\(\int \frac {e^{-\frac {e^2 x}{3 x-e^x x+\log (\log (\frac {x^2}{\log ^2(2) \log ^2(x)}))}} (-2 e^2+2 e^2 \log (x)-e^{2+x} x^2 \log (x) \log (\frac {x^2}{\log ^2(2) \log ^2(x)})-e^2 \log (x) \log (\frac {x^2}{\log ^2(2) \log ^2(x)}) \log (\log (\frac {x^2}{\log ^2(2) \log ^2(x)})))}{(9 x^2-6 e^x x^2+e^{2 x} x^2) \log (x) \log (\frac {x^2}{\log ^2(2) \log ^2(x)})+(6 x-2 e^x x) \log (x) \log (\frac {x^2}{\log ^2(2) \log ^2(x)}) \log (\log (\frac {x^2}{\log ^2(2) \log ^2(x)}))+\log (x) \log (\frac {x^2}{\log ^2(2) \log ^2(x)}) \log ^2(\log (\frac {x^2}{\log ^2(2) \log ^2(x)}))} \, dx\) [493]

Optimal result

Integrand size = 221, antiderivative size = 33 \[ \int \frac {e^{-\frac {e^2 x}{3 x-e^x x+\log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )}} \left (-2 e^2+2 e^2 \log (x)-e^{2+x} x^2 \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )-e^2 \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )\right )}{\left (9 x^2-6 e^x x^2+e^{2 x} x^2\right ) \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )+\left (6 x-2 e^x x\right ) \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )+\log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \log ^2\left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )} \, dx=e^{\frac {e^2 x}{\left (-3+e^x\right ) x-\log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )}} \] Output:

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

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.03 \[ \int \frac {e^{-\frac {e^2 x}{3 x-e^x x+\log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )}} \left (-2 e^2+2 e^2 \log (x)-e^{2+x} x^2 \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )-e^2 \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )\right )}{\left (9 x^2-6 e^x x^2+e^{2 x} x^2\right ) \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )+\left (6 x-2 e^x x\right ) \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )+\log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \log ^2\left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )} \, dx=e^{-\frac {e^2 x}{3 x-e^x x+\log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 3.84 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.03, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.009, Rules used = {7239, 7257}

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.

\(\displaystyle \int \frac {\left (-e^{x+2} x^2 \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )-e^2 \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )+2 e^2 \log (x)-2 e^2\right ) \exp \left (-\frac {e^2 x}{\log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )-e^x x+3 x}\right )}{\log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \log ^2\left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )+\left (6 x-2 e^x x\right ) \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )+\left (-6 e^x x^2+e^{2 x} x^2+9 x^2\right ) \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )} \, dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7257

\(\displaystyle \exp \left (-\frac {e^2 x}{\log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )+\left (3-e^x\right ) x}\right )\)

Input:

Int[(-2*E^2 + 2*E^2*Log[x] - E^(2 + x)*x^2*Log[x]*Log[x^2/(Log[2]^2*Log[x] 
^2)] - E^2*Log[x]*Log[x^2/(Log[2]^2*Log[x]^2)]*Log[Log[x^2/(Log[2]^2*Log[x 
]^2)]])/(E^((E^2*x)/(3*x - E^x*x + Log[Log[x^2/(Log[2]^2*Log[x]^2)]]))*((9 
*x^2 - 6*E^x*x^2 + E^(2*x)*x^2)*Log[x]*Log[x^2/(Log[2]^2*Log[x]^2)] + (6*x 
 - 2*E^x*x)*Log[x]*Log[x^2/(Log[2]^2*Log[x]^2)]*Log[Log[x^2/(Log[2]^2*Log[ 
x]^2)]] + Log[x]*Log[x^2/(Log[2]^2*Log[x]^2)]*Log[Log[x^2/(Log[2]^2*Log[x] 
^2)]]^2)),x]
 

Output:

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

Defintions of rubi rules used

rule 7239

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl 
erIntegrandQ[v, u, x]]
 

rule 7257

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Sim 
p[q*(F^v/Log[F]), x] /;  !FalseQ[q]] /; FreeQ[F, x]
 

Maple [C] (warning: unable to verify)

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

Time = 0.54 (sec) , antiderivative size = 151, normalized size of antiderivative = 4.58

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

Input:

int((-exp(2)*ln(x)*ln(x^2/ln(2)^2/ln(x)^2)*ln(ln(x^2/ln(2)^2/ln(x)^2))-x^2 
*exp(2)*exp(x)*ln(x)*ln(x^2/ln(2)^2/ln(x)^2)+2*exp(2)*ln(x)-2*exp(2))*exp( 
-x*exp(2)/(ln(ln(x^2/ln(2)^2/ln(x)^2))-exp(x)*x+3*x))/(ln(x)*ln(x^2/ln(2)^ 
2/ln(x)^2)*ln(ln(x^2/ln(2)^2/ln(x)^2))^2+(-2*exp(x)*x+6*x)*ln(x)*ln(x^2/ln 
(2)^2/ln(x)^2)*ln(ln(x^2/ln(2)^2/ln(x)^2))+(exp(x)^2*x^2-6*exp(x)*x^2+9*x^ 
2)*ln(x)*ln(x^2/ln(2)^2/ln(x)^2)),x)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.15 \[ \int \frac {e^{-\frac {e^2 x}{3 x-e^x x+\log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )}} \left (-2 e^2+2 e^2 \log (x)-e^{2+x} x^2 \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )-e^2 \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )\right )}{\left (9 x^2-6 e^x x^2+e^{2 x} x^2\right ) \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )+\left (6 x-2 e^x x\right ) \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )+\log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \log ^2\left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )} \, dx=e^{\left (-\frac {x e^{4}}{3 \, x e^{2} - x e^{\left (x + 2\right )} + e^{2} \log \left (\log \left (\frac {x^{2}}{\log \left (2\right )^{2} \log \left (x\right )^{2}}\right )\right )}\right )} \] Input:

integrate((-exp(2)*log(x)*log(x^2/log(2)^2/log(x)^2)*log(log(x^2/log(2)^2/ 
log(x)^2))-x^2*exp(2)*exp(x)*log(x)*log(x^2/log(2)^2/log(x)^2)+2*exp(2)*lo 
g(x)-2*exp(2))*exp(-x*exp(2)/(log(log(x^2/log(2)^2/log(x)^2))-exp(x)*x+3*x 
))/(log(x)*log(x^2/log(2)^2/log(x)^2)*log(log(x^2/log(2)^2/log(x)^2))^2+(- 
2*exp(x)*x+6*x)*log(x)*log(x^2/log(2)^2/log(x)^2)*log(log(x^2/log(2)^2/log 
(x)^2))+(exp(x)^2*x^2-6*exp(x)*x^2+9*x^2)*log(x)*log(x^2/log(2)^2/log(x)^2 
)),x, algorithm="fricas")
 

Output:

e^(-x*e^4/(3*x*e^2 - x*e^(x + 2) + e^2*log(log(x^2/(log(2)^2*log(x)^2)))))
 

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{-\frac {e^2 x}{3 x-e^x x+\log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )}} \left (-2 e^2+2 e^2 \log (x)-e^{2+x} x^2 \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )-e^2 \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )\right )}{\left (9 x^2-6 e^x x^2+e^{2 x} x^2\right ) \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )+\left (6 x-2 e^x x\right ) \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )+\log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \log ^2\left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )} \, dx=\text {Timed out} \] Input:

integrate((-exp(2)*ln(x)*ln(x**2/ln(2)**2/ln(x)**2)*ln(ln(x**2/ln(2)**2/ln 
(x)**2))-x**2*exp(2)*exp(x)*ln(x)*ln(x**2/ln(2)**2/ln(x)**2)+2*exp(2)*ln(x 
)-2*exp(2))*exp(-x*exp(2)/(ln(ln(x**2/ln(2)**2/ln(x)**2))-exp(x)*x+3*x))/( 
ln(x)*ln(x**2/ln(2)**2/ln(x)**2)*ln(ln(x**2/ln(2)**2/ln(x)**2))**2+(-2*exp 
(x)*x+6*x)*ln(x)*ln(x**2/ln(2)**2/ln(x)**2)*ln(ln(x**2/ln(2)**2/ln(x)**2)) 
+(exp(x)**2*x**2-6*exp(x)*x**2+9*x**2)*ln(x)*ln(x**2/ln(2)**2/ln(x)**2)),x 
)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {e^{-\frac {e^2 x}{3 x-e^x x+\log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )}} \left (-2 e^2+2 e^2 \log (x)-e^{2+x} x^2 \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )-e^2 \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )\right )}{\left (9 x^2-6 e^x x^2+e^{2 x} x^2\right ) \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )+\left (6 x-2 e^x x\right ) \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )+\log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \log ^2\left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )} \, dx=\int { \frac {{\left (x^{2} e^{\left (x + 2\right )} \log \left (x\right ) \log \left (\frac {x^{2}}{\log \left (2\right )^{2} \log \left (x\right )^{2}}\right ) + e^{2} \log \left (x\right ) \log \left (\frac {x^{2}}{\log \left (2\right )^{2} \log \left (x\right )^{2}}\right ) \log \left (\log \left (\frac {x^{2}}{\log \left (2\right )^{2} \log \left (x\right )^{2}}\right )\right ) - 2 \, e^{2} \log \left (x\right ) + 2 \, e^{2}\right )} e^{\left (\frac {x e^{2}}{x e^{x} - 3 \, x - \log \left (\log \left (\frac {x^{2}}{\log \left (2\right )^{2} \log \left (x\right )^{2}}\right )\right )}\right )}}{2 \, {\left (x e^{x} - 3 \, x\right )} \log \left (x\right ) \log \left (\frac {x^{2}}{\log \left (2\right )^{2} \log \left (x\right )^{2}}\right ) \log \left (\log \left (\frac {x^{2}}{\log \left (2\right )^{2} \log \left (x\right )^{2}}\right )\right ) - \log \left (x\right ) \log \left (\frac {x^{2}}{\log \left (2\right )^{2} \log \left (x\right )^{2}}\right ) \log \left (\log \left (\frac {x^{2}}{\log \left (2\right )^{2} \log \left (x\right )^{2}}\right )\right )^{2} - {\left (x^{2} e^{\left (2 \, x\right )} - 6 \, x^{2} e^{x} + 9 \, x^{2}\right )} \log \left (x\right ) \log \left (\frac {x^{2}}{\log \left (2\right )^{2} \log \left (x\right )^{2}}\right )} \,d x } \] Input:

integrate((-exp(2)*log(x)*log(x^2/log(2)^2/log(x)^2)*log(log(x^2/log(2)^2/ 
log(x)^2))-x^2*exp(2)*exp(x)*log(x)*log(x^2/log(2)^2/log(x)^2)+2*exp(2)*lo 
g(x)-2*exp(2))*exp(-x*exp(2)/(log(log(x^2/log(2)^2/log(x)^2))-exp(x)*x+3*x 
))/(log(x)*log(x^2/log(2)^2/log(x)^2)*log(log(x^2/log(2)^2/log(x)^2))^2+(- 
2*exp(x)*x+6*x)*log(x)*log(x^2/log(2)^2/log(x)^2)*log(log(x^2/log(2)^2/log 
(x)^2))+(exp(x)^2*x^2-6*exp(x)*x^2+9*x^2)*log(x)*log(x^2/log(2)^2/log(x)^2 
)),x, algorithm="maxima")
 

Output:

integrate((x^2*e^(x + 2)*log(x)*log(x^2/(log(2)^2*log(x)^2)) + e^2*log(x)* 
log(x^2/(log(2)^2*log(x)^2))*log(log(x^2/(log(2)^2*log(x)^2))) - 2*e^2*log 
(x) + 2*e^2)*e^(x*e^2/(x*e^x - 3*x - log(log(x^2/(log(2)^2*log(x)^2)))))/( 
2*(x*e^x - 3*x)*log(x)*log(x^2/(log(2)^2*log(x)^2))*log(log(x^2/(log(2)^2* 
log(x)^2))) - log(x)*log(x^2/(log(2)^2*log(x)^2))*log(log(x^2/(log(2)^2*lo 
g(x)^2)))^2 - (x^2*e^(2*x) - 6*x^2*e^x + 9*x^2)*log(x)*log(x^2/(log(2)^2*l 
og(x)^2))), x)
 

Giac [F]

\[ \int \frac {e^{-\frac {e^2 x}{3 x-e^x x+\log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )}} \left (-2 e^2+2 e^2 \log (x)-e^{2+x} x^2 \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )-e^2 \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )\right )}{\left (9 x^2-6 e^x x^2+e^{2 x} x^2\right ) \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )+\left (6 x-2 e^x x\right ) \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )+\log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \log ^2\left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )} \, dx=\int { \frac {{\left (x^{2} e^{\left (x + 2\right )} \log \left (x\right ) \log \left (\frac {x^{2}}{\log \left (2\right )^{2} \log \left (x\right )^{2}}\right ) + e^{2} \log \left (x\right ) \log \left (\frac {x^{2}}{\log \left (2\right )^{2} \log \left (x\right )^{2}}\right ) \log \left (\log \left (\frac {x^{2}}{\log \left (2\right )^{2} \log \left (x\right )^{2}}\right )\right ) - 2 \, e^{2} \log \left (x\right ) + 2 \, e^{2}\right )} e^{\left (\frac {x e^{2}}{x e^{x} - 3 \, x - \log \left (\log \left (\frac {x^{2}}{\log \left (2\right )^{2} \log \left (x\right )^{2}}\right )\right )}\right )}}{2 \, {\left (x e^{x} - 3 \, x\right )} \log \left (x\right ) \log \left (\frac {x^{2}}{\log \left (2\right )^{2} \log \left (x\right )^{2}}\right ) \log \left (\log \left (\frac {x^{2}}{\log \left (2\right )^{2} \log \left (x\right )^{2}}\right )\right ) - \log \left (x\right ) \log \left (\frac {x^{2}}{\log \left (2\right )^{2} \log \left (x\right )^{2}}\right ) \log \left (\log \left (\frac {x^{2}}{\log \left (2\right )^{2} \log \left (x\right )^{2}}\right )\right )^{2} - {\left (x^{2} e^{\left (2 \, x\right )} - 6 \, x^{2} e^{x} + 9 \, x^{2}\right )} \log \left (x\right ) \log \left (\frac {x^{2}}{\log \left (2\right )^{2} \log \left (x\right )^{2}}\right )} \,d x } \] Input:

integrate((-exp(2)*log(x)*log(x^2/log(2)^2/log(x)^2)*log(log(x^2/log(2)^2/ 
log(x)^2))-x^2*exp(2)*exp(x)*log(x)*log(x^2/log(2)^2/log(x)^2)+2*exp(2)*lo 
g(x)-2*exp(2))*exp(-x*exp(2)/(log(log(x^2/log(2)^2/log(x)^2))-exp(x)*x+3*x 
))/(log(x)*log(x^2/log(2)^2/log(x)^2)*log(log(x^2/log(2)^2/log(x)^2))^2+(- 
2*exp(x)*x+6*x)*log(x)*log(x^2/log(2)^2/log(x)^2)*log(log(x^2/log(2)^2/log 
(x)^2))+(exp(x)^2*x^2-6*exp(x)*x^2+9*x^2)*log(x)*log(x^2/log(2)^2/log(x)^2 
)),x, algorithm="giac")
 

Output:

undef
 

Mupad [B] (verification not implemented)

Time = 3.56 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-\frac {e^2 x}{3 x-e^x x+\log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )}} \left (-2 e^2+2 e^2 \log (x)-e^{2+x} x^2 \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )-e^2 \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )\right )}{\left (9 x^2-6 e^x x^2+e^{2 x} x^2\right ) \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )+\left (6 x-2 e^x x\right ) \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )+\log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \log ^2\left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )} \, dx={\mathrm {e}}^{-\frac {x\,{\mathrm {e}}^2}{3\,x+\ln \left (\ln \left (x^2\right )+\ln \left (\frac {1}{{\ln \left (x\right )}^2}\right )-2\,\ln \left (\ln \left (2\right )\right )\right )-x\,{\mathrm {e}}^x}} \] Input:

int(-(exp(-(x*exp(2))/(3*x + log(log(x^2/(log(2)^2*log(x)^2))) - x*exp(x)) 
)*(2*exp(2) - 2*exp(2)*log(x) + log(log(x^2/(log(2)^2*log(x)^2)))*exp(2)*l 
og(x^2/(log(2)^2*log(x)^2))*log(x) + x^2*exp(2)*exp(x)*log(x^2/(log(2)^2*l 
og(x)^2))*log(x)))/(log(x^2/(log(2)^2*log(x)^2))*log(x)*(x^2*exp(2*x) - 6* 
x^2*exp(x) + 9*x^2) + log(log(x^2/(log(2)^2*log(x)^2)))^2*log(x^2/(log(2)^ 
2*log(x)^2))*log(x) + log(log(x^2/(log(2)^2*log(x)^2)))*log(x^2/(log(2)^2* 
log(x)^2))*log(x)*(6*x - 2*x*exp(x))),x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.81 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.03 \[ \int \frac {e^{-\frac {e^2 x}{3 x-e^x x+\log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )}} \left (-2 e^2+2 e^2 \log (x)-e^{2+x} x^2 \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )-e^2 \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )\right )}{\left (9 x^2-6 e^x x^2+e^{2 x} x^2\right ) \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )+\left (6 x-2 e^x x\right ) \log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \log \left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )+\log (x) \log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right ) \log ^2\left (\log \left (\frac {x^2}{\log ^2(2) \log ^2(x)}\right )\right )} \, dx=e^{\frac {e^{2} x}{e^{x} x -\mathrm {log}\left (\mathrm {log}\left (\frac {x^{2}}{\mathrm {log}\left (x \right )^{2} \mathrm {log}\left (2\right )^{2}}\right )\right )-3 x}} \] Input:

int((-exp(2)*log(x)*log(x^2/log(2)^2/log(x)^2)*log(log(x^2/log(2)^2/log(x) 
^2))-x^2*exp(2)*exp(x)*log(x)*log(x^2/log(2)^2/log(x)^2)+2*exp(2)*log(x)-2 
*exp(2))*exp(-x*exp(2)/(log(log(x^2/log(2)^2/log(x)^2))-exp(x)*x+3*x))/(lo 
g(x)*log(x^2/log(2)^2/log(x)^2)*log(log(x^2/log(2)^2/log(x)^2))^2+(-2*exp( 
x)*x+6*x)*log(x)*log(x^2/log(2)^2/log(x)^2)*log(log(x^2/log(2)^2/log(x)^2) 
)+(exp(x)^2*x^2-6*exp(x)*x^2+9*x^2)*log(x)*log(x^2/log(2)^2/log(x)^2)),x)
 

Output:

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