\(\int e^{-e^{4 x}} (e^{2 e^{-e^{4 x}} (x+e^{e^{4 x}} \log (2))} (2-8 e^{4 x} x)+e^{e^{4 x}} (6 e^{6 x}+8 x+e^{3 x} (4+12 x))+e^{e^{-e^{4 x}} (x+e^{e^{4 x}} \log (2))} (-2 e^{3 x}+e^{e^{4 x}} (-4-6 e^{3 x})-4 x+e^{4 x} (8 e^{3 x} x+16 x^2))) \, dx\) [2522]

Optimal result

Integrand size = 150, antiderivative size = 26 \[ \int e^{-e^{4 x}} \left (e^{2 e^{-e^{4 x}} \left (x+e^{e^{4 x}} \log (2)\right )} \left (2-8 e^{4 x} x\right )+e^{e^{4 x}} \left (6 e^{6 x}+8 x+e^{3 x} (4+12 x)\right )+e^{e^{-e^{4 x}} \left (x+e^{e^{4 x}} \log (2)\right )} \left (-2 e^{3 x}+e^{e^{4 x}} \left (-4-6 e^{3 x}\right )-4 x+e^{4 x} \left (8 e^{3 x} x+16 x^2\right )\right )\right ) \, dx=\left (e^{3 x}-2 e^{e^{-e^{4 x}} x}+2 x\right )^2 \] Output:

(2*x+exp(3*x)-exp(x/exp(exp(4*x))+ln(2)))^2
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int e^{-e^{4 x}} \left (e^{2 e^{-e^{4 x}} \left (x+e^{e^{4 x}} \log (2)\right )} \left (2-8 e^{4 x} x\right )+e^{e^{4 x}} \left (6 e^{6 x}+8 x+e^{3 x} (4+12 x)\right )+e^{e^{-e^{4 x}} \left (x+e^{e^{4 x}} \log (2)\right )} \left (-2 e^{3 x}+e^{e^{4 x}} \left (-4-6 e^{3 x}\right )-4 x+e^{4 x} \left (8 e^{3 x} x+16 x^2\right )\right )\right ) \, dx=\left (e^{3 x}-2 e^{e^{-e^{4 x}} x}+2 x\right )^2 \] Input:

Integrate[(E^((2*(x + E^E^(4*x)*Log[2]))/E^E^(4*x))*(2 - 8*E^(4*x)*x) + E^ 
E^(4*x)*(6*E^(6*x) + 8*x + E^(3*x)*(4 + 12*x)) + E^((x + E^E^(4*x)*Log[2]) 
/E^E^(4*x))*(-2*E^(3*x) + E^E^(4*x)*(-4 - 6*E^(3*x)) - 4*x + E^(4*x)*(8*E^ 
(3*x)*x + 16*x^2)))/E^E^(4*x),x]
 

Output:

(E^(3*x) - 2*E^(x/E^E^(4*x)) + 2*x)^2
 

Rubi [A] (verified)

Time = 1.23 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, Rules used = {7239, 27, 7237}

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.

\(\displaystyle \int e^{-e^{4 x}} \left (\left (e^{4 x} \left (16 x^2+8 e^{3 x} x\right )+e^{e^{4 x}} \left (-6 e^{3 x}-4\right )-2 e^{3 x}-4 x\right ) e^{e^{-e^{4 x}} \left (x+e^{e^{4 x}} \log (2)\right )}+e^{e^{4 x}} \left (8 x+6 e^{6 x}+e^{3 x} (12 x+4)\right )+\left (2-8 e^{4 x} x\right ) e^{2 e^{-e^{4 x}} \left (x+e^{e^{4 x}} \log (2)\right )}\right ) \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int 2 e^{-e^{4 x}} \left (2 x+e^{3 x}-2 e^{e^{-e^{4 x}} x}\right ) \left (8 e^{\left (e^{-e^{4 x}}+4\right ) x} x+2 e^{e^{4 x}}-2 e^{e^{-e^{4 x}} x}+3 e^{3 x+e^{4 x}}\right )dx\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \int e^{-e^{4 x}} \left (2 x+e^{3 x}-2 e^{e^{-e^{4 x}} x}\right ) \left (8 e^{\left (4+e^{-e^{4 x}}\right ) x} x+2 e^{e^{4 x}}-2 e^{e^{-e^{4 x}} x}+3 e^{3 x+e^{4 x}}\right )dx\)

\(\Big \downarrow \) 7237

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

Input:

Int[(E^((2*(x + E^E^(4*x)*Log[2]))/E^E^(4*x))*(2 - 8*E^(4*x)*x) + E^E^(4*x 
)*(6*E^(6*x) + 8*x + E^(3*x)*(4 + 12*x)) + E^((x + E^E^(4*x)*Log[2])/E^E^( 
4*x))*(-2*E^(3*x) + E^E^(4*x)*(-4 - 6*E^(3*x)) - 4*x + E^(4*x)*(8*E^(3*x)* 
x + 16*x^2)))/E^E^(4*x),x]
 

Output:

(E^(3*x) - 2*E^(x/E^E^(4*x)) + 2*x)^2
 

Defintions of rubi rules used

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 7237

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Si 
mp[q*(y^(m + 1)/(m + 1)), x] /;  !FalseQ[q]] /; FreeQ[m, 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. \(52\) vs. \(2(25)=50\).

Time = 0.56 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.04

method	result	size
risch	\({\mathrm e}^{6 x}+4 x \,{\mathrm e}^{3 x}+4 x^{2}+4 \,{\mathrm e}^{2 \,{\mathrm e}^{-{\mathrm e}^{4 x}} x}+2 \left (-2 \,{\mathrm e}^{3 x}-4 x \right ) {\mathrm e}^{{\mathrm e}^{-{\mathrm e}^{4 x}} x}\)	\(53\)
parallelrisch	\(4 x^{2}+4 x \,{\mathrm e}^{3 x}-4 \,{\mathrm e}^{\left (\ln \left (2\right ) {\mathrm e}^{{\mathrm e}^{4 x}}+x \right ) {\mathrm e}^{-{\mathrm e}^{4 x}}} x +{\mathrm e}^{6 x}-2 \,{\mathrm e}^{\left (\ln \left (2\right ) {\mathrm e}^{{\mathrm e}^{4 x}}+x \right ) {\mathrm e}^{-{\mathrm e}^{4 x}}} {\mathrm e}^{3 x}+{\mathrm e}^{2 \left (\ln \left (2\right ) {\mathrm e}^{{\mathrm e}^{4 x}}+x \right ) {\mathrm e}^{-{\mathrm e}^{4 x}}}\)	\(88\)

Input:

int(((-8*x*exp(4*x)+2)*exp((ln(2)*exp(exp(4*x))+x)/exp(exp(4*x)))^2+((-6*e 
xp(3*x)-4)*exp(exp(4*x))+(8*x*exp(3*x)+16*x^2)*exp(4*x)-2*exp(3*x)-4*x)*ex 
p((ln(2)*exp(exp(4*x))+x)/exp(exp(4*x)))+(6*exp(3*x)^2+(12*x+4)*exp(3*x)+8 
*x)*exp(exp(4*x)))/exp(exp(4*x)),x,method=_RETURNVERBOSE)
 

Output:

exp(6*x)+4*x*exp(3*x)+4*x^2+4*exp(2*exp(-exp(4*x))*x)+2*(-2*exp(3*x)-4*x)* 
exp(exp(-exp(4*x))*x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (25) = 50\).

Time = 0.10 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.54 \[ \int e^{-e^{4 x}} \left (e^{2 e^{-e^{4 x}} \left (x+e^{e^{4 x}} \log (2)\right )} \left (2-8 e^{4 x} x\right )+e^{e^{4 x}} \left (6 e^{6 x}+8 x+e^{3 x} (4+12 x)\right )+e^{e^{-e^{4 x}} \left (x+e^{e^{4 x}} \log (2)\right )} \left (-2 e^{3 x}+e^{e^{4 x}} \left (-4-6 e^{3 x}\right )-4 x+e^{4 x} \left (8 e^{3 x} x+16 x^2\right )\right )\right ) \, dx=4 \, x^{2} - 2 \, {\left (2 \, x + e^{\left (3 \, x\right )}\right )} e^{\left ({\left (e^{\left (e^{\left (4 \, x\right )}\right )} \log \left (2\right ) + x\right )} e^{\left (-e^{\left (4 \, x\right )}\right )}\right )} + 4 \, x e^{\left (3 \, x\right )} + e^{\left (2 \, {\left (e^{\left (e^{\left (4 \, x\right )}\right )} \log \left (2\right ) + x\right )} e^{\left (-e^{\left (4 \, x\right )}\right )}\right )} + e^{\left (6 \, x\right )} \] Input:

integrate(((-8*x*exp(4*x)+2)*exp((log(2)*exp(exp(4*x))+x)/exp(exp(4*x)))^2 
+((-6*exp(3*x)-4)*exp(exp(4*x))+(8*x*exp(3*x)+16*x^2)*exp(4*x)-2*exp(3*x)- 
4*x)*exp((log(2)*exp(exp(4*x))+x)/exp(exp(4*x)))+(6*exp(3*x)^2+(12*x+4)*ex 
p(3*x)+8*x)*exp(exp(4*x)))/exp(exp(4*x)),x, algorithm="fricas")
 

Output:

4*x^2 - 2*(2*x + e^(3*x))*e^((e^(e^(4*x))*log(2) + x)*e^(-e^(4*x))) + 4*x* 
e^(3*x) + e^(2*(e^(e^(4*x))*log(2) + x)*e^(-e^(4*x))) + e^(6*x)
 

Sympy [F(-1)]

Timed out. \[ \int e^{-e^{4 x}} \left (e^{2 e^{-e^{4 x}} \left (x+e^{e^{4 x}} \log (2)\right )} \left (2-8 e^{4 x} x\right )+e^{e^{4 x}} \left (6 e^{6 x}+8 x+e^{3 x} (4+12 x)\right )+e^{e^{-e^{4 x}} \left (x+e^{e^{4 x}} \log (2)\right )} \left (-2 e^{3 x}+e^{e^{4 x}} \left (-4-6 e^{3 x}\right )-4 x+e^{4 x} \left (8 e^{3 x} x+16 x^2\right )\right )\right ) \, dx=\text {Timed out} \] Input:

integrate(((-8*x*exp(4*x)+2)*exp((ln(2)*exp(exp(4*x))+x)/exp(exp(4*x)))**2 
+((-6*exp(3*x)-4)*exp(exp(4*x))+(8*x*exp(3*x)+16*x**2)*exp(4*x)-2*exp(3*x) 
-4*x)*exp((ln(2)*exp(exp(4*x))+x)/exp(exp(4*x)))+(6*exp(3*x)**2+(12*x+4)*e 
xp(3*x)+8*x)*exp(exp(4*x)))/exp(exp(4*x)),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int e^{-e^{4 x}} \left (e^{2 e^{-e^{4 x}} \left (x+e^{e^{4 x}} \log (2)\right )} \left (2-8 e^{4 x} x\right )+e^{e^{4 x}} \left (6 e^{6 x}+8 x+e^{3 x} (4+12 x)\right )+e^{e^{-e^{4 x}} \left (x+e^{e^{4 x}} \log (2)\right )} \left (-2 e^{3 x}+e^{e^{4 x}} \left (-4-6 e^{3 x}\right )-4 x+e^{4 x} \left (8 e^{3 x} x+16 x^2\right )\right )\right ) \, dx=\int { -2 \, {\left ({\left (4 \, x e^{\left (4 \, x\right )} - 1\right )} e^{\left (2 \, {\left (e^{\left (e^{\left (4 \, x\right )}\right )} \log \left (2\right ) + x\right )} e^{\left (-e^{\left (4 \, x\right )}\right )}\right )} - {\left (4 \, {\left (2 \, x^{2} + x e^{\left (3 \, x\right )}\right )} e^{\left (4 \, x\right )} - {\left (3 \, e^{\left (3 \, x\right )} + 2\right )} e^{\left (e^{\left (4 \, x\right )}\right )} - 2 \, x - e^{\left (3 \, x\right )}\right )} e^{\left ({\left (e^{\left (e^{\left (4 \, x\right )}\right )} \log \left (2\right ) + x\right )} e^{\left (-e^{\left (4 \, x\right )}\right )}\right )} - {\left (2 \, {\left (3 \, x + 1\right )} e^{\left (3 \, x\right )} + 4 \, x + 3 \, e^{\left (6 \, x\right )}\right )} e^{\left (e^{\left (4 \, x\right )}\right )}\right )} e^{\left (-e^{\left (4 \, x\right )}\right )} \,d x } \] Input:

integrate(((-8*x*exp(4*x)+2)*exp((log(2)*exp(exp(4*x))+x)/exp(exp(4*x)))^2 
+((-6*exp(3*x)-4)*exp(exp(4*x))+(8*x*exp(3*x)+16*x^2)*exp(4*x)-2*exp(3*x)- 
4*x)*exp((log(2)*exp(exp(4*x))+x)/exp(exp(4*x)))+(6*exp(3*x)^2+(12*x+4)*ex 
p(3*x)+8*x)*exp(exp(4*x)))/exp(exp(4*x)),x, algorithm="maxima")
 

Output:

4*x^2 + 4/3*(3*x - 1)*e^(3*x) + 4*e^(2*x*e^(-e^(4*x))) + e^(6*x) + 4/3*e^( 
3*x) - 2*integrate(-2*(8*x^2*e^(4*x) + 4*x*e^(7*x) - (3*e^(3*x) + 2)*e^(e^ 
(4*x)) - 2*x - e^(3*x))*e^(x*e^(-e^(4*x)) - e^(4*x)), x)
 

Giac [F]

\[ \int e^{-e^{4 x}} \left (e^{2 e^{-e^{4 x}} \left (x+e^{e^{4 x}} \log (2)\right )} \left (2-8 e^{4 x} x\right )+e^{e^{4 x}} \left (6 e^{6 x}+8 x+e^{3 x} (4+12 x)\right )+e^{e^{-e^{4 x}} \left (x+e^{e^{4 x}} \log (2)\right )} \left (-2 e^{3 x}+e^{e^{4 x}} \left (-4-6 e^{3 x}\right )-4 x+e^{4 x} \left (8 e^{3 x} x+16 x^2\right )\right )\right ) \, dx=\int { -2 \, {\left ({\left (4 \, x e^{\left (4 \, x\right )} - 1\right )} e^{\left (2 \, {\left (e^{\left (e^{\left (4 \, x\right )}\right )} \log \left (2\right ) + x\right )} e^{\left (-e^{\left (4 \, x\right )}\right )}\right )} - {\left (4 \, {\left (2 \, x^{2} + x e^{\left (3 \, x\right )}\right )} e^{\left (4 \, x\right )} - {\left (3 \, e^{\left (3 \, x\right )} + 2\right )} e^{\left (e^{\left (4 \, x\right )}\right )} - 2 \, x - e^{\left (3 \, x\right )}\right )} e^{\left ({\left (e^{\left (e^{\left (4 \, x\right )}\right )} \log \left (2\right ) + x\right )} e^{\left (-e^{\left (4 \, x\right )}\right )}\right )} - {\left (2 \, {\left (3 \, x + 1\right )} e^{\left (3 \, x\right )} + 4 \, x + 3 \, e^{\left (6 \, x\right )}\right )} e^{\left (e^{\left (4 \, x\right )}\right )}\right )} e^{\left (-e^{\left (4 \, x\right )}\right )} \,d x } \] Input:

integrate(((-8*x*exp(4*x)+2)*exp((log(2)*exp(exp(4*x))+x)/exp(exp(4*x)))^2 
+((-6*exp(3*x)-4)*exp(exp(4*x))+(8*x*exp(3*x)+16*x^2)*exp(4*x)-2*exp(3*x)- 
4*x)*exp((log(2)*exp(exp(4*x))+x)/exp(exp(4*x)))+(6*exp(3*x)^2+(12*x+4)*ex 
p(3*x)+8*x)*exp(exp(4*x)))/exp(exp(4*x)),x, algorithm="giac")
 

Output:

integrate(-2*((4*x*e^(4*x) - 1)*e^(2*(e^(e^(4*x))*log(2) + x)*e^(-e^(4*x)) 
) - (4*(2*x^2 + x*e^(3*x))*e^(4*x) - (3*e^(3*x) + 2)*e^(e^(4*x)) - 2*x - e 
^(3*x))*e^((e^(e^(4*x))*log(2) + x)*e^(-e^(4*x))) - (2*(3*x + 1)*e^(3*x) + 
 4*x + 3*e^(6*x))*e^(e^(4*x)))*e^(-e^(4*x)), x)
 

Mupad [F(-1)]

Timed out. \[ \int e^{-e^{4 x}} \left (e^{2 e^{-e^{4 x}} \left (x+e^{e^{4 x}} \log (2)\right )} \left (2-8 e^{4 x} x\right )+e^{e^{4 x}} \left (6 e^{6 x}+8 x+e^{3 x} (4+12 x)\right )+e^{e^{-e^{4 x}} \left (x+e^{e^{4 x}} \log (2)\right )} \left (-2 e^{3 x}+e^{e^{4 x}} \left (-4-6 e^{3 x}\right )-4 x+e^{4 x} \left (8 e^{3 x} x+16 x^2\right )\right )\right ) \, dx=-\int {\mathrm {e}}^{-{\mathrm {e}}^{4\,x}}\,\left ({\mathrm {e}}^{{\mathrm {e}}^{-{\mathrm {e}}^{4\,x}}\,\left (x+{\mathrm {e}}^{{\mathrm {e}}^{4\,x}}\,\ln \left (2\right )\right )}\,\left (4\,x+2\,{\mathrm {e}}^{3\,x}-{\mathrm {e}}^{4\,x}\,\left (8\,x\,{\mathrm {e}}^{3\,x}+16\,x^2\right )+{\mathrm {e}}^{{\mathrm {e}}^{4\,x}}\,\left (6\,{\mathrm {e}}^{3\,x}+4\right )\right )-{\mathrm {e}}^{{\mathrm {e}}^{4\,x}}\,\left (8\,x+6\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{3\,x}\,\left (12\,x+4\right )\right )+{\mathrm {e}}^{2\,{\mathrm {e}}^{-{\mathrm {e}}^{4\,x}}\,\left (x+{\mathrm {e}}^{{\mathrm {e}}^{4\,x}}\,\ln \left (2\right )\right )}\,\left (8\,x\,{\mathrm {e}}^{4\,x}-2\right )\right ) \,d x \] Input:

int(-exp(-exp(4*x))*(exp(exp(-exp(4*x))*(x + exp(exp(4*x))*log(2)))*(4*x + 
 2*exp(3*x) - exp(4*x)*(8*x*exp(3*x) + 16*x^2) + exp(exp(4*x))*(6*exp(3*x) 
 + 4)) - exp(exp(4*x))*(8*x + 6*exp(6*x) + exp(3*x)*(12*x + 4)) + exp(2*ex 
p(-exp(4*x))*(x + exp(exp(4*x))*log(2)))*(8*x*exp(4*x) - 2)),x)
 

Output:

-int(exp(-exp(4*x))*(exp(exp(-exp(4*x))*(x + exp(exp(4*x))*log(2)))*(4*x + 
 2*exp(3*x) - exp(4*x)*(8*x*exp(3*x) + 16*x^2) + exp(exp(4*x))*(6*exp(3*x) 
 + 4)) - exp(exp(4*x))*(8*x + 6*exp(6*x) + exp(3*x)*(12*x + 4)) + exp(2*ex 
p(-exp(4*x))*(x + exp(exp(4*x))*log(2)))*(8*x*exp(4*x) - 2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.96 \[ \int e^{-e^{4 x}} \left (e^{2 e^{-e^{4 x}} \left (x+e^{e^{4 x}} \log (2)\right )} \left (2-8 e^{4 x} x\right )+e^{e^{4 x}} \left (6 e^{6 x}+8 x+e^{3 x} (4+12 x)\right )+e^{e^{-e^{4 x}} \left (x+e^{e^{4 x}} \log (2)\right )} \left (-2 e^{3 x}+e^{e^{4 x}} \left (-4-6 e^{3 x}\right )-4 x+e^{4 x} \left (8 e^{3 x} x+16 x^2\right )\right )\right ) \, dx=4 e^{\frac {2 x}{e^{e^{4 x}}}}-4 e^{\frac {3 e^{e^{4 x}} x +x}{e^{e^{4 x}}}}-8 e^{\frac {x}{e^{e^{4 x}}}} x +e^{6 x}+4 e^{3 x} x +4 x^{2} \] Input:

int(((-8*x*exp(4*x)+2)*exp((log(2)*exp(exp(4*x))+x)/exp(exp(4*x)))^2+((-6* 
exp(3*x)-4)*exp(exp(4*x))+(8*x*exp(3*x)+16*x^2)*exp(4*x)-2*exp(3*x)-4*x)*e 
xp((log(2)*exp(exp(4*x))+x)/exp(exp(4*x)))+(6*exp(3*x)^2+(12*x+4)*exp(3*x) 
+8*x)*exp(exp(4*x)))/exp(exp(4*x)),x)
 

Output:

4*e**((2*x)/e**(e**(4*x))) - 4*e**((3*e**(e**(4*x))*x + x)/e**(e**(4*x))) 
- 8*e**(x/e**(e**(4*x)))*x + e**(6*x) + 4*e**(3*x)*x + 4*x**2