\(\int \frac {e^{\frac {1}{4} (4 x+(e^{4+4 x} x-x^2) \log (\log (x+\log (x))))} (-x-x^2+e^{4+4 x} (1+x)+(4 x+4 \log (x)) \log (x+\log (x))+(-2 x^2+e^{4+4 x} (x+4 x^2)+(-2 x+e^{4+4 x} (1+4 x)) \log (x)) \log (x+\log (x)) \log (\log (x+\log (x))))}{(4 x+4 \log (x)) \log (x+\log (x))} \, dx\) [2100]

Optimal result

Integrand size = 137, antiderivative size = 28 \[ \int \frac {e^{\frac {1}{4} \left (4 x+\left (e^{4+4 x} x-x^2\right ) \log (\log (x+\log (x)))\right )} \left (-x-x^2+e^{4+4 x} (1+x)+(4 x+4 \log (x)) \log (x+\log (x))+\left (-2 x^2+e^{4+4 x} \left (x+4 x^2\right )+\left (-2 x+e^{4+4 x} (1+4 x)\right ) \log (x)\right ) \log (x+\log (x)) \log (\log (x+\log (x)))\right )}{(4 x+4 \log (x)) \log (x+\log (x))} \, dx=e^{\frac {1}{4} x \left (4-\left (-e^{4+4 x}+x\right ) \log (\log (x+\log (x)))\right )} \] Output:

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

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {e^{\frac {1}{4} \left (4 x+\left (e^{4+4 x} x-x^2\right ) \log (\log (x+\log (x)))\right )} \left (-x-x^2+e^{4+4 x} (1+x)+(4 x+4 \log (x)) \log (x+\log (x))+\left (-2 x^2+e^{4+4 x} \left (x+4 x^2\right )+\left (-2 x+e^{4+4 x} (1+4 x)\right ) \log (x)\right ) \log (x+\log (x)) \log (\log (x+\log (x)))\right )}{(4 x+4 \log (x)) \log (x+\log (x))} \, dx=e^x \log ^{\frac {1}{4} \left (e^{4+4 x}-x\right ) x}(x+\log (x)) \] Input:

Integrate[(E^((4*x + (E^(4 + 4*x)*x - x^2)*Log[Log[x + Log[x]]])/4)*(-x - 
x^2 + E^(4 + 4*x)*(1 + x) + (4*x + 4*Log[x])*Log[x + Log[x]] + (-2*x^2 + E 
^(4 + 4*x)*(x + 4*x^2) + (-2*x + E^(4 + 4*x)*(1 + 4*x))*Log[x])*Log[x + Lo 
g[x]]*Log[Log[x + Log[x]]]))/((4*x + 4*Log[x])*Log[x + Log[x]]),x]
 

Output:

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

Rubi [A] (verified)

Time = 4.47 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {7292, 27, 25, 2726}

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.

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2726

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

Input:

Int[(E^((4*x + (E^(4 + 4*x)*x - x^2)*Log[Log[x + Log[x]]])/4)*(-x - x^2 + 
E^(4 + 4*x)*(1 + x) + (4*x + 4*Log[x])*Log[x + Log[x]] + (-2*x^2 + E^(4 + 
4*x)*(x + 4*x^2) + (-2*x + E^(4 + 4*x)*(1 + 4*x))*Log[x])*Log[x + Log[x]]* 
Log[Log[x + Log[x]]]))/((4*x + 4*Log[x])*Log[x + Log[x]]),x]
 

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2726

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, 
 x]))}, Simp[F^u*z, x] /; EqQ[D[z, x], w*y]] /; FreeQ[F, x]
 

rule 7292

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

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.82

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {e^{\frac {1}{4} \left (4 x+\left (e^{4+4 x} x-x^2\right ) \log (\log (x+\log (x)))\right )} \left (-x-x^2+e^{4+4 x} (1+x)+(4 x+4 \log (x)) \log (x+\log (x))+\left (-2 x^2+e^{4+4 x} \left (x+4 x^2\right )+\left (-2 x+e^{4+4 x} (1+4 x)\right ) \log (x)\right ) \log (x+\log (x)) \log (\log (x+\log (x)))\right )}{(4 x+4 \log (x)) \log (x+\log (x))} \, dx=e^{\left (-\frac {1}{4} \, {\left (x^{2} - x e^{\left (4 \, x + 4\right )}\right )} \log \left (\log \left (x + \log \left (x\right )\right )\right ) + x\right )} \] Input:

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

Output:

e^(-1/4*(x^2 - x*e^(4*x + 4))*log(log(x + log(x))) + x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{\frac {1}{4} \left (4 x+\left (e^{4+4 x} x-x^2\right ) \log (\log (x+\log (x)))\right )} \left (-x-x^2+e^{4+4 x} (1+x)+(4 x+4 \log (x)) \log (x+\log (x))+\left (-2 x^2+e^{4+4 x} \left (x+4 x^2\right )+\left (-2 x+e^{4+4 x} (1+4 x)\right ) \log (x)\right ) \log (x+\log (x)) \log (\log (x+\log (x)))\right )}{(4 x+4 \log (x)) \log (x+\log (x))} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {e^{\frac {1}{4} \left (4 x+\left (e^{4+4 x} x-x^2\right ) \log (\log (x+\log (x)))\right )} \left (-x-x^2+e^{4+4 x} (1+x)+(4 x+4 \log (x)) \log (x+\log (x))+\left (-2 x^2+e^{4+4 x} \left (x+4 x^2\right )+\left (-2 x+e^{4+4 x} (1+4 x)\right ) \log (x)\right ) \log (x+\log (x)) \log (\log (x+\log (x)))\right )}{(4 x+4 \log (x)) \log (x+\log (x))} \, dx=e^{\left (-\frac {1}{4} \, x^{2} \log \left (\log \left (x + \log \left (x\right )\right )\right ) + \frac {1}{4} \, x e^{\left (4 \, x + 4\right )} \log \left (\log \left (x + \log \left (x\right )\right )\right ) + x\right )} \] Input:

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

Output:

e^(-1/4*x^2*log(log(x + log(x))) + 1/4*x*e^(4*x + 4)*log(log(x + log(x))) 
+ x)
 

Giac [F]

\[ \int \frac {e^{\frac {1}{4} \left (4 x+\left (e^{4+4 x} x-x^2\right ) \log (\log (x+\log (x)))\right )} \left (-x-x^2+e^{4+4 x} (1+x)+(4 x+4 \log (x)) \log (x+\log (x))+\left (-2 x^2+e^{4+4 x} \left (x+4 x^2\right )+\left (-2 x+e^{4+4 x} (1+4 x)\right ) \log (x)\right ) \log (x+\log (x)) \log (\log (x+\log (x)))\right )}{(4 x+4 \log (x)) \log (x+\log (x))} \, dx=\int { -\frac {{\left ({\left (2 \, x^{2} - {\left (4 \, x^{2} + x\right )} e^{\left (4 \, x + 4\right )} - {\left ({\left (4 \, x + 1\right )} e^{\left (4 \, x + 4\right )} - 2 \, x\right )} \log \left (x\right )\right )} \log \left (x + \log \left (x\right )\right ) \log \left (\log \left (x + \log \left (x\right )\right )\right ) + x^{2} - {\left (x + 1\right )} e^{\left (4 \, x + 4\right )} - 4 \, {\left (x + \log \left (x\right )\right )} \log \left (x + \log \left (x\right )\right ) + x\right )} e^{\left (-\frac {1}{4} \, {\left (x^{2} - x e^{\left (4 \, x + 4\right )}\right )} \log \left (\log \left (x + \log \left (x\right )\right )\right ) + x\right )}}{4 \, {\left (x + \log \left (x\right )\right )} \log \left (x + \log \left (x\right )\right )} \,d x } \] Input:

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

Output:

undef
 

Mupad [B] (verification not implemented)

Time = 3.55 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {e^{\frac {1}{4} \left (4 x+\left (e^{4+4 x} x-x^2\right ) \log (\log (x+\log (x)))\right )} \left (-x-x^2+e^{4+4 x} (1+x)+(4 x+4 \log (x)) \log (x+\log (x))+\left (-2 x^2+e^{4+4 x} \left (x+4 x^2\right )+\left (-2 x+e^{4+4 x} (1+4 x)\right ) \log (x)\right ) \log (x+\log (x)) \log (\log (x+\log (x)))\right )}{(4 x+4 \log (x)) \log (x+\log (x))} \, dx={\ln \left (x+\ln \left (x\right )\right )}^{\frac {x\,{\mathrm {e}}^{4\,x+4}}{4}-\frac {x^2}{4}}\,{\mathrm {e}}^x \] Input:

int(-(exp(x + (log(log(x + log(x)))*(x*exp(4*x + 4) - x^2))/4)*(x - log(x 
+ log(x))*(4*x + 4*log(x)) - exp(4*x + 4)*(x + 1) + x^2 + log(x + log(x))* 
log(log(x + log(x)))*(log(x)*(2*x - exp(4*x + 4)*(4*x + 1)) - exp(4*x + 4) 
*(x + 4*x^2) + 2*x^2)))/(log(x + log(x))*(4*x + 4*log(x))),x)
 

Output:

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

Reduce [F]

\[ \int \frac {e^{\frac {1}{4} \left (4 x+\left (e^{4+4 x} x-x^2\right ) \log (\log (x+\log (x)))\right )} \left (-x-x^2+e^{4+4 x} (1+x)+(4 x+4 \log (x)) \log (x+\log (x))+\left (-2 x^2+e^{4+4 x} \left (x+4 x^2\right )+\left (-2 x+e^{4+4 x} (1+4 x)\right ) \log (x)\right ) \log (x+\log (x)) \log (\log (x+\log (x)))\right )}{(4 x+4 \log (x)) \log (x+\log (x))} \, dx =\text {Too large to display} \] Input:

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

Output:

(int(e**((e**(4*x)*log(log(log(x) + x))*e**4*x + 20*x)/4)/(log(log(x) + x) 
**(x**2/4)*log(log(x) + x)*log(x) + log(log(x) + x)**(x**2/4)*log(log(x) + 
 x)*x),x)*e**4 + 4*int((e**((e**(4*x)*log(log(log(x) + x))*e**4*x + 20*x)/ 
4)*log(log(log(x) + x))*x**2)/(log(log(x) + x)**(x**2/4)*log(x) + log(log( 
x) + x)**(x**2/4)*x),x)*e**4 + 4*int((e**((e**(4*x)*log(log(log(x) + x))*e 
**4*x + 20*x)/4)*log(log(log(x) + x))*log(x)*x)/(log(log(x) + x)**(x**2/4) 
*log(x) + log(log(x) + x)**(x**2/4)*x),x)*e**4 + int((e**((e**(4*x)*log(lo 
g(log(x) + x))*e**4*x + 20*x)/4)*log(log(log(x) + x))*log(x))/(log(log(x) 
+ x)**(x**2/4)*log(x) + log(log(x) + x)**(x**2/4)*x),x)*e**4 + int((e**((e 
**(4*x)*log(log(log(x) + x))*e**4*x + 20*x)/4)*log(log(log(x) + x))*x)/(lo 
g(log(x) + x)**(x**2/4)*log(x) + log(log(x) + x)**(x**2/4)*x),x)*e**4 + in 
t((e**((e**(4*x)*log(log(log(x) + x))*e**4*x + 20*x)/4)*x)/(log(log(x) + x 
)**(x**2/4)*log(log(x) + x)*log(x) + log(log(x) + x)**(x**2/4)*log(log(x) 
+ x)*x),x)*e**4 - int((e**((e**(4*x)*log(log(log(x) + x))*e**4*x + 4*x)/4) 
*x**2)/(log(log(x) + x)**(x**2/4)*log(log(x) + x)*log(x) + log(log(x) + x) 
**(x**2/4)*log(log(x) + x)*x),x) - 2*int((e**((e**(4*x)*log(log(log(x) + x 
))*e**4*x + 4*x)/4)*log(log(log(x) + x))*x**2)/(log(log(x) + x)**(x**2/4)* 
log(x) + log(log(x) + x)**(x**2/4)*x),x) - 2*int((e**((e**(4*x)*log(log(lo 
g(x) + x))*e**4*x + 4*x)/4)*log(log(log(x) + x))*log(x)*x)/(log(log(x) + x 
)**(x**2/4)*log(x) + log(log(x) + x)**(x**2/4)*x),x) + 4*int((e**((e**(...