\(\int \frac {e^{\frac {4 (e^4 x+x^2)}{e^3}} (4 e^4+8 x)+e^{\frac {3 (e^4 x+x^2)}{e^3}} (-4 e^3+e^4 (-12-12 x)-24 x-24 x^2)+e^3 (4+12 x+12 x^2+4 x^3)+e^{\frac {2 (e^4 x+x^2)}{e^3}} (24 x+48 x^2+24 x^3+e^3 (12+12 x)+e^4 (12+24 x+12 x^2))+e^{\frac {e^4 x+x^2}{e^3}} (-8 x-24 x^2-24 x^3-8 x^4+e^3 (-12-24 x-12 x^2)+e^4 (-4-12 x-12 x^2-4 x^3))}{e^3} \, dx\) [1014]

Optimal result

Integrand size = 212, antiderivative size = 19 \[ \int \frac {e^{\frac {4 \left (e^4 x+x^2\right )}{e^3}} \left (4 e^4+8 x\right )+e^{\frac {3 \left (e^4 x+x^2\right )}{e^3}} \left (-4 e^3+e^4 (-12-12 x)-24 x-24 x^2\right )+e^3 \left (4+12 x+12 x^2+4 x^3\right )+e^{\frac {2 \left (e^4 x+x^2\right )}{e^3}} \left (24 x+48 x^2+24 x^3+e^3 (12+12 x)+e^4 \left (12+24 x+12 x^2\right )\right )+e^{\frac {e^4 x+x^2}{e^3}} \left (-8 x-24 x^2-24 x^3-8 x^4+e^3 \left (-12-24 x-12 x^2\right )+e^4 \left (-4-12 x-12 x^2-4 x^3\right )\right )}{e^3} \, dx=\left (1-e^{\frac {x \left (e^4+x\right )}{e^3}}+x\right )^4 \] Output:

(1-exp(x/exp(3)*(x+exp(4)))+x)^4
 

Mathematica [A] (verified)

Time = 1.78 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {4 \left (e^4 x+x^2\right )}{e^3}} \left (4 e^4+8 x\right )+e^{\frac {3 \left (e^4 x+x^2\right )}{e^3}} \left (-4 e^3+e^4 (-12-12 x)-24 x-24 x^2\right )+e^3 \left (4+12 x+12 x^2+4 x^3\right )+e^{\frac {2 \left (e^4 x+x^2\right )}{e^3}} \left (24 x+48 x^2+24 x^3+e^3 (12+12 x)+e^4 \left (12+24 x+12 x^2\right )\right )+e^{\frac {e^4 x+x^2}{e^3}} \left (-8 x-24 x^2-24 x^3-8 x^4+e^3 \left (-12-24 x-12 x^2\right )+e^4 \left (-4-12 x-12 x^2-4 x^3\right )\right )}{e^3} \, dx=\left (-1+e^{\frac {x \left (e^4+x\right )}{e^3}}-x\right )^4 \] Input:

Integrate[(E^((4*(E^4*x + x^2))/E^3)*(4*E^4 + 8*x) + E^((3*(E^4*x + x^2))/ 
E^3)*(-4*E^3 + E^4*(-12 - 12*x) - 24*x - 24*x^2) + E^3*(4 + 12*x + 12*x^2 
+ 4*x^3) + E^((2*(E^4*x + x^2))/E^3)*(24*x + 48*x^2 + 24*x^3 + E^3*(12 + 1 
2*x) + E^4*(12 + 24*x + 12*x^2)) + E^((E^4*x + x^2)/E^3)*(-8*x - 24*x^2 - 
24*x^3 - 8*x^4 + E^3*(-12 - 24*x - 12*x^2) + E^4*(-4 - 12*x - 12*x^2 - 4*x 
^3)))/E^3,x]
 

Output:

(-1 + E^((x*(E^4 + x))/E^3) - x)^4
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(194\) vs. \(2(19)=38\).

Time = 0.62 (sec) , antiderivative size = 194, normalized size of antiderivative = 10.21, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.009, Rules used = {27, 2009}

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.

\(\displaystyle \int \frac {e^{\frac {4 \left (x^2+e^4 x\right )}{e^3}} \left (8 x+4 e^4\right )+e^{\frac {3 \left (x^2+e^4 x\right )}{e^3}} \left (-24 x^2-24 x+e^4 (-12 x-12)-4 e^3\right )+e^3 \left (4 x^3+12 x^2+12 x+4\right )+e^{\frac {2 \left (x^2+e^4 x\right )}{e^3}} \left (24 x^3+48 x^2+e^4 \left (12 x^2+24 x+12\right )+24 x+e^3 (12 x+12)\right )+e^{\frac {x^2+e^4 x}{e^3}} \left (-8 x^4-24 x^3-24 x^2+e^3 \left (-12 x^2-24 x-12\right )+e^4 \left (-4 x^3-12 x^2-12 x-4\right )-8 x\right )}{e^3} \, dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \left (4 e^{\frac {4 \left (x^2+e^4 x\right )}{e^3}} \left (2 x+e^4\right )+4 e^3 \left (x^3+3 x^2+3 x+1\right )-4 e^{\frac {3 \left (x^2+e^4 x\right )}{e^3}} \left (6 x^2+6 x+3 e^4 (x+1)+e^3\right )+12 e^{\frac {2 \left (x^2+e^4 x\right )}{e^3}} \left (2 x^3+4 x^2+2 x+e^3 (x+1)+e^4 \left (x^2+2 x+1\right )\right )-4 e^{\frac {x^2+e^4 x}{e^3}} \left (2 x^4+6 x^3+6 x^2+2 x+3 e^3 \left (x^2+2 x+1\right )+e^4 \left (x^3+3 x^2+3 x+1\right )\right )\right )dx}{e^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {e^{\frac {4 x^2}{e^3}+4 e x+3}-\frac {4 e^{\frac {3 \left (x^2+e^4 x\right )}{e^3}+3} \left (2 x^2+2 x+e^4 (x+1)\right )}{2 x+e^4}+\frac {6 e^{\frac {2 \left (x^2+e^4 x\right )}{e^3}+3} \left (2 x^3+4 x^2+e^4 \left (x^2+2 x+1\right )+2 x\right )}{2 x+e^4}-\frac {4 e^{\frac {x^2+e^4 x}{e^3}+3} \left (2 x^4+6 x^3+6 x^2+e^4 \left (x^3+3 x^2+3 x+1\right )+2 x\right )}{2 x+e^4}+e^3 (x+1)^4}{e^3}\)

Input:

Int[(E^((4*(E^4*x + x^2))/E^3)*(4*E^4 + 8*x) + E^((3*(E^4*x + x^2))/E^3)*( 
-4*E^3 + E^4*(-12 - 12*x) - 24*x - 24*x^2) + E^3*(4 + 12*x + 12*x^2 + 4*x^ 
3) + E^((2*(E^4*x + x^2))/E^3)*(24*x + 48*x^2 + 24*x^3 + E^3*(12 + 12*x) + 
 E^4*(12 + 24*x + 12*x^2)) + E^((E^4*x + x^2)/E^3)*(-8*x - 24*x^2 - 24*x^3 
 - 8*x^4 + E^3*(-12 - 24*x - 12*x^2) + E^4*(-4 - 12*x - 12*x^2 - 4*x^3)))/ 
E^3,x]
 

Output:

(E^(3 + 4*E*x + (4*x^2)/E^3) + E^3*(1 + x)^4 - (4*E^(3 + (3*(E^4*x + x^2)) 
/E^3)*(2*x + 2*x^2 + E^4*(1 + x)))/(E^4 + 2*x) + (6*E^(3 + (2*(E^4*x + x^2 
))/E^3)*(2*x + 4*x^2 + 2*x^3 + E^4*(1 + 2*x + x^2)))/(E^4 + 2*x) - (4*E^(3 
 + (E^4*x + x^2)/E^3)*(2*x + 6*x^2 + 6*x^3 + 2*x^4 + E^4*(1 + 3*x + 3*x^2 
+ x^3)))/(E^4 + 2*x))/E^3
 

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 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3.

Time = 0.05 (sec) , antiderivative size = 2892, normalized size of antiderivative = 152.21

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

Input:

int(((4*exp(4)+8*x)*exp((x*exp(4)+x^2)/exp(3))^4+((-12*x-12)*exp(4)-4*exp( 
3)-24*x^2-24*x)*exp((x*exp(4)+x^2)/exp(3))^3+((12*x^2+24*x+12)*exp(4)+(12* 
x+12)*exp(3)+24*x^3+48*x^2+24*x)*exp((x*exp(4)+x^2)/exp(3))^2+((-4*x^3-12* 
x^2-12*x-4)*exp(4)+(-12*x^2-24*x-12)*exp(3)-8*x^4-24*x^3-24*x^2-8*x)*exp(( 
x*exp(4)+x^2)/exp(3))+(4*x^3+12*x^2+12*x+4)*exp(3))/exp(3),x)
 

Output:

1/exp(3)*(-4/exp(-3)*exp(exp(-3)*x^2+x*exp(1))+24/exp(-3)*(1/2/exp(-3)*exp 
(exp(-3)*x^2+x*exp(1))+1/4*I*exp(1)/exp(-3)*Pi^(1/2)*exp(-1/4*exp(1)^2/exp 
(-3))/exp(-3/2)*erf(I*exp(-3/2)*x+1/2*I*exp(1)/exp(-3/2)))+12/exp(-3)*(1/2 
/exp(-3)*x*exp(exp(-3)*x^2+x*exp(1))-1/2*exp(1)/exp(-3)*(1/2/exp(-3)*exp(e 
xp(-3)*x^2+x*exp(1))+1/4*I*exp(1)/exp(-3)*Pi^(1/2)*exp(-1/4*exp(1)^2/exp(- 
3))/exp(-3/2)*erf(I*exp(-3/2)*x+1/2*I*exp(1)/exp(-3/2)))+1/4*I/exp(-3)*Pi^ 
(1/2)*exp(-1/4*exp(1)^2/exp(-3))/exp(-3/2)*erf(I*exp(-3/2)*x+1/2*I*exp(1)/ 
exp(-3/2)))-12*exp(4)*(1/2/exp(-3)*exp(exp(-3)*x^2+x*exp(1))+1/4*I*exp(1)/ 
exp(-3)*Pi^(1/2)*exp(-1/4*exp(1)^2/exp(-3))/exp(-3/2)*erf(I*exp(-3/2)*x+1/ 
2*I*exp(1)/exp(-3/2)))-12*exp(3)*(1/2/exp(-3)*x*exp(exp(-3)*x^2+x*exp(1))- 
1/2*exp(1)/exp(-3)*(1/2/exp(-3)*exp(exp(-3)*x^2+x*exp(1))+1/4*I*exp(1)/exp 
(-3)*Pi^(1/2)*exp(-1/4*exp(1)^2/exp(-3))/exp(-3/2)*erf(I*exp(-3/2)*x+1/2*I 
*exp(1)/exp(-3/2)))+1/4*I/exp(-3)*Pi^(1/2)*exp(-1/4*exp(1)^2/exp(-3))/exp( 
-3/2)*erf(I*exp(-3/2)*x+1/2*I*exp(1)/exp(-3/2)))-24*exp(3)*(1/2/exp(-3)*ex 
p(exp(-3)*x^2+x*exp(1))+1/4*I*exp(1)/exp(-3)*Pi^(1/2)*exp(-1/4*exp(1)^2/ex 
p(-3))/exp(-3/2)*erf(I*exp(-3/2)*x+1/2*I*exp(1)/exp(-3/2)))-12*exp(4)*(1/2 
/exp(-3)*x*exp(exp(-3)*x^2+x*exp(1))-1/2*exp(1)/exp(-3)*(1/2/exp(-3)*exp(e 
xp(-3)*x^2+x*exp(1))+1/4*I*exp(1)/exp(-3)*Pi^(1/2)*exp(-1/4*exp(1)^2/exp(- 
3))/exp(-3/2)*erf(I*exp(-3/2)*x+1/2*I*exp(1)/exp(-3/2)))+1/4*I/exp(-3)*Pi^ 
(1/2)*exp(-1/4*exp(1)^2/exp(-3))/exp(-3/2)*erf(I*exp(-3/2)*x+1/2*I*exp(...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (16) = 32\).

Time = 0.08 (sec) , antiderivative size = 98, normalized size of antiderivative = 5.16 \[ \int \frac {e^{\frac {4 \left (e^4 x+x^2\right )}{e^3}} \left (4 e^4+8 x\right )+e^{\frac {3 \left (e^4 x+x^2\right )}{e^3}} \left (-4 e^3+e^4 (-12-12 x)-24 x-24 x^2\right )+e^3 \left (4+12 x+12 x^2+4 x^3\right )+e^{\frac {2 \left (e^4 x+x^2\right )}{e^3}} \left (24 x+48 x^2+24 x^3+e^3 (12+12 x)+e^4 \left (12+24 x+12 x^2\right )\right )+e^{\frac {e^4 x+x^2}{e^3}} \left (-8 x-24 x^2-24 x^3-8 x^4+e^3 \left (-12-24 x-12 x^2\right )+e^4 \left (-4-12 x-12 x^2-4 x^3\right )\right )}{e^3} \, dx=x^{4} + 4 \, x^{3} + 6 \, x^{2} - 4 \, {\left (x + 1\right )} e^{\left (3 \, {\left (x^{2} + x e^{4}\right )} e^{\left (-3\right )}\right )} + 6 \, {\left (x^{2} + 2 \, x + 1\right )} e^{\left (2 \, {\left (x^{2} + x e^{4}\right )} e^{\left (-3\right )}\right )} - 4 \, {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )} e^{\left ({\left (x^{2} + x e^{4}\right )} e^{\left (-3\right )}\right )} + 4 \, x + e^{\left (4 \, {\left (x^{2} + x e^{4}\right )} e^{\left (-3\right )}\right )} \] Input:

integrate(((4*exp(4)+8*x)*exp((x*exp(4)+x^2)/exp(3))^4+((-12*x-12)*exp(4)- 
4*exp(3)-24*x^2-24*x)*exp((x*exp(4)+x^2)/exp(3))^3+((12*x^2+24*x+12)*exp(4 
)+(12*x+12)*exp(3)+24*x^3+48*x^2+24*x)*exp((x*exp(4)+x^2)/exp(3))^2+((-4*x 
^3-12*x^2-12*x-4)*exp(4)+(-12*x^2-24*x-12)*exp(3)-8*x^4-24*x^3-24*x^2-8*x) 
*exp((x*exp(4)+x^2)/exp(3))+(4*x^3+12*x^2+12*x+4)*exp(3))/exp(3),x, algori 
thm="fricas")
 

Output:

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (15) = 30\).

Time = 0.16 (sec) , antiderivative size = 114, normalized size of antiderivative = 6.00 \[ \int \frac {e^{\frac {4 \left (e^4 x+x^2\right )}{e^3}} \left (4 e^4+8 x\right )+e^{\frac {3 \left (e^4 x+x^2\right )}{e^3}} \left (-4 e^3+e^4 (-12-12 x)-24 x-24 x^2\right )+e^3 \left (4+12 x+12 x^2+4 x^3\right )+e^{\frac {2 \left (e^4 x+x^2\right )}{e^3}} \left (24 x+48 x^2+24 x^3+e^3 (12+12 x)+e^4 \left (12+24 x+12 x^2\right )\right )+e^{\frac {e^4 x+x^2}{e^3}} \left (-8 x-24 x^2-24 x^3-8 x^4+e^3 \left (-12-24 x-12 x^2\right )+e^4 \left (-4-12 x-12 x^2-4 x^3\right )\right )}{e^3} \, dx=x^{4} + 4 x^{3} + 6 x^{2} + 4 x + \left (- 4 x - 4\right ) e^{\frac {3 \left (x^{2} + x e^{4}\right )}{e^{3}}} + \left (6 x^{2} + 12 x + 6\right ) e^{\frac {2 \left (x^{2} + x e^{4}\right )}{e^{3}}} + \left (- 4 x^{3} - 12 x^{2} - 12 x - 4\right ) e^{\frac {x^{2} + x e^{4}}{e^{3}}} + e^{\frac {4 \left (x^{2} + x e^{4}\right )}{e^{3}}} \] Input:

integrate(((4*exp(4)+8*x)*exp((x*exp(4)+x**2)/exp(3))**4+((-12*x-12)*exp(4 
)-4*exp(3)-24*x**2-24*x)*exp((x*exp(4)+x**2)/exp(3))**3+((12*x**2+24*x+12) 
*exp(4)+(12*x+12)*exp(3)+24*x**3+48*x**2+24*x)*exp((x*exp(4)+x**2)/exp(3)) 
**2+((-4*x**3-12*x**2-12*x-4)*exp(4)+(-12*x**2-24*x-12)*exp(3)-8*x**4-24*x 
**3-24*x**2-8*x)*exp((x*exp(4)+x**2)/exp(3))+(4*x**3+12*x**2+12*x+4)*exp(3 
))/exp(3),x)
 

Output:

x**4 + 4*x**3 + 6*x**2 + 4*x + (-4*x - 4)*exp(3*(x**2 + x*exp(4))*exp(-3)) 
 + (6*x**2 + 12*x + 6)*exp(2*(x**2 + x*exp(4))*exp(-3)) + (-4*x**3 - 12*x* 
*2 - 12*x - 4)*exp((x**2 + x*exp(4))*exp(-3)) + exp(4*(x**2 + x*exp(4))*ex 
p(-3))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 127 vs. \(2 (16) = 32\).

Time = 0.04 (sec) , antiderivative size = 127, normalized size of antiderivative = 6.68 \[ \int \frac {e^{\frac {4 \left (e^4 x+x^2\right )}{e^3}} \left (4 e^4+8 x\right )+e^{\frac {3 \left (e^4 x+x^2\right )}{e^3}} \left (-4 e^3+e^4 (-12-12 x)-24 x-24 x^2\right )+e^3 \left (4+12 x+12 x^2+4 x^3\right )+e^{\frac {2 \left (e^4 x+x^2\right )}{e^3}} \left (24 x+48 x^2+24 x^3+e^3 (12+12 x)+e^4 \left (12+24 x+12 x^2\right )\right )+e^{\frac {e^4 x+x^2}{e^3}} \left (-8 x-24 x^2-24 x^3-8 x^4+e^3 \left (-12-24 x-12 x^2\right )+e^4 \left (-4-12 x-12 x^2-4 x^3\right )\right )}{e^3} \, dx={\left ({\left (x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x\right )} e^{3} - 4 \, {\left (x e^{3} + e^{3}\right )} e^{\left (3 \, x^{2} e^{\left (-3\right )} + 3 \, x e\right )} + 6 \, {\left (x^{2} e^{3} + 2 \, x e^{3} + e^{3}\right )} e^{\left (2 \, x^{2} e^{\left (-3\right )} + 2 \, x e\right )} - 4 \, {\left (x^{3} e^{3} + 3 \, x^{2} e^{3} + 3 \, x e^{3} + e^{3}\right )} e^{\left (x^{2} e^{\left (-3\right )} + x e\right )} + e^{\left (4 \, {\left (x^{2} + x e^{4}\right )} e^{\left (-3\right )} + 3\right )}\right )} e^{\left (-3\right )} \] Input:

integrate(((4*exp(4)+8*x)*exp((x*exp(4)+x^2)/exp(3))^4+((-12*x-12)*exp(4)- 
4*exp(3)-24*x^2-24*x)*exp((x*exp(4)+x^2)/exp(3))^3+((12*x^2+24*x+12)*exp(4 
)+(12*x+12)*exp(3)+24*x^3+48*x^2+24*x)*exp((x*exp(4)+x^2)/exp(3))^2+((-4*x 
^3-12*x^2-12*x-4)*exp(4)+(-12*x^2-24*x-12)*exp(3)-8*x^4-24*x^3-24*x^2-8*x) 
*exp((x*exp(4)+x^2)/exp(3))+(4*x^3+12*x^2+12*x+4)*exp(3))/exp(3),x, algori 
thm="maxima")
 

Output:

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

Giac [C] (verification not implemented)

Result contains higher order function than in optimal. Order 4 vs. order 3.

Time = 0.17 (sec) , antiderivative size = 796, normalized size of antiderivative = 41.89 \[ \int \frac {e^{\frac {4 \left (e^4 x+x^2\right )}{e^3}} \left (4 e^4+8 x\right )+e^{\frac {3 \left (e^4 x+x^2\right )}{e^3}} \left (-4 e^3+e^4 (-12-12 x)-24 x-24 x^2\right )+e^3 \left (4+12 x+12 x^2+4 x^3\right )+e^{\frac {2 \left (e^4 x+x^2\right )}{e^3}} \left (24 x+48 x^2+24 x^3+e^3 (12+12 x)+e^4 \left (12+24 x+12 x^2\right )\right )+e^{\frac {e^4 x+x^2}{e^3}} \left (-8 x-24 x^2-24 x^3-8 x^4+e^3 \left (-12-24 x-12 x^2\right )+e^4 \left (-4-12 x-12 x^2-4 x^3\right )\right )}{e^3} \, dx=\text {Too large to display} \] Input:

integrate(((4*exp(4)+8*x)*exp((x*exp(4)+x^2)/exp(3))^4+((-12*x-12)*exp(4)- 
4*exp(3)-24*x^2-24*x)*exp((x*exp(4)+x^2)/exp(3))^3+((12*x^2+24*x+12)*exp(4 
)+(12*x+12)*exp(3)+24*x^3+48*x^2+24*x)*exp((x*exp(4)+x^2)/exp(3))^2+((-4*x 
^3-12*x^2-12*x-4)*exp(4)+(-12*x^2-24*x-12)*exp(3)-8*x^4-24*x^3-24*x^2-8*x) 
*exp((x*exp(4)+x^2)/exp(3))+(4*x^3+12*x^2+12*x+4)*exp(3))/exp(3),x, algori 
thm="giac")
 

Output:

-1/12*(-9*I*sqrt(2)*sqrt(pi)*(e^8 - 4*e^4 - e^3 + 4)*erf(-1/2*I*sqrt(2)*(2 
*x + e^4)*e^(-3/2))*e^(-1/2*e^5 + 11/2) + 18*I*sqrt(2)*sqrt(pi)*(e^4 - 2)* 
erf(-1/2*I*sqrt(2)*(2*x + e^4)*e^(-3/2))*e^(-1/2*e^5 + 9/2) + 9*I*sqrt(2)* 
sqrt(pi)*(e^12 - 4*e^8 - 3*e^7 + 4*e^4 + 4*e^3)*erf(-1/2*I*sqrt(2)*(2*x + 
e^4)*e^(-3/2))*e^(-1/2*e^5 + 3/2) - 12*I*sqrt(3)*sqrt(pi)*(e^4 - 2)*erf(-1 
/2*I*sqrt(3)*(2*x + e^4)*e^(-3/2))*e^(-3/4*e^5 + 11/2) + 4*I*sqrt(3)*sqrt( 
pi)*(3*e^8 - 6*e^4 - 2*e^3)*erf(-1/2*I*sqrt(3)*(2*x + e^4)*e^(-3/2))*e^(-3 
/4*e^5 + 3/2) - 3*I*sqrt(pi)*(e^12 - 6*e^8 - 6*e^7 + 12*e^4 + 12*e^3 - 8)* 
erf(-1/2*I*(2*x + e^4)*e^(-3/2))*e^(-1/4*e^5 + 11/2) + 18*I*sqrt(pi)*(e^8 
- 4*e^4 - 2*e^3 + 4)*erf(-1/2*I*(2*x + e^4)*e^(-3/2))*e^(-1/4*e^5 + 9/2) + 
 3*I*sqrt(pi)*(e^16 - 6*e^12 - 12*e^11 + 12*e^8 + 36*e^7 + 12*e^6 - 8*e^4 
- 24*e^3)*erf(-1/2*I*(2*x + e^4)*e^(-3/2))*e^(-1/4*e^5 + 3/2) + 8*I*sqrt(3 
)*sqrt(pi)*erf(-1/2*I*sqrt(3)*(2*x + e^4)*e^(-3/2))*e^(-3/4*e^5 + 9/2) - 1 
2*(x^4 + 4*x^3 + 6*x^2 + 4*x)*e^3 + 6*((2*x + e^4)^2*e^3 - 3*(2*x + e^4)*e 
^7 + 6*(2*x + e^4)*e^3 + 3*e^11 - 12*e^7 - 4*e^6 + 12*e^3)*e^((x^2 + x*e^4 
 + 4*e^3)*e^(-3)) + 36*((2*x + e^4)*e^3 - 2*e^7 + 4*e^3)*e^((x^2 + x*e^4 + 
 3*e^3)*e^(-3)) - 18*((2*x + e^4)*e^3 - 2*e^7 + 4*e^3)*e^(2*(x^2 + x*e^4 + 
 2*e^3)*e^(-3)) + 24*((2*x + e^4)*e^3 - 2*e^7 + 2*e^3)*e^(3*(x^2 + x*e^4)* 
e^(-3)) - 18*((2*x + e^4)^2*e^3 - 3*(2*x + e^4)*e^7 + 4*(2*x + e^4)*e^3 + 
3*e^11 - 8*e^7 - 2*e^6 + 4*e^3)*e^(2*(x^2 + x*e^4)*e^(-3)) + 6*((2*x + ...
 

Mupad [B] (verification not implemented)

Time = 1.41 (sec) , antiderivative size = 179, normalized size of antiderivative = 9.42 \[ \int \frac {e^{\frac {4 \left (e^4 x+x^2\right )}{e^3}} \left (4 e^4+8 x\right )+e^{\frac {3 \left (e^4 x+x^2\right )}{e^3}} \left (-4 e^3+e^4 (-12-12 x)-24 x-24 x^2\right )+e^3 \left (4+12 x+12 x^2+4 x^3\right )+e^{\frac {2 \left (e^4 x+x^2\right )}{e^3}} \left (24 x+48 x^2+24 x^3+e^3 (12+12 x)+e^4 \left (12+24 x+12 x^2\right )\right )+e^{\frac {e^4 x+x^2}{e^3}} \left (-8 x-24 x^2-24 x^3-8 x^4+e^3 \left (-12-24 x-12 x^2\right )+e^4 \left (-4-12 x-12 x^2-4 x^3\right )\right )}{e^3} \, dx=4\,x-4\,{\mathrm {e}}^{{\mathrm {e}}^{-3}\,x^2+\mathrm {e}\,x}+6\,{\mathrm {e}}^{2\,{\mathrm {e}}^{-3}\,x^2+2\,\mathrm {e}\,x}-4\,{\mathrm {e}}^{3\,{\mathrm {e}}^{-3}\,x^2+3\,\mathrm {e}\,x}+{\mathrm {e}}^{4\,{\mathrm {e}}^{-3}\,x^2+4\,\mathrm {e}\,x}-12\,x\,{\mathrm {e}}^{{\mathrm {e}}^{-3}\,x^2+\mathrm {e}\,x}+12\,x\,{\mathrm {e}}^{2\,{\mathrm {e}}^{-3}\,x^2+2\,\mathrm {e}\,x}-4\,x\,{\mathrm {e}}^{3\,{\mathrm {e}}^{-3}\,x^2+3\,\mathrm {e}\,x}-12\,x^2\,{\mathrm {e}}^{{\mathrm {e}}^{-3}\,x^2+\mathrm {e}\,x}-4\,x^3\,{\mathrm {e}}^{{\mathrm {e}}^{-3}\,x^2+\mathrm {e}\,x}+6\,x^2\,{\mathrm {e}}^{2\,{\mathrm {e}}^{-3}\,x^2+2\,\mathrm {e}\,x}+6\,x^2+4\,x^3+x^4 \] Input:

int(exp(-3)*(exp(3)*(12*x + 12*x^2 + 4*x^3 + 4) - exp(3*exp(-3)*(x*exp(4) 
+ x^2))*(24*x + 4*exp(3) + 24*x^2 + exp(4)*(12*x + 12)) - exp(exp(-3)*(x*e 
xp(4) + x^2))*(8*x + exp(3)*(24*x + 12*x^2 + 12) + exp(4)*(12*x + 12*x^2 + 
 4*x^3 + 4) + 24*x^2 + 24*x^3 + 8*x^4) + exp(2*exp(-3)*(x*exp(4) + x^2))*( 
24*x + exp(4)*(24*x + 12*x^2 + 12) + 48*x^2 + 24*x^3 + exp(3)*(12*x + 12)) 
 + exp(4*exp(-3)*(x*exp(4) + x^2))*(8*x + 4*exp(4))),x)
 

Output:

4*x - 4*exp(x*exp(1) + x^2*exp(-3)) + 6*exp(2*x*exp(1) + 2*x^2*exp(-3)) - 
4*exp(3*x*exp(1) + 3*x^2*exp(-3)) + exp(4*x*exp(1) + 4*x^2*exp(-3)) - 12*x 
*exp(x*exp(1) + x^2*exp(-3)) + 12*x*exp(2*x*exp(1) + 2*x^2*exp(-3)) - 4*x* 
exp(3*x*exp(1) + 3*x^2*exp(-3)) - 12*x^2*exp(x*exp(1) + x^2*exp(-3)) - 4*x 
^3*exp(x*exp(1) + x^2*exp(-3)) + 6*x^2*exp(2*x*exp(1) + 2*x^2*exp(-3)) + 6 
*x^2 + 4*x^3 + x^4
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 215, normalized size of antiderivative = 11.32 \[ \int \frac {e^{\frac {4 \left (e^4 x+x^2\right )}{e^3}} \left (4 e^4+8 x\right )+e^{\frac {3 \left (e^4 x+x^2\right )}{e^3}} \left (-4 e^3+e^4 (-12-12 x)-24 x-24 x^2\right )+e^3 \left (4+12 x+12 x^2+4 x^3\right )+e^{\frac {2 \left (e^4 x+x^2\right )}{e^3}} \left (24 x+48 x^2+24 x^3+e^3 (12+12 x)+e^4 \left (12+24 x+12 x^2\right )\right )+e^{\frac {e^4 x+x^2}{e^3}} \left (-8 x-24 x^2-24 x^3-8 x^4+e^3 \left (-12-24 x-12 x^2\right )+e^4 \left (-4-12 x-12 x^2-4 x^3\right )\right )}{e^3} \, dx=e^{\frac {4 e^{4} x +4 x^{2}}{e^{3}}}-4 e^{\frac {3 e^{4} x +3 x^{2}}{e^{3}}} x -4 e^{\frac {3 e^{4} x +3 x^{2}}{e^{3}}}+6 e^{\frac {2 e^{4} x +2 x^{2}}{e^{3}}} x^{2}+12 e^{\frac {2 e^{4} x +2 x^{2}}{e^{3}}} x +6 e^{\frac {2 e^{4} x +2 x^{2}}{e^{3}}}-4 e^{\frac {e^{4} x +x^{2}}{e^{3}}} x^{3}-12 e^{\frac {e^{4} x +x^{2}}{e^{3}}} x^{2}-12 e^{\frac {e^{4} x +x^{2}}{e^{3}}} x -4 e^{\frac {e^{4} x +x^{2}}{e^{3}}}+x^{4}+4 x^{3}+6 x^{2}+4 x \] Input:

int(((4*exp(4)+8*x)*exp((x*exp(4)+x^2)/exp(3))^4+((-12*x-12)*exp(4)-4*exp( 
3)-24*x^2-24*x)*exp((x*exp(4)+x^2)/exp(3))^3+((12*x^2+24*x+12)*exp(4)+(12* 
x+12)*exp(3)+24*x^3+48*x^2+24*x)*exp((x*exp(4)+x^2)/exp(3))^2+((-4*x^3-12* 
x^2-12*x-4)*exp(4)+(-12*x^2-24*x-12)*exp(3)-8*x^4-24*x^3-24*x^2-8*x)*exp(( 
x*exp(4)+x^2)/exp(3))+(4*x^3+12*x^2+12*x+4)*exp(3))/exp(3),x)
 

Output:

e**((4*e**4*x + 4*x**2)/e**3) - 4*e**((3*e**4*x + 3*x**2)/e**3)*x - 4*e**( 
(3*e**4*x + 3*x**2)/e**3) + 6*e**((2*e**4*x + 2*x**2)/e**3)*x**2 + 12*e**( 
(2*e**4*x + 2*x**2)/e**3)*x + 6*e**((2*e**4*x + 2*x**2)/e**3) - 4*e**((e** 
4*x + x**2)/e**3)*x**3 - 12*e**((e**4*x + x**2)/e**3)*x**2 - 12*e**((e**4* 
x + x**2)/e**3)*x - 4*e**((e**4*x + x**2)/e**3) + x**4 + 4*x**3 + 6*x**2 + 
 4*x