\(\int \frac {e^{-2+e^{5 e^x}} (-50 x^2+50 e^{-3+x} x^2)+e^{-4+2 e^{5 e^x}} (50 e^{-3+x} x-50 x^2+e^{5 e^x+x} (250 e^{-3+x} x^2-250 x^3)) \log (e^{-3+x}-x) \log (\log (e^{-3+x}-x))+(-50 x^2+50 e^{-3+x} x^2+e^{-2+e^{5 e^x}} (100 e^{-3+x} x-100 x^2+e^{5 e^x+x} (250 e^{-3+x} x^2-250 x^3)) \log (e^{-3+x}-x) \log (\log (e^{-3+x}-x))) \log (\log (\log (e^{-3+x}-x)))+(50 e^{-3+x} x-50 x^2) \log (e^{-3+x}-x) \log (\log (e^{-3+x}-x)) \log ^2(\log (\log (e^{-3+x}-x)))}{(e^{-3+x}-x) \log (e^{-3+x}-x) \log (\log (e^{-3+x}-x))} \, dx\) [817]

Optimal result

Integrand size = 293, antiderivative size = 31 \[ \int \frac {e^{-2+e^{5 e^x}} \left (-50 x^2+50 e^{-3+x} x^2\right )+e^{-4+2 e^{5 e^x}} \left (50 e^{-3+x} x-50 x^2+e^{5 e^x+x} \left (250 e^{-3+x} x^2-250 x^3\right )\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right )+\left (-50 x^2+50 e^{-3+x} x^2+e^{-2+e^{5 e^x}} \left (100 e^{-3+x} x-100 x^2+e^{5 e^x+x} \left (250 e^{-3+x} x^2-250 x^3\right )\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right )\right ) \log \left (\log \left (\log \left (e^{-3+x}-x\right )\right )\right )+\left (50 e^{-3+x} x-50 x^2\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right ) \log ^2\left (\log \left (\log \left (e^{-3+x}-x\right )\right )\right )}{\left (e^{-3+x}-x\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right )} \, dx=25 x^2 \left (e^{-2+e^{5 e^x}}+\log \left (\log \left (\log \left (e^{-3+x}-x\right )\right )\right )\right )^2 \] Output:

25*x^2*(exp(exp(5*exp(x))-2)+ln(ln(ln(exp(-3+x)-x))))^2
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int \frac {e^{-2+e^{5 e^x}} \left (-50 x^2+50 e^{-3+x} x^2\right )+e^{-4+2 e^{5 e^x}} \left (50 e^{-3+x} x-50 x^2+e^{5 e^x+x} \left (250 e^{-3+x} x^2-250 x^3\right )\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right )+\left (-50 x^2+50 e^{-3+x} x^2+e^{-2+e^{5 e^x}} \left (100 e^{-3+x} x-100 x^2+e^{5 e^x+x} \left (250 e^{-3+x} x^2-250 x^3\right )\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right )\right ) \log \left (\log \left (\log \left (e^{-3+x}-x\right )\right )\right )+\left (50 e^{-3+x} x-50 x^2\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right ) \log ^2\left (\log \left (\log \left (e^{-3+x}-x\right )\right )\right )}{\left (e^{-3+x}-x\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right )} \, dx=\frac {25 x^2 \left (e^{e^{5 e^x}}+e^2 \log \left (\log \left (\log \left (e^{-3+x}-x\right )\right )\right )\right )^2}{e^4} \] Input:

Integrate[(E^(-2 + E^(5*E^x))*(-50*x^2 + 50*E^(-3 + x)*x^2) + E^(-4 + 2*E^ 
(5*E^x))*(50*E^(-3 + x)*x - 50*x^2 + E^(5*E^x + x)*(250*E^(-3 + x)*x^2 - 2 
50*x^3))*Log[E^(-3 + x) - x]*Log[Log[E^(-3 + x) - x]] + (-50*x^2 + 50*E^(- 
3 + x)*x^2 + E^(-2 + E^(5*E^x))*(100*E^(-3 + x)*x - 100*x^2 + E^(5*E^x + x 
)*(250*E^(-3 + x)*x^2 - 250*x^3))*Log[E^(-3 + x) - x]*Log[Log[E^(-3 + x) - 
 x]])*Log[Log[Log[E^(-3 + x) - x]]] + (50*E^(-3 + x)*x - 50*x^2)*Log[E^(-3 
 + x) - x]*Log[Log[E^(-3 + x) - x]]*Log[Log[Log[E^(-3 + x) - x]]]^2)/((E^( 
-3 + x) - x)*Log[E^(-3 + x) - x]*Log[Log[E^(-3 + x) - x]]),x]
 

Output:

(25*x^2*(E^E^(5*E^x) + E^2*Log[Log[Log[E^(-3 + x) - x]]])^2)/E^4
 

Rubi [A] (verified)

Time = 3.81 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {7239, 27, 25, 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.

\(\displaystyle \int \frac {e^{e^{5 e^x}-2} \left (50 e^{x-3} x^2-50 x^2\right )+\left (50 e^{x-3} x-50 x^2\right ) \log \left (e^{x-3}-x\right ) \log \left (\log \left (e^{x-3}-x\right )\right ) \log ^2\left (\log \left (\log \left (e^{x-3}-x\right )\right )\right )+\left (50 e^{x-3} x^2-50 x^2+e^{e^{5 e^x}-2} \left (-100 x^2+e^{x+5 e^x} \left (250 e^{x-3} x^2-250 x^3\right )+100 e^{x-3} x\right ) \log \left (e^{x-3}-x\right ) \log \left (\log \left (e^{x-3}-x\right )\right )\right ) \log \left (\log \left (\log \left (e^{x-3}-x\right )\right )\right )+e^{2 e^{5 e^x}-4} \left (-50 x^2+e^{x+5 e^x} \left (250 e^{x-3} x^2-250 x^3\right )+50 e^{x-3} x\right ) \log \left (e^{x-3}-x\right ) \log \left (\log \left (e^{x-3}-x\right )\right )}{\left (e^{x-3}-x\right ) \log \left (e^{x-3}-x\right ) \log \left (\log \left (e^{x-3}-x\right )\right )} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {50 x \left (e^{e^{5 e^x}}+e^2 \log \left (\log \left (\log \left (e^{x-3}-x\right )\right )\right )\right ) \left (e^{x+2} x-e^5 x+\left (e^x-e^3 x\right ) \log \left (e^{x-3}-x\right ) \log \left (\log \left (e^{x-3}-x\right )\right ) \left (e^{e^{5 e^x}} \left (5 e^{x+5 e^x} x+1\right )+e^2 \log \left (\log \left (\log \left (e^{x-3}-x\right )\right )\right )\right )\right )}{e^4 \left (e^x-e^3 x\right ) \log \left (e^{x-3}-x\right ) \log \left (\log \left (e^{x-3}-x\right )\right )}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {50 \int -\frac {x \left (e^2 \log \left (\log \left (\log \left (e^{x-3}-x\right )\right )\right )+e^{e^{5 e^x}}\right ) \left (-e^{x+2} x+e^5 x-\left (e^x-e^3 x\right ) \log \left (e^{x-3}-x\right ) \log \left (\log \left (e^{x-3}-x\right )\right ) \left (e^{e^{5 e^x}} \left (5 e^{x+5 e^x} x+1\right )+e^2 \log \left (\log \left (\log \left (e^{x-3}-x\right )\right )\right )\right )\right )}{\left (e^x-e^3 x\right ) \log \left (e^{x-3}-x\right ) \log \left (\log \left (e^{x-3}-x\right )\right )}dx}{e^4}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {50 \int \frac {x \left (e^2 \log \left (\log \left (\log \left (e^{x-3}-x\right )\right )\right )+e^{e^{5 e^x}}\right ) \left (-e^{x+2} x+e^5 x-\left (e^x-e^3 x\right ) \log \left (e^{x-3}-x\right ) \log \left (\log \left (e^{x-3}-x\right )\right ) \left (e^{e^{5 e^x}} \left (5 e^{x+5 e^x} x+1\right )+e^2 \log \left (\log \left (\log \left (e^{x-3}-x\right )\right )\right )\right )\right )}{\left (e^x-e^3 x\right ) \log \left (e^{x-3}-x\right ) \log \left (\log \left (e^{x-3}-x\right )\right )}dx}{e^4}\)

\(\Big \downarrow \) 7238

\(\displaystyle \frac {25 x^2 \left (e^{e^{5 e^x}}+e^2 \log \left (\log \left (\log \left (e^{x-3}-x\right )\right )\right )\right )^2}{e^4}\)

Input:

Int[(E^(-2 + E^(5*E^x))*(-50*x^2 + 50*E^(-3 + x)*x^2) + E^(-4 + 2*E^(5*E^x 
))*(50*E^(-3 + x)*x - 50*x^2 + E^(5*E^x + x)*(250*E^(-3 + x)*x^2 - 250*x^3 
))*Log[E^(-3 + x) - x]*Log[Log[E^(-3 + x) - x]] + (-50*x^2 + 50*E^(-3 + x) 
*x^2 + E^(-2 + E^(5*E^x))*(100*E^(-3 + x)*x - 100*x^2 + E^(5*E^x + x)*(250 
*E^(-3 + x)*x^2 - 250*x^3))*Log[E^(-3 + x) - x]*Log[Log[E^(-3 + x) - x]])* 
Log[Log[Log[E^(-3 + x) - x]]] + (50*E^(-3 + x)*x - 50*x^2)*Log[E^(-3 + x) 
- x]*Log[Log[E^(-3 + x) - x]]*Log[Log[Log[E^(-3 + x) - x]]]^2)/((E^(-3 + x 
) - x)*Log[E^(-3 + x) - x]*Log[Log[E^(-3 + x) - x]]),x]
 

Output:

(25*x^2*(E^E^(5*E^x) + E^2*Log[Log[Log[E^(-3 + x) - x]]])^2)/E^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 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 7239

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(58\) vs. \(2(27)=54\).

Time = 0.10 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.90

\[25 x^{2} {\mathrm e}^{2 \,{\mathrm e}^{5 \,{\mathrm e}^{x}}-4}+50 x^{2} {\mathrm e}^{{\mathrm e}^{5 \,{\mathrm e}^{x}}-2} \ln \left (\ln \left (\ln \left ({\mathrm e}^{-3+x}-x \right )\right )\right )+25 {\ln \left (\ln \left (\ln \left ({\mathrm e}^{-3+x}-x \right )\right )\right )}^{2} x^{2}\]

Input:

int(((50*x*exp(-3+x)-50*x^2)*ln(exp(-3+x)-x)*ln(ln(exp(-3+x)-x))*ln(ln(ln( 
exp(-3+x)-x)))^2+(((250*x^2*exp(-3+x)-250*x^3)*exp(x)*exp(5*exp(x))+100*x* 
exp(-3+x)-100*x^2)*ln(exp(-3+x)-x)*exp(exp(5*exp(x))-2)*ln(ln(exp(-3+x)-x) 
)+50*x^2*exp(-3+x)-50*x^2)*ln(ln(ln(exp(-3+x)-x)))+((250*x^2*exp(-3+x)-250 
*x^3)*exp(x)*exp(5*exp(x))+50*x*exp(-3+x)-50*x^2)*ln(exp(-3+x)-x)*exp(exp( 
5*exp(x))-2)^2*ln(ln(exp(-3+x)-x))+(50*x^2*exp(-3+x)-50*x^2)*exp(exp(5*exp 
(x))-2))/(exp(-3+x)-x)/ln(exp(-3+x)-x)/ln(ln(exp(-3+x)-x)),x)
 

Output:

25*x^2*exp(2*exp(5*exp(x))-4)+50*x^2*exp(exp(5*exp(x))-2)*ln(ln(ln(exp(-3+ 
x)-x)))+25*ln(ln(ln(exp(-3+x)-x)))^2*x^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (27) = 54\).

Time = 0.10 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.81 \[ \int \frac {e^{-2+e^{5 e^x}} \left (-50 x^2+50 e^{-3+x} x^2\right )+e^{-4+2 e^{5 e^x}} \left (50 e^{-3+x} x-50 x^2+e^{5 e^x+x} \left (250 e^{-3+x} x^2-250 x^3\right )\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right )+\left (-50 x^2+50 e^{-3+x} x^2+e^{-2+e^{5 e^x}} \left (100 e^{-3+x} x-100 x^2+e^{5 e^x+x} \left (250 e^{-3+x} x^2-250 x^3\right )\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right )\right ) \log \left (\log \left (\log \left (e^{-3+x}-x\right )\right )\right )+\left (50 e^{-3+x} x-50 x^2\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right ) \log ^2\left (\log \left (\log \left (e^{-3+x}-x\right )\right )\right )}{\left (e^{-3+x}-x\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right )} \, dx=50 \, x^{2} e^{\left ({\left (e^{\left (x + 5 \, e^{x}\right )} - 2 \, e^{x}\right )} e^{\left (-x\right )}\right )} \log \left (\log \left (\log \left (-{\left (x e^{3} - e^{x}\right )} e^{\left (-3\right )}\right )\right )\right ) + 25 \, x^{2} \log \left (\log \left (\log \left (-{\left (x e^{3} - e^{x}\right )} e^{\left (-3\right )}\right )\right )\right )^{2} + 25 \, x^{2} e^{\left (2 \, {\left (e^{\left (x + 5 \, e^{x}\right )} - 2 \, e^{x}\right )} e^{\left (-x\right )}\right )} \] Input:

integrate(((50*x*exp(-3+x)-50*x^2)*log(exp(-3+x)-x)*log(log(exp(-3+x)-x))* 
log(log(log(exp(-3+x)-x)))^2+(((250*x^2*exp(-3+x)-250*x^3)*exp(x)*exp(5*ex 
p(x))+100*x*exp(-3+x)-100*x^2)*log(exp(-3+x)-x)*exp(exp(5*exp(x))-2)*log(l 
og(exp(-3+x)-x))+50*x^2*exp(-3+x)-50*x^2)*log(log(log(exp(-3+x)-x)))+((250 
*x^2*exp(-3+x)-250*x^3)*exp(x)*exp(5*exp(x))+50*x*exp(-3+x)-50*x^2)*log(ex 
p(-3+x)-x)*exp(exp(5*exp(x))-2)^2*log(log(exp(-3+x)-x))+(50*x^2*exp(-3+x)- 
50*x^2)*exp(exp(5*exp(x))-2))/(exp(-3+x)-x)/log(exp(-3+x)-x)/log(log(exp(- 
3+x)-x)),x, algorithm="fricas")
 

Output:

50*x^2*e^((e^(x + 5*e^x) - 2*e^x)*e^(-x))*log(log(log(-(x*e^3 - e^x)*e^(-3 
)))) + 25*x^2*log(log(log(-(x*e^3 - e^x)*e^(-3))))^2 + 25*x^2*e^(2*(e^(x + 
 5*e^x) - 2*e^x)*e^(-x))
 

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{-2+e^{5 e^x}} \left (-50 x^2+50 e^{-3+x} x^2\right )+e^{-4+2 e^{5 e^x}} \left (50 e^{-3+x} x-50 x^2+e^{5 e^x+x} \left (250 e^{-3+x} x^2-250 x^3\right )\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right )+\left (-50 x^2+50 e^{-3+x} x^2+e^{-2+e^{5 e^x}} \left (100 e^{-3+x} x-100 x^2+e^{5 e^x+x} \left (250 e^{-3+x} x^2-250 x^3\right )\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right )\right ) \log \left (\log \left (\log \left (e^{-3+x}-x\right )\right )\right )+\left (50 e^{-3+x} x-50 x^2\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right ) \log ^2\left (\log \left (\log \left (e^{-3+x}-x\right )\right )\right )}{\left (e^{-3+x}-x\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right )} \, dx=\text {Timed out} \] Input:

integrate(((50*x*exp(-3+x)-50*x**2)*ln(exp(-3+x)-x)*ln(ln(exp(-3+x)-x))*ln 
(ln(ln(exp(-3+x)-x)))**2+(((250*x**2*exp(-3+x)-250*x**3)*exp(x)*exp(5*exp( 
x))+100*x*exp(-3+x)-100*x**2)*ln(exp(-3+x)-x)*exp(exp(5*exp(x))-2)*ln(ln(e 
xp(-3+x)-x))+50*x**2*exp(-3+x)-50*x**2)*ln(ln(ln(exp(-3+x)-x)))+((250*x**2 
*exp(-3+x)-250*x**3)*exp(x)*exp(5*exp(x))+50*x*exp(-3+x)-50*x**2)*ln(exp(- 
3+x)-x)*exp(exp(5*exp(x))-2)**2*ln(ln(exp(-3+x)-x))+(50*x**2*exp(-3+x)-50* 
x**2)*exp(exp(5*exp(x))-2))/(exp(-3+x)-x)/ln(exp(-3+x)-x)/ln(ln(exp(-3+x)- 
x)),x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (27) = 54\).

Time = 0.30 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.06 \[ \int \frac {e^{-2+e^{5 e^x}} \left (-50 x^2+50 e^{-3+x} x^2\right )+e^{-4+2 e^{5 e^x}} \left (50 e^{-3+x} x-50 x^2+e^{5 e^x+x} \left (250 e^{-3+x} x^2-250 x^3\right )\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right )+\left (-50 x^2+50 e^{-3+x} x^2+e^{-2+e^{5 e^x}} \left (100 e^{-3+x} x-100 x^2+e^{5 e^x+x} \left (250 e^{-3+x} x^2-250 x^3\right )\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right )\right ) \log \left (\log \left (\log \left (e^{-3+x}-x\right )\right )\right )+\left (50 e^{-3+x} x-50 x^2\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right ) \log ^2\left (\log \left (\log \left (e^{-3+x}-x\right )\right )\right )}{\left (e^{-3+x}-x\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right )} \, dx=25 \, {\left (x^{2} e^{4} \log \left (\log \left (\log \left (-x e^{3} + e^{x}\right ) - 3\right )\right )^{2} + 2 \, x^{2} e^{\left (e^{\left (5 \, e^{x}\right )} + 2\right )} \log \left (\log \left (\log \left (-x e^{3} + e^{x}\right ) - 3\right )\right ) + x^{2} e^{\left (2 \, e^{\left (5 \, e^{x}\right )}\right )}\right )} e^{\left (-4\right )} \] Input:

integrate(((50*x*exp(-3+x)-50*x^2)*log(exp(-3+x)-x)*log(log(exp(-3+x)-x))* 
log(log(log(exp(-3+x)-x)))^2+(((250*x^2*exp(-3+x)-250*x^3)*exp(x)*exp(5*ex 
p(x))+100*x*exp(-3+x)-100*x^2)*log(exp(-3+x)-x)*exp(exp(5*exp(x))-2)*log(l 
og(exp(-3+x)-x))+50*x^2*exp(-3+x)-50*x^2)*log(log(log(exp(-3+x)-x)))+((250 
*x^2*exp(-3+x)-250*x^3)*exp(x)*exp(5*exp(x))+50*x*exp(-3+x)-50*x^2)*log(ex 
p(-3+x)-x)*exp(exp(5*exp(x))-2)^2*log(log(exp(-3+x)-x))+(50*x^2*exp(-3+x)- 
50*x^2)*exp(exp(5*exp(x))-2))/(exp(-3+x)-x)/log(exp(-3+x)-x)/log(log(exp(- 
3+x)-x)),x, algorithm="maxima")
 

Output:

25*(x^2*e^4*log(log(log(-x*e^3 + e^x) - 3))^2 + 2*x^2*e^(e^(5*e^x) + 2)*lo 
g(log(log(-x*e^3 + e^x) - 3)) + x^2*e^(2*e^(5*e^x)))*e^(-4)
 

Giac [F]

\[ \int \frac {e^{-2+e^{5 e^x}} \left (-50 x^2+50 e^{-3+x} x^2\right )+e^{-4+2 e^{5 e^x}} \left (50 e^{-3+x} x-50 x^2+e^{5 e^x+x} \left (250 e^{-3+x} x^2-250 x^3\right )\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right )+\left (-50 x^2+50 e^{-3+x} x^2+e^{-2+e^{5 e^x}} \left (100 e^{-3+x} x-100 x^2+e^{5 e^x+x} \left (250 e^{-3+x} x^2-250 x^3\right )\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right )\right ) \log \left (\log \left (\log \left (e^{-3+x}-x\right )\right )\right )+\left (50 e^{-3+x} x-50 x^2\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right ) \log ^2\left (\log \left (\log \left (e^{-3+x}-x\right )\right )\right )}{\left (e^{-3+x}-x\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right )} \, dx=\int { \frac {50 \, {\left ({\left (x^{2} - x e^{\left (x - 3\right )}\right )} \log \left (-x + e^{\left (x - 3\right )}\right ) \log \left (\log \left (-x + e^{\left (x - 3\right )}\right )\right ) \log \left (\log \left (\log \left (-x + e^{\left (x - 3\right )}\right )\right )\right )^{2} + {\left (x^{2} + 5 \, {\left (x^{3} - x^{2} e^{\left (x - 3\right )}\right )} e^{\left (x + 5 \, e^{x}\right )} - x e^{\left (x - 3\right )}\right )} e^{\left (2 \, e^{\left (5 \, e^{x}\right )} - 4\right )} \log \left (-x + e^{\left (x - 3\right )}\right ) \log \left (\log \left (-x + e^{\left (x - 3\right )}\right )\right ) - {\left (x^{2} e^{\left (x - 3\right )} - x^{2}\right )} e^{\left (e^{\left (5 \, e^{x}\right )} - 2\right )} + {\left ({\left (2 \, x^{2} + 5 \, {\left (x^{3} - x^{2} e^{\left (x - 3\right )}\right )} e^{\left (x + 5 \, e^{x}\right )} - 2 \, x e^{\left (x - 3\right )}\right )} e^{\left (e^{\left (5 \, e^{x}\right )} - 2\right )} \log \left (-x + e^{\left (x - 3\right )}\right ) \log \left (\log \left (-x + e^{\left (x - 3\right )}\right )\right ) - x^{2} e^{\left (x - 3\right )} + x^{2}\right )} \log \left (\log \left (\log \left (-x + e^{\left (x - 3\right )}\right )\right )\right )\right )}}{{\left (x - e^{\left (x - 3\right )}\right )} \log \left (-x + e^{\left (x - 3\right )}\right ) \log \left (\log \left (-x + e^{\left (x - 3\right )}\right )\right )} \,d x } \] Input:

integrate(((50*x*exp(-3+x)-50*x^2)*log(exp(-3+x)-x)*log(log(exp(-3+x)-x))* 
log(log(log(exp(-3+x)-x)))^2+(((250*x^2*exp(-3+x)-250*x^3)*exp(x)*exp(5*ex 
p(x))+100*x*exp(-3+x)-100*x^2)*log(exp(-3+x)-x)*exp(exp(5*exp(x))-2)*log(l 
og(exp(-3+x)-x))+50*x^2*exp(-3+x)-50*x^2)*log(log(log(exp(-3+x)-x)))+((250 
*x^2*exp(-3+x)-250*x^3)*exp(x)*exp(5*exp(x))+50*x*exp(-3+x)-50*x^2)*log(ex 
p(-3+x)-x)*exp(exp(5*exp(x))-2)^2*log(log(exp(-3+x)-x))+(50*x^2*exp(-3+x)- 
50*x^2)*exp(exp(5*exp(x))-2))/(exp(-3+x)-x)/log(exp(-3+x)-x)/log(log(exp(- 
3+x)-x)),x, algorithm="giac")
 

Output:

integrate(50*((x^2 - x*e^(x - 3))*log(-x + e^(x - 3))*log(log(-x + e^(x - 
3)))*log(log(log(-x + e^(x - 3))))^2 + (x^2 + 5*(x^3 - x^2*e^(x - 3))*e^(x 
 + 5*e^x) - x*e^(x - 3))*e^(2*e^(5*e^x) - 4)*log(-x + e^(x - 3))*log(log(- 
x + e^(x - 3))) - (x^2*e^(x - 3) - x^2)*e^(e^(5*e^x) - 2) + ((2*x^2 + 5*(x 
^3 - x^2*e^(x - 3))*e^(x + 5*e^x) - 2*x*e^(x - 3))*e^(e^(5*e^x) - 2)*log(- 
x + e^(x - 3))*log(log(-x + e^(x - 3))) - x^2*e^(x - 3) + x^2)*log(log(log 
(-x + e^(x - 3)))))/((x - e^(x - 3))*log(-x + e^(x - 3))*log(log(-x + e^(x 
 - 3)))), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{-2+e^{5 e^x}} \left (-50 x^2+50 e^{-3+x} x^2\right )+e^{-4+2 e^{5 e^x}} \left (50 e^{-3+x} x-50 x^2+e^{5 e^x+x} \left (250 e^{-3+x} x^2-250 x^3\right )\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right )+\left (-50 x^2+50 e^{-3+x} x^2+e^{-2+e^{5 e^x}} \left (100 e^{-3+x} x-100 x^2+e^{5 e^x+x} \left (250 e^{-3+x} x^2-250 x^3\right )\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right )\right ) \log \left (\log \left (\log \left (e^{-3+x}-x\right )\right )\right )+\left (50 e^{-3+x} x-50 x^2\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right ) \log ^2\left (\log \left (\log \left (e^{-3+x}-x\right )\right )\right )}{\left (e^{-3+x}-x\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right )} \, dx=-\int \frac {\ln \left (\ln \left ({\mathrm {e}}^{x-3}-x\right )\right )\,\ln \left ({\mathrm {e}}^{x-3}-x\right )\,\left (50\,x\,{\mathrm {e}}^{x-3}-50\,x^2\right )\,{\ln \left (\ln \left (\ln \left ({\mathrm {e}}^{x-3}-x\right )\right )\right )}^2+\left (50\,x^2\,{\mathrm {e}}^{x-3}-50\,x^2+\ln \left (\ln \left ({\mathrm {e}}^{x-3}-x\right )\right )\,{\mathrm {e}}^{{\mathrm {e}}^{5\,{\mathrm {e}}^x}-2}\,\ln \left ({\mathrm {e}}^{x-3}-x\right )\,\left (100\,x\,{\mathrm {e}}^{x-3}-100\,x^2+{\mathrm {e}}^{x+5\,{\mathrm {e}}^x}\,\left (250\,x^2\,{\mathrm {e}}^{x-3}-250\,x^3\right )\right )\right )\,\ln \left (\ln \left (\ln \left ({\mathrm {e}}^{x-3}-x\right )\right )\right )+{\mathrm {e}}^{{\mathrm {e}}^{5\,{\mathrm {e}}^x}-2}\,\left (50\,x^2\,{\mathrm {e}}^{x-3}-50\,x^2\right )+\ln \left (\ln \left ({\mathrm {e}}^{x-3}-x\right )\right )\,{\mathrm {e}}^{2\,{\mathrm {e}}^{5\,{\mathrm {e}}^x}-4}\,\ln \left ({\mathrm {e}}^{x-3}-x\right )\,\left (50\,x\,{\mathrm {e}}^{x-3}-50\,x^2+{\mathrm {e}}^{x+5\,{\mathrm {e}}^x}\,\left (250\,x^2\,{\mathrm {e}}^{x-3}-250\,x^3\right )\right )}{\ln \left (\ln \left ({\mathrm {e}}^{x-3}-x\right )\right )\,\ln \left ({\mathrm {e}}^{x-3}-x\right )\,\left (x-{\mathrm {e}}^{x-3}\right )} \,d x \] Input:

int(-(log(log(log(exp(x - 3) - x)))*(50*x^2*exp(x - 3) - 50*x^2 + log(log( 
exp(x - 3) - x))*exp(exp(5*exp(x)) - 2)*log(exp(x - 3) - x)*(100*x*exp(x - 
 3) - 100*x^2 + exp(5*exp(x))*exp(x)*(250*x^2*exp(x - 3) - 250*x^3))) + ex 
p(exp(5*exp(x)) - 2)*(50*x^2*exp(x - 3) - 50*x^2) + log(log(exp(x - 3) - x 
))*log(log(log(exp(x - 3) - x)))^2*log(exp(x - 3) - x)*(50*x*exp(x - 3) - 
50*x^2) + log(log(exp(x - 3) - x))*exp(2*exp(5*exp(x)) - 4)*log(exp(x - 3) 
 - x)*(50*x*exp(x - 3) - 50*x^2 + exp(5*exp(x))*exp(x)*(250*x^2*exp(x - 3) 
 - 250*x^3)))/(log(log(exp(x - 3) - x))*log(exp(x - 3) - x)*(x - exp(x - 3 
))),x)
 

Output:

-int((log(log(log(exp(x - 3) - x)))*(50*x^2*exp(x - 3) - 50*x^2 + log(log( 
exp(x - 3) - x))*exp(exp(5*exp(x)) - 2)*log(exp(x - 3) - x)*(100*x*exp(x - 
 3) - 100*x^2 + exp(x + 5*exp(x))*(250*x^2*exp(x - 3) - 250*x^3))) + exp(e 
xp(5*exp(x)) - 2)*(50*x^2*exp(x - 3) - 50*x^2) + log(log(exp(x - 3) - x))* 
log(log(log(exp(x - 3) - x)))^2*log(exp(x - 3) - x)*(50*x*exp(x - 3) - 50* 
x^2) + log(log(exp(x - 3) - x))*exp(2*exp(5*exp(x)) - 4)*log(exp(x - 3) - 
x)*(50*x*exp(x - 3) - 50*x^2 + exp(x + 5*exp(x))*(250*x^2*exp(x - 3) - 250 
*x^3)))/(log(log(exp(x - 3) - x))*log(exp(x - 3) - x)*(x - exp(x - 3))), x 
)
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.39 \[ \int \frac {e^{-2+e^{5 e^x}} \left (-50 x^2+50 e^{-3+x} x^2\right )+e^{-4+2 e^{5 e^x}} \left (50 e^{-3+x} x-50 x^2+e^{5 e^x+x} \left (250 e^{-3+x} x^2-250 x^3\right )\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right )+\left (-50 x^2+50 e^{-3+x} x^2+e^{-2+e^{5 e^x}} \left (100 e^{-3+x} x-100 x^2+e^{5 e^x+x} \left (250 e^{-3+x} x^2-250 x^3\right )\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right )\right ) \log \left (\log \left (\log \left (e^{-3+x}-x\right )\right )\right )+\left (50 e^{-3+x} x-50 x^2\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right ) \log ^2\left (\log \left (\log \left (e^{-3+x}-x\right )\right )\right )}{\left (e^{-3+x}-x\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right )} \, dx=\frac {25 x^{2} \left (e^{2 e^{5 e^{x}}}+2 e^{e^{5 e^{x}}} \mathrm {log}\left (\mathrm {log}\left (\mathrm {log}\left (\frac {e^{x}-e^{3} x}{e^{3}}\right )\right )\right ) e^{2}+{\mathrm {log}\left (\mathrm {log}\left (\mathrm {log}\left (\frac {e^{x}-e^{3} x}{e^{3}}\right )\right )\right )}^{2} e^{4}\right )}{e^{4}} \] Input:

int(((50*x*exp(-3+x)-50*x^2)*log(exp(-3+x)-x)*log(log(exp(-3+x)-x))*log(lo 
g(log(exp(-3+x)-x)))^2+(((250*x^2*exp(-3+x)-250*x^3)*exp(x)*exp(5*exp(x))+ 
100*x*exp(-3+x)-100*x^2)*log(exp(-3+x)-x)*exp(exp(5*exp(x))-2)*log(log(exp 
(-3+x)-x))+50*x^2*exp(-3+x)-50*x^2)*log(log(log(exp(-3+x)-x)))+((250*x^2*e 
xp(-3+x)-250*x^3)*exp(x)*exp(5*exp(x))+50*x*exp(-3+x)-50*x^2)*log(exp(-3+x 
)-x)*exp(exp(5*exp(x))-2)^2*log(log(exp(-3+x)-x))+(50*x^2*exp(-3+x)-50*x^2 
)*exp(exp(5*exp(x))-2))/(exp(-3+x)-x)/log(exp(-3+x)-x)/log(log(exp(-3+x)-x 
)),x)
 

Output:

(25*x**2*(e**(2*e**(5*e**x)) + 2*e**(e**(5*e**x))*log(log(log((e**x - e**3 
*x)/e**3)))*e**2 + log(log(log((e**x - e**3*x)/e**3)))**2*e**4))/e**4