + 9*x^2*e^(2*x)*log(2)^2*log(x) - 6*x^2*e^x* 
log(2)^2*log(x) + x*e^(2*x)*log(2)^3*log(x) - 27*x^3*e^(2*x)*log(x)^2 + 36 
*x^3*e^x*log(x)^2 - 18*x^2*e^(2*x)*log(2)*log(x)^2 + 12*x^2*e^x*log(2)*log 
(x)^2 - 3*x*e^(2*x)*log(2)^2*log(x)^2 + 9*x^2*e^(2*x)*log(x)^3 - 6*x^2*e^x 
*log(x)^3 + 3*x*e^(2*x)*log(2)*log(x)^3 - x*e^(2*x)*log(x)^4 + 27*x^3*e^(2 
*x)*log(x)*log(log(x)) - 36*x^3*e^x*log(x)*log(log(x)) + 18*x^2*e^(2*x)*lo 
g(2)*log(x)*log(log(x)) - 12*x^2*e^x*log(2)*log(x)*log(log(x)) + 3*x*e^(2* 
x)*log(2)^2*log(x)*log(log(x)) - 18*x^2*e^(2*x)*log(x)^2*log(log(x)) + 12* 
x^2*e^x*log(x)^2*log(log(x)) - 6*x*e^(2*x)*log(2)*log(x)^2*log(log(x)) + 3 
*x*e^(2*x)*log(x)^3*log(log(x)) + 9*x^2*e^(2*x)*log(x)*log(log(x))^2 - 6*x 
^2*e^x*log(x)*log(log(x))^2 + 3*x*e^(2*x)*log(2)*log(x)*log(log(x))^2 - 3* 
x*e^(2*x)*log(x)^2*log(log(x))^2 + x*e^(2*x)*log(x)*log(log(x))^3 - 27*x^3 
*e^(2*x)*log(x) + 18*x^3*e^x*log(x) - 27*x^2*e^(2*x)*log(2)*log(x) + 12*x^ 
2*e^x*log(2)*log(x) - 9*x*e^(2*x)*log(2)^2*log(x) + 2*x*e^x*log(2)^2*log(x 
) - e^(2*x)*log(2)^3*log(x) + 27*x^2*e^(2*x)*log(x)^2 - 12*x^2*e^x*log(x)^ 
2 + 18*x*e^(2*x)*log(2)*log(x)^2 - 4*x*e^x*log(2)*log(x)^2 + 3*e^(2*x)*log 
(2)^2*log(x)^2 - 9*x*e^(2*x)*log(x)^3 + 2*x*e^x*log(x)^3 - 3*e^(2*x)*log(2 
)*log(x)^3 + e^(2*x)*log(x)^4 - 27*x^2*e^(2*x)*log(x)*log(log(x)) + 12*x^2 
*e^x*log(x)*log(log(x)) - 18*x*e^(2*x)*log(2)*log(x)*log(log(x)) + 4*x*...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {-108 x^2+e^{\frac {e^x}{x}} \left (e^x (162-162 x)-162 x\right ) \log (x)+\left (108 x^2-486 x^3\right ) \log (x)+\left (36 x+\left (-36 x+216 x^2\right ) \log (x)\right ) \log \left (\frac {x}{2 \log (x)}\right )-18 x \log (x) \log ^2\left (\frac {x}{2 \log (x)}\right )}{81 e^{\frac {2 e^x}{x}} x^3 \log (x)+162 e^{\frac {e^x}{x}} x^5 \log (x)+81 x^7 \log (x)+\left (-108 e^{\frac {e^x}{x}} x^4 \log (x)-108 x^6 \log (x)\right ) \log \left (\frac {x}{2 \log (x)}\right )+\left (18 e^{\frac {e^x}{x}} x^3 \log (x)+54 x^5 \log (x)\right ) \log ^2\left (\frac {x}{2 \log (x)}\right )-12 x^4 \log (x) \log ^3\left (\frac {x}{2 \log (x)}\right )+x^3 \log (x) \log ^4\left (\frac {x}{2 \log (x)}\right )} \, dx=\int -\frac {108\,x^2-\ln \left (\frac {x}{2\,\ln \left (x\right )}\right )\,\left (36\,x-\ln \left (x\right )\,\left (36\,x-216\,x^2\right )\right )-\ln \left (x\right )\,\left (108\,x^2-486\,x^3\right )+{\mathrm {e}}^{\frac {{\mathrm {e}}^x}{x}}\,\ln \left (x\right )\,\left (162\,x+{\mathrm {e}}^x\,\left (162\,x-162\right )\right )+18\,x\,{\ln \left (\frac {x}{2\,\ln \left (x\right )}\right )}^2\,\ln \left (x\right )}{81\,x^7\,\ln \left (x\right )+{\ln \left (\frac {x}{2\,\ln \left (x\right )}\right )}^2\,\left (54\,x^5\,\ln \left (x\right )+18\,x^3\,{\mathrm {e}}^{\frac {{\mathrm {e}}^x}{x}}\,\ln \left (x\right )\right )-\ln \left (\frac {x}{2\,\ln \left (x\right )}\right )\,\left (108\,x^6\,\ln \left (x\right )+108\,x^4\,{\mathrm {e}}^{\frac {{\mathrm {e}}^x}{x}}\,\ln \left (x\right )\right )+x^3\,{\ln \left (\frac {x}{2\,\ln \left (x\right )}\right )}^4\,\ln \left (x\right )-12\,x^4\,{\ln \left (\frac {x}{2\,\ln \left (x\right )}\right )}^3\,\ln \left (x\right )+81\,x^3\,{\mathrm {e}}^{\frac {2\,{\mathrm {e}}^x}{x}}\,\ln \left (x\right )+162\,x^5\,{\mathrm {e}}^{\frac {{\mathrm {e}}^x}{x}}\,\ln \left (x\right )} \,d x \] Input:

int(-(108*x^2 - log(x/(2*log(x)))*(36*x - log(x)*(36*x - 216*x^2)) - log(x 
)*(108*x^2 - 486*x^3) + exp(exp(x)/x)*log(x)*(162*x + exp(x)*(162*x - 162) 
) + 18*x*log(x/(2*log(x)))^2*log(x))/(81*x^7*log(x) + log(x/(2*log(x)))^2* 
(54*x^5*log(x) + 18*x^3*exp(exp(x)/x)*log(x)) - log(x/(2*log(x)))*(108*x^6 
*log(x) + 108*x^4*exp(exp(x)/x)*log(x)) + x^3*log(x/(2*log(x)))^4*log(x) - 
 12*x^4*log(x/(2*log(x)))^3*log(x) + 81*x^3*exp((2*exp(x))/x)*log(x) + 162 
*x^5*exp(exp(x)/x)*log(x)),x)
 

Output:

int(-(108*x^2 - log(x/(2*log(x)))*(36*x - log(x)*(36*x - 216*x^2)) - log(x 
)*(108*x^2 - 486*x^3) + exp(exp(x)/x)*log(x)*(162*x + exp(x)*(162*x - 162) 
) + 18*x*log(x/(2*log(x)))^2*log(x))/(81*x^7*log(x) + log(x/(2*log(x)))^2* 
(54*x^5*log(x) + 18*x^3*exp(exp(x)/x)*log(x)) - log(x/(2*log(x)))*(108*x^6 
*log(x) + 108*x^4*exp(exp(x)/x)*log(x)) + x^3*log(x/(2*log(x)))^4*log(x) - 
 12*x^4*log(x/(2*log(x)))^3*log(x) + 81*x^3*exp((2*exp(x))/x)*log(x) + 162 
*x^5*exp(exp(x)/x)*log(x)), x)
 

Reduce [B] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.29 \[ \int \frac {-108 x^2+e^{\frac {e^x}{x}} \left (e^x (162-162 x)-162 x\right ) \log (x)+\left (108 x^2-486 x^3\right ) \log (x)+\left (36 x+\left (-36 x+216 x^2\right ) \log (x)\right ) \log \left (\frac {x}{2 \log (x)}\right )-18 x \log (x) \log ^2\left (\frac {x}{2 \log (x)}\right )}{81 e^{\frac {2 e^x}{x}} x^3 \log (x)+162 e^{\frac {e^x}{x}} x^5 \log (x)+81 x^7 \log (x)+\left (-108 e^{\frac {e^x}{x}} x^4 \log (x)-108 x^6 \log (x)\right ) \log \left (\frac {x}{2 \log (x)}\right )+\left (18 e^{\frac {e^x}{x}} x^3 \log (x)+54 x^5 \log (x)\right ) \log ^2\left (\frac {x}{2 \log (x)}\right )-12 x^4 \log (x) \log ^3\left (\frac {x}{2 \log (x)}\right )+x^3 \log (x) \log ^4\left (\frac {x}{2 \log (x)}\right )} \, dx=\frac {18}{x \left (9 e^{\frac {e^{x}}{x}}+\mathrm {log}\left (\frac {x}{2 \,\mathrm {log}\left (x \right )}\right )^{2}-6 \,\mathrm {log}\left (\frac {x}{2 \,\mathrm {log}\left (x \right )}\right ) x +9 x^{2}\right )} \] Input:

int((-18*x*log(x)*log(1/2*x/log(x))^2+((216*x^2-36*x)*log(x)+36*x)*log(1/2 
*x/log(x))+((-162*x+162)*exp(x)-162*x)*log(x)*exp(exp(x)/x)+(-486*x^3+108* 
x^2)*log(x)-108*x^2)/(x^3*log(x)*log(1/2*x/log(x))^4-12*x^4*log(x)*log(1/2 
*x/log(x))^3+(18*x^3*log(x)*exp(exp(x)/x)+54*x^5*log(x))*log(1/2*x/log(x)) 
^2+(-108*x^4*log(x)*exp(exp(x)/x)-108*x^6*log(x))*log(1/2*x/log(x))+81*x^3 
*log(x)*exp(exp(x)/x)^2+162*x^5*log(x)*exp(exp(x)/x)+81*x^7*log(x)),x)
 

Output:

18/(x*(9*e**(e**x/x) + log(x/(2*log(x)))**2 - 6*log(x/(2*log(x)))*x + 9*x* 
*2))