\(\int \frac {-16-4 e^2+e (-16-8 x)-16 x-4 x^2+e^2 (10 x^2+2 e^2 x^2+8 x^3+2 x^4+e (9 x^2+4 x^3))+e^{2+x^2} (8 x^3+2 e^2 x^3+8 x^4+2 x^5+e (8 x^3+4 x^4))}{e^2 (4 x^2+e^2 x^2+4 x^3+x^4+e (4 x^2+2 x^3))} \, dx\) [201]

Optimal result

Integrand size = 148, antiderivative size = 25 \[ \int \frac {-16-4 e^2+e (-16-8 x)-16 x-4 x^2+e^2 \left (10 x^2+2 e^2 x^2+8 x^3+2 x^4+e \left (9 x^2+4 x^3\right )\right )+e^{2+x^2} \left (8 x^3+2 e^2 x^3+8 x^4+2 x^5+e \left (8 x^3+4 x^4\right )\right )}{e^2 \left (4 x^2+e^2 x^2+4 x^3+x^4+e \left (4 x^2+2 x^3\right )\right )} \, dx=e^{x^2}+\frac {4}{e^2 x}+2 x+\frac {x}{2+e+x} \] Output:

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

Mathematica [A] (verified)

Time = 2.13 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48 \[ \int \frac {-16-4 e^2+e (-16-8 x)-16 x-4 x^2+e^2 \left (10 x^2+2 e^2 x^2+8 x^3+2 x^4+e \left (9 x^2+4 x^3\right )\right )+e^{2+x^2} \left (8 x^3+2 e^2 x^3+8 x^4+2 x^5+e \left (8 x^3+4 x^4\right )\right )}{e^2 \left (4 x^2+e^2 x^2+4 x^3+x^4+e \left (4 x^2+2 x^3\right )\right )} \, dx=\frac {e^{2+x^2}+\frac {4}{x}+2 e^2 x-\frac {e^2 (2+e)}{2+e+x}}{e^2} \] Input:

Integrate[(-16 - 4*E^2 + E*(-16 - 8*x) - 16*x - 4*x^2 + E^2*(10*x^2 + 2*E^ 
2*x^2 + 8*x^3 + 2*x^4 + E*(9*x^2 + 4*x^3)) + E^(2 + x^2)*(8*x^3 + 2*E^2*x^ 
3 + 8*x^4 + 2*x^5 + E*(8*x^3 + 4*x^4)))/(E^2*(4*x^2 + E^2*x^2 + 4*x^3 + x^ 
4 + E*(4*x^2 + 2*x^3))),x]
 

Output:

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

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(200\) vs. \(2(25)=50\).

Time = 1.17 (sec) , antiderivative size = 200, normalized size of antiderivative = 8.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.047, Rules used = {6, 27, 25, 2026, 2007, 7293, 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.

\(\displaystyle \int \frac {-4 x^2+e^2 \left (2 x^4+8 x^3+2 e^2 x^2+10 x^2+e \left (4 x^3+9 x^2\right )\right )+e^{x^2+2} \left (2 x^5+8 x^4+2 e^2 x^3+8 x^3+e \left (4 x^4+8 x^3\right )\right )-16 x+e (-8 x-16)-4 e^2-16}{e^2 \left (x^4+4 x^3+e^2 x^2+4 x^2+e \left (2 x^3+4 x^2\right )\right )} \, dx\)

\(\Big \downarrow \) 6

\(\displaystyle \int \frac {-4 x^2+e^2 \left (2 x^4+8 x^3+2 e^2 x^2+10 x^2+e \left (4 x^3+9 x^2\right )\right )+e^{x^2+2} \left (2 x^5+8 x^4+2 e^2 x^3+8 x^3+e \left (4 x^4+8 x^3\right )\right )-16 x+e (-8 x-16)-4 e^2-16}{e^2 \left (x^4+4 x^3+\left (4+e^2\right ) x^2+e \left (2 x^3+4 x^2\right )\right )}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int -\frac {4 x^2+16 x+8 e (x+2)-e^2 \left (2 x^4+8 x^3+2 e^2 x^2+10 x^2+e \left (4 x^3+9 x^2\right )\right )-2 e^{x^2+2} \left (x^5+4 x^4+e^2 x^3+4 x^3+2 e \left (x^4+2 x^3\right )\right )+4 \left (4+e^2\right )}{x^4+4 x^3+\left (4+e^2\right ) x^2+2 e \left (x^3+2 x^2\right )}dx}{e^2}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\int \frac {4 x^2+16 x+8 e (x+2)-e^2 \left (2 x^4+8 x^3+2 e^2 x^2+10 x^2+e \left (4 x^3+9 x^2\right )\right )-2 e^{x^2+2} \left (x^5+4 x^4+e^2 x^3+4 x^3+2 e \left (x^4+2 x^3\right )\right )+4 \left (4+e^2\right )}{x^4+4 x^3+\left (4+e^2\right ) x^2+2 e \left (x^3+2 x^2\right )}dx}{e^2}\)

\(\Big \downarrow \) 2026

\(\displaystyle -\frac {\int \frac {4 x^2+16 x+8 e (x+2)-e^2 \left (2 x^4+8 x^3+2 e^2 x^2+10 x^2+e \left (4 x^3+9 x^2\right )\right )-2 e^{x^2+2} \left (x^5+4 x^4+e^2 x^3+4 x^3+2 e \left (x^4+2 x^3\right )\right )+4 \left (4+e^2\right )}{x^2 \left (x^2+2 (2+e) x+(2+e)^2\right )}dx}{e^2}\)

\(\Big \downarrow \) 2007

\(\displaystyle -\frac {\int \frac {4 x^2+16 x+8 e (x+2)-e^2 \left (2 x^4+8 x^3+2 e^2 x^2+10 x^2+e \left (4 x^3+9 x^2\right )\right )-2 e^{x^2+2} \left (x^5+4 x^4+e^2 x^3+4 x^3+2 e \left (x^4+2 x^3\right )\right )+4 \left (4+e^2\right )}{x^2 (x+e+2)^2}dx}{e^2}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\int \left (-2 e^{x^2+2} x+\frac {e^2 \left (-2 x^2-4 (2+e) x-(2+e) (5+2 e)\right )}{(x+e+2)^2}+\frac {4}{(x+e+2)^2}+\frac {16}{(x+e+2)^2 x}+\frac {8 e (x+2)}{(x+e+2)^2 x^2}+\frac {4 \left (4+e^2\right )}{(x+e+2)^2 x^2}\right )dx}{e^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {-e^{x^2+2}-2 e^2 x-\frac {4 \left (4+e^2\right )}{(2+e)^2 (x+e+2)}+\frac {e^2 (2+e)}{x+e+2}+\frac {16}{(2+e) (x+e+2)}+\frac {8 e^2}{(2+e)^2 (x+e+2)}-\frac {4}{x+e+2}-\frac {4 \left (4+e^2\right )}{(2+e)^2 x}-\frac {16 e}{(2+e)^2 x}-\frac {8 \left (4+e^2\right ) \log (x)}{(2+e)^3}+\frac {16 \log (x)}{(2+e)^2}-\frac {8 (2-e) e \log (x)}{(2+e)^3}+\frac {8 \left (4+e^2\right ) \log (x+e+2)}{(2+e)^3}-\frac {16 \log (x+e+2)}{(2+e)^2}+\frac {8 (2-e) e \log (x+e+2)}{(2+e)^3}}{e^2}\)

Input:

Int[(-16 - 4*E^2 + E*(-16 - 8*x) - 16*x - 4*x^2 + E^2*(10*x^2 + 2*E^2*x^2 
+ 8*x^3 + 2*x^4 + E*(9*x^2 + 4*x^3)) + E^(2 + x^2)*(8*x^3 + 2*E^2*x^3 + 8* 
x^4 + 2*x^5 + E*(8*x^3 + 4*x^4)))/(E^2*(4*x^2 + E^2*x^2 + 4*x^3 + x^4 + E* 
(4*x^2 + 2*x^3))),x]
 

Output:

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

Defintions of rubi rules used

rule 6

Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v 
+ (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] &&  !FreeQ[Fx, x]
 

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 2007

Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, 
x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex 
pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol 
yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
 

rule 2009

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

rule 2026

Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p 
*r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ 
erQ[p] &&  !MonomialQ[Px, x] && (ILtQ[p, 0] ||  !PolyQ[u, x])
 

rule 7293

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

Maple [A] (verified)

Time = 3.79 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.60

method	result	size
risch	\(2 x +\frac {{\mathrm e}^{-2} \left (\left (-{\mathrm e}^{3}-2 \,{\mathrm e}^{2}+4\right ) x +4 \,{\mathrm e}+8\right )}{\left (2+x +{\mathrm e}\right ) x}+{\mathrm e}^{x^{2}}\)	\(40\)
parts	\(\frac {-\left (2 \left ({\mathrm e}^{2}\right )^{2}+9 \,{\mathrm e} \,{\mathrm e}^{2}+10 \,{\mathrm e}^{2}-4\right ) {\mathrm e}^{-2} x +2 x^{3}+4 \left ({\mathrm e}+2\right ) {\mathrm e}^{-2}}{x \left (2+x +{\mathrm e}\right )}+{\mathrm e}^{x^{2}}\)	\(60\)
norman	\(\frac {x^{2} {\mathrm e}^{x^{2}}-\left (2 \left ({\mathrm e}^{2}\right )^{2}+9 \,{\mathrm e} \,{\mathrm e}^{2}+10 \,{\mathrm e}^{2}-4\right ) {\mathrm e}^{-2} x +x \left ({\mathrm e}+2\right ) {\mathrm e}^{x^{2}}+2 x^{3}+4 \left ({\mathrm e}+2\right ) {\mathrm e}^{-2}}{x \left (2+x +{\mathrm e}\right )}\)	\(73\)
parallelrisch	\(-\frac {{\mathrm e}^{-2} \left (2 x \left ({\mathrm e}^{2}\right )^{2}-{\mathrm e} \,{\mathrm e}^{2} x \,{\mathrm e}^{x^{2}}-2 x^{3} {\mathrm e}^{2}-{\mathrm e}^{2} x^{2} {\mathrm e}^{x^{2}}+9 x \,{\mathrm e} \,{\mathrm e}^{2}-8-2 \,{\mathrm e}^{2} {\mathrm e}^{x^{2}} x +10 \,{\mathrm e}^{2} x -4 \,{\mathrm e}-4 x \right )}{x \left (2+x +{\mathrm e}\right )}\)	\(85\)

Input:

int(((2*x^3*exp(1)^2+(4*x^4+8*x^3)*exp(1)+2*x^5+8*x^4+8*x^3)*exp(2)*exp(x^ 
2)+(2*x^2*exp(1)^2+(4*x^3+9*x^2)*exp(1)+2*x^4+8*x^3+10*x^2)*exp(2)-4*exp(1 
)^2+(-8*x-16)*exp(1)-4*x^2-16*x-16)/(x^2*exp(1)^2+(2*x^3+4*x^2)*exp(1)+x^4 
+4*x^3+4*x^2)/exp(2),x,method=_RETURNVERBOSE)
 

Output:

2*x+exp(-2)*((-exp(3)-2*exp(2)+4)*x+4*exp(1)+8)/(2+x+exp(1))/x+exp(x^2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (24) = 48\).

Time = 0.09 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.92 \[ \int \frac {-16-4 e^2+e (-16-8 x)-16 x-4 x^2+e^2 \left (10 x^2+2 e^2 x^2+8 x^3+2 x^4+e \left (9 x^2+4 x^3\right )\right )+e^{2+x^2} \left (8 x^3+2 e^2 x^3+8 x^4+2 x^5+e \left (8 x^3+4 x^4\right )\right )}{e^2 \left (4 x^2+e^2 x^2+4 x^3+x^4+e \left (4 x^2+2 x^3\right )\right )} \, dx=\frac {{\left (2 \, x^{2} - x\right )} e^{3} + 2 \, {\left (x^{3} + 2 \, x^{2} - x\right )} e^{2} + {\left (x^{2} + x e + 2 \, x\right )} e^{\left (x^{2} + 2\right )} + 4 \, x + 4 \, e + 8}{x e^{3} + {\left (x^{2} + 2 \, x\right )} e^{2}} \] Input:

integrate(((2*x^3*exp(1)^2+(4*x^4+8*x^3)*exp(1)+2*x^5+8*x^4+8*x^3)*exp(2)* 
exp(x^2)+(2*x^2*exp(1)^2+(4*x^3+9*x^2)*exp(1)+2*x^4+8*x^3+10*x^2)*exp(2)-4 
*exp(1)^2+(-8*x-16)*exp(1)-4*x^2-16*x-16)/(x^2*exp(1)^2+(2*x^3+4*x^2)*exp( 
1)+x^4+4*x^3+4*x^2)/exp(2),x, algorithm="fricas")
 

Output:

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

Sympy [A] (verification not implemented)

Time = 0.74 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.68 \[ \int \frac {-16-4 e^2+e (-16-8 x)-16 x-4 x^2+e^2 \left (10 x^2+2 e^2 x^2+8 x^3+2 x^4+e \left (9 x^2+4 x^3\right )\right )+e^{2+x^2} \left (8 x^3+2 e^2 x^3+8 x^4+2 x^5+e \left (8 x^3+4 x^4\right )\right )}{e^2 \left (4 x^2+e^2 x^2+4 x^3+x^4+e \left (4 x^2+2 x^3\right )\right )} \, dx=2 x + e^{x^{2}} + \frac {x \left (- e^{3} - 2 e^{2} + 4\right ) + 8 + 4 e}{x^{2} e^{2} + x \left (2 e^{2} + e^{3}\right )} \] Input:

integrate(((2*x**3*exp(1)**2+(4*x**4+8*x**3)*exp(1)+2*x**5+8*x**4+8*x**3)* 
exp(2)*exp(x**2)+(2*x**2*exp(1)**2+(4*x**3+9*x**2)*exp(1)+2*x**4+8*x**3+10 
*x**2)*exp(2)-4*exp(1)**2+(-8*x-16)*exp(1)-4*x**2-16*x-16)/(x**2*exp(1)**2 
+(2*x**3+4*x**2)*exp(1)+x**4+4*x**3+4*x**2)/exp(2),x)
 

Output:

2*x + exp(x**2) + (x*(-exp(3) - 2*exp(2) + 4) + 8 + 4*E)/(x**2*exp(2) + x* 
(2*exp(2) + exp(3)))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 480 vs. \(2 (24) = 48\).

Time = 0.11 (sec) , antiderivative size = 480, normalized size of antiderivative = 19.20 \[ \int \frac {-16-4 e^2+e (-16-8 x)-16 x-4 x^2+e^2 \left (10 x^2+2 e^2 x^2+8 x^3+2 x^4+e \left (9 x^2+4 x^3\right )\right )+e^{2+x^2} \left (8 x^3+2 e^2 x^3+8 x^4+2 x^5+e \left (8 x^3+4 x^4\right )\right )}{e^2 \left (4 x^2+e^2 x^2+4 x^3+x^4+e \left (4 x^2+2 x^3\right )\right )} \, dx={\left (4 \, {\left (\frac {e + 2}{x + e + 2} + \log \left (x + e + 2\right )\right )} e^{3} - 2 \, {\left (2 \, {\left (e + 2\right )} \log \left (x + e + 2\right ) - x + \frac {e^{2} + 4 \, e + 4}{x + e + 2}\right )} e^{2} + 4 \, {\left (\frac {2 \, x + e + 2}{x^{2} {\left (e^{2} + 4 \, e + 4\right )} + x {\left (e^{3} + 6 \, e^{2} + 12 \, e + 8\right )}} - \frac {2 \, \log \left (x + e + 2\right )}{e^{3} + 6 \, e^{2} + 12 \, e + 8} + \frac {2 \, \log \left (x\right )}{e^{3} + 6 \, e^{2} + 12 \, e + 8}\right )} e^{2} + 8 \, {\left (\frac {e + 2}{x + e + 2} + \log \left (x + e + 2\right )\right )} e^{2} + 16 \, {\left (\frac {2 \, x + e + 2}{x^{2} {\left (e^{2} + 4 \, e + 4\right )} + x {\left (e^{3} + 6 \, e^{2} + 12 \, e + 8\right )}} - \frac {2 \, \log \left (x + e + 2\right )}{e^{3} + 6 \, e^{2} + 12 \, e + 8} + \frac {2 \, \log \left (x\right )}{e^{3} + 6 \, e^{2} + 12 \, e + 8}\right )} e + 8 \, {\left (\frac {\log \left (x + e + 2\right )}{e^{2} + 4 \, e + 4} - \frac {\log \left (x\right )}{e^{2} + 4 \, e + 4} - \frac {1}{x {\left (e + 2\right )} + e^{2} + 4 \, e + 4}\right )} e + \frac {16 \, {\left (2 \, x + e + 2\right )}}{x^{2} {\left (e^{2} + 4 \, e + 4\right )} + x {\left (e^{3} + 6 \, e^{2} + 12 \, e + 8\right )}} - \frac {2 \, e^{4}}{x + e + 2} - \frac {9 \, e^{3}}{x + e + 2} - \frac {10 \, e^{2}}{x + e + 2} - \frac {32 \, \log \left (x + e + 2\right )}{e^{3} + 6 \, e^{2} + 12 \, e + 8} + \frac {16 \, \log \left (x + e + 2\right )}{e^{2} + 4 \, e + 4} + \frac {32 \, \log \left (x\right )}{e^{3} + 6 \, e^{2} + 12 \, e + 8} - \frac {16 \, \log \left (x\right )}{e^{2} + 4 \, e + 4} - \frac {16}{x {\left (e + 2\right )} + e^{2} + 4 \, e + 4} + \frac {4}{x + e + 2} + e^{\left (x^{2} + 2\right )}\right )} e^{\left (-2\right )} \] Input:

integrate(((2*x^3*exp(1)^2+(4*x^4+8*x^3)*exp(1)+2*x^5+8*x^4+8*x^3)*exp(2)* 
exp(x^2)+(2*x^2*exp(1)^2+(4*x^3+9*x^2)*exp(1)+2*x^4+8*x^3+10*x^2)*exp(2)-4 
*exp(1)^2+(-8*x-16)*exp(1)-4*x^2-16*x-16)/(x^2*exp(1)^2+(2*x^3+4*x^2)*exp( 
1)+x^4+4*x^3+4*x^2)/exp(2),x, algorithm="maxima")
 

Output:

(4*((e + 2)/(x + e + 2) + log(x + e + 2))*e^3 - 2*(2*(e + 2)*log(x + e + 2 
) - x + (e^2 + 4*e + 4)/(x + e + 2))*e^2 + 4*((2*x + e + 2)/(x^2*(e^2 + 4* 
e + 4) + x*(e^3 + 6*e^2 + 12*e + 8)) - 2*log(x + e + 2)/(e^3 + 6*e^2 + 12* 
e + 8) + 2*log(x)/(e^3 + 6*e^2 + 12*e + 8))*e^2 + 8*((e + 2)/(x + e + 2) + 
 log(x + e + 2))*e^2 + 16*((2*x + e + 2)/(x^2*(e^2 + 4*e + 4) + x*(e^3 + 6 
*e^2 + 12*e + 8)) - 2*log(x + e + 2)/(e^3 + 6*e^2 + 12*e + 8) + 2*log(x)/( 
e^3 + 6*e^2 + 12*e + 8))*e + 8*(log(x + e + 2)/(e^2 + 4*e + 4) - log(x)/(e 
^2 + 4*e + 4) - 1/(x*(e + 2) + e^2 + 4*e + 4))*e + 16*(2*x + e + 2)/(x^2*( 
e^2 + 4*e + 4) + x*(e^3 + 6*e^2 + 12*e + 8)) - 2*e^4/(x + e + 2) - 9*e^3/( 
x + e + 2) - 10*e^2/(x + e + 2) - 32*log(x + e + 2)/(e^3 + 6*e^2 + 12*e + 
8) + 16*log(x + e + 2)/(e^2 + 4*e + 4) + 32*log(x)/(e^3 + 6*e^2 + 12*e + 8 
) - 16*log(x)/(e^2 + 4*e + 4) - 16/(x*(e + 2) + e^2 + 4*e + 4) + 4/(x + e 
+ 2) + e^(x^2 + 2))*e^(-2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (24) = 48\).

Time = 0.13 (sec) , antiderivative size = 83, normalized size of antiderivative = 3.32 \[ \int \frac {-16-4 e^2+e (-16-8 x)-16 x-4 x^2+e^2 \left (10 x^2+2 e^2 x^2+8 x^3+2 x^4+e \left (9 x^2+4 x^3\right )\right )+e^{2+x^2} \left (8 x^3+2 e^2 x^3+8 x^4+2 x^5+e \left (8 x^3+4 x^4\right )\right )}{e^2 \left (4 x^2+e^2 x^2+4 x^3+x^4+e \left (4 x^2+2 x^3\right )\right )} \, dx=\frac {{\left (2 \, x^{3} e^{2} + 2 \, x^{2} e^{3} + 4 \, x^{2} e^{2} + x^{2} e^{\left (x^{2} + 2\right )} - x e^{3} - 2 \, x e^{2} + x e^{\left (x^{2} + 3\right )} + 2 \, x e^{\left (x^{2} + 2\right )} + 4 \, x + 4 \, e + 8\right )} e^{\left (-2\right )}}{x^{2} + x e + 2 \, x} \] Input:

integrate(((2*x^3*exp(1)^2+(4*x^4+8*x^3)*exp(1)+2*x^5+8*x^4+8*x^3)*exp(2)* 
exp(x^2)+(2*x^2*exp(1)^2+(4*x^3+9*x^2)*exp(1)+2*x^4+8*x^3+10*x^2)*exp(2)-4 
*exp(1)^2+(-8*x-16)*exp(1)-4*x^2-16*x-16)/(x^2*exp(1)^2+(2*x^3+4*x^2)*exp( 
1)+x^4+4*x^3+4*x^2)/exp(2),x, algorithm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 2.75 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.76 \[ \int \frac {-16-4 e^2+e (-16-8 x)-16 x-4 x^2+e^2 \left (10 x^2+2 e^2 x^2+8 x^3+2 x^4+e \left (9 x^2+4 x^3\right )\right )+e^{2+x^2} \left (8 x^3+2 e^2 x^3+8 x^4+2 x^5+e \left (8 x^3+4 x^4\right )\right )}{e^2 \left (4 x^2+e^2 x^2+4 x^3+x^4+e \left (4 x^2+2 x^3\right )\right )} \, dx=2\,x+{\mathrm {e}}^{x^2}+\frac {4\,\mathrm {e}-x\,\left (2\,{\mathrm {e}}^2+{\mathrm {e}}^3-4\right )+8}{{\mathrm {e}}^2\,x^2+\left (2\,{\mathrm {e}}^2+{\mathrm {e}}^3\right )\,x} \] Input:

int(-(exp(-2)*(16*x + 4*exp(2) + 4*x^2 - exp(2)*(exp(1)*(9*x^2 + 4*x^3) + 
2*x^2*exp(2) + 10*x^2 + 8*x^3 + 2*x^4) + exp(1)*(8*x + 16) - exp(x^2)*exp( 
2)*(exp(1)*(8*x^3 + 4*x^4) + 2*x^3*exp(2) + 8*x^3 + 8*x^4 + 2*x^5) + 16))/ 
(exp(1)*(4*x^2 + 2*x^3) + x^2*exp(2) + 4*x^2 + 4*x^3 + x^4),x)
 

Output:

2*x + exp(x^2) + (4*exp(1) - x*(2*exp(2) + exp(3) - 4) + 8)/(x*(2*exp(2) + 
 exp(3)) + x^2*exp(2))
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 135, normalized size of antiderivative = 5.40 \[ \int \frac {-16-4 e^2+e (-16-8 x)-16 x-4 x^2+e^2 \left (10 x^2+2 e^2 x^2+8 x^3+2 x^4+e \left (9 x^2+4 x^3\right )\right )+e^{2+x^2} \left (8 x^3+2 e^2 x^3+8 x^4+2 x^5+e \left (8 x^3+4 x^4\right )\right )}{e^2 \left (4 x^2+e^2 x^2+4 x^3+x^4+e \left (4 x^2+2 x^3\right )\right )} \, dx=\frac {e^{x^{2}} e^{4} x +e^{x^{2}} e^{3} x^{2}+4 e^{x^{2}} e^{3} x +2 e^{x^{2}} e^{2} x^{2}+4 e^{x^{2}} e^{2} x +2 e^{4} x^{2}+2 e^{3} x^{3}+9 e^{3} x^{2}+4 e^{2} x^{3}+10 e^{2} x^{2}+4 e^{2}+16 e -4 x^{2}+16}{e^{2} x \left (e^{2}+e x +4 e +2 x +4\right )} \] Input:

int(((2*x^3*exp(1)^2+(4*x^4+8*x^3)*exp(1)+2*x^5+8*x^4+8*x^3)*exp(2)*exp(x^ 
2)+(2*x^2*exp(1)^2+(4*x^3+9*x^2)*exp(1)+2*x^4+8*x^3+10*x^2)*exp(2)-4*exp(1 
)^2+(-8*x-16)*exp(1)-4*x^2-16*x-16)/(x^2*exp(1)^2+(2*x^3+4*x^2)*exp(1)+x^4 
+4*x^3+4*x^2)/exp(2),x)
 

Output:

(e**(x**2)*e**4*x + e**(x**2)*e**3*x**2 + 4*e**(x**2)*e**3*x + 2*e**(x**2) 
*e**2*x**2 + 4*e**(x**2)*e**2*x + 2*e**4*x**2 + 2*e**3*x**3 + 9*e**3*x**2 
+ 4*e**2*x**3 + 10*e**2*x**2 + 4*e**2 + 16*e - 4*x**2 + 16)/(e**2*x*(e**2 
+ e*x + 4*e + 2*x + 4))