\(\int 3 e^{-2 x^3-x^6} x^2 (1+x^3)^2 \, dx\) [173]

Optimal result

Integrand size = 25, antiderivative size = 39 \[ \int 3 e^{-2 x^3-x^6} x^2 \left (1+x^3\right )^2 \, dx=-\frac {1}{2} e^{-2 x^3-x^6} \left (1+x^3\right )+\frac {1}{4} e \sqrt {\pi } \text {erf}\left (1+x^3\right ) \] Output:

-1/2*exp(-x^6-2*x^3)*(x^3+1)+1/4*exp(1)*Pi^(1/2)*erf(x^3+1)
 

Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.95 \[ \int 3 e^{-2 x^3-x^6} x^2 \left (1+x^3\right )^2 \, dx=\frac {1}{4} \left (-2 e^{-x^3 \left (2+x^3\right )} \left (1+x^3\right )+e \sqrt {\pi } \text {erf}\left (1+x^3\right )\right ) \] Input:

Integrate[3*E^(-2*x^3 - x^6)*x^2*(1 + x^3)^2,x]
 

Output:

((-2*(1 + x^3))/E^(x^3*(2 + x^3)) + E*Sqrt[Pi]*Erf[1 + x^3])/4
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {27, 7266, 2667, 2664, 2634}

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.

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 7266

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

\(\Big \downarrow \) 2667

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

\(\Big \downarrow \) 2664

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

\(\Big \downarrow \) 2634

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

Input:

Int[3*E^(-2*x^3 - x^6)*x^2*(1 + x^3)^2,x]
 

Output:

-1/2*(E^(-2*x^3 - x^6)*(1 + x^3)) + (E*Sqrt[Pi]*Erf[1 + x^3])/4
 

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 2634

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt 
[Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F], 2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; Fr 
eeQ[{F, a, b, c, d}, x] && NegQ[b]
 

rule 2664

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[F^(a - b^2/ 
(4*c))   Int[F^((b + 2*c*x)^2/(4*c)), x], x] /; FreeQ[{F, a, b, c}, x]
 

rule 2667

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_))^(m_), x_S 
ymbol] :> Simp[e*(d + e*x)^(m - 1)*(F^(a + b*x + c*x^2)/(2*c*Log[F])), x] - 
 Simp[(m - 1)*(e^2/(2*c*Log[F]))   Int[(d + e*x)^(m - 2)*F^(a + b*x + c*x^2 
), x], x] /; FreeQ[{F, a, b, c, d, e}, x] && EqQ[b*e - 2*c*d, 0] && GtQ[m, 
1]
 

rule 7266

Int[(u_)*(x_)^(m_.), x_Symbol] :> Simp[1/(m + 1)   Subst[Int[SubstFor[x^(m 
+ 1), u, x], x], x, x^(m + 1)], x] /; FreeQ[m, x] && NeQ[m, -1] && Function 
OfQ[x^(m + 1), u, x]
 

Maple [F]

\[\int 3 \,{\mathrm e}^{-x^{6}-2 x^{3}} x^{2} \left (x^{3}+1\right )^{2}d x\]

Input:

int(3*exp(-x^6-2*x^3)*x^2*(x^3+1)^2,x)
 

Output:

int(3*exp(-x^6-2*x^3)*x^2*(x^3+1)^2,x)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.85 \[ \int 3 e^{-2 x^3-x^6} x^2 \left (1+x^3\right )^2 \, dx=\frac {1}{4} \, \sqrt {\pi } \operatorname {erf}\left (x^{3} + 1\right ) e - \frac {1}{2} \, {\left (x^{3} + 1\right )} e^{\left (-x^{6} - 2 \, x^{3}\right )} \] Input:

integrate(3*exp(-x^6-2*x^3)*x^2*(x^3+1)^2,x, algorithm="fricas")
 

Output:

1/4*sqrt(pi)*erf(x^3 + 1)*e - 1/2*(x^3 + 1)*e^(-x^6 - 2*x^3)
 

Sympy [F]

\[ \int 3 e^{-2 x^3-x^6} x^2 \left (1+x^3\right )^2 \, dx=3 \left (\int x^{2} e^{- 2 x^{3}} e^{- x^{6}}\, dx + \int 2 x^{5} e^{- 2 x^{3}} e^{- x^{6}}\, dx + \int x^{8} e^{- 2 x^{3}} e^{- x^{6}}\, dx\right ) \] Input:

integrate(3*exp(-x**6-2*x**3)*x**2*(x**3+1)**2,x)
 

Output:

3*(Integral(x**2*exp(-2*x**3)*exp(-x**6), x) + Integral(2*x**5*exp(-2*x**3 
)*exp(-x**6), x) + Integral(x**8*exp(-2*x**3)*exp(-x**6), x))
 

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.17 (sec) , antiderivative size = 137, normalized size of antiderivative = 3.51 \[ \int 3 e^{-2 x^3-x^6} x^2 \left (1+x^3\right )^2 \, dx=\frac {1}{2} \, \sqrt {\pi } \operatorname {erf}\left (x^{3} + 1\right ) e + \frac {1}{2} i \, {\left (\frac {i \, {\left (x^{3} + 1\right )}^{3} \Gamma \left (\frac {3}{2}, {\left (x^{3} + 1\right )}^{2}\right )}{{\left ({\left (x^{3} + 1\right )}^{2}\right )}^{\frac {3}{2}}} - \frac {i \, \sqrt {\pi } {\left (x^{3} + 1\right )} {\left (\operatorname {erf}\left (\sqrt {{\left (x^{3} + 1\right )}^{2}}\right ) - 1\right )}}{\sqrt {{\left (x^{3} + 1\right )}^{2}}} - 2 i \, e^{\left (-{\left (x^{3} + 1\right )}^{2}\right )}\right )} e + i \, {\left (\frac {i \, \sqrt {\pi } {\left (x^{3} + 1\right )} {\left (\operatorname {erf}\left (\sqrt {{\left (x^{3} + 1\right )}^{2}}\right ) - 1\right )}}{\sqrt {{\left (x^{3} + 1\right )}^{2}}} + i \, e^{\left (-{\left (x^{3} + 1\right )}^{2}\right )}\right )} e \] Input:

integrate(3*exp(-x^6-2*x^3)*x^2*(x^3+1)^2,x, algorithm="maxima")
 

Output:

1/2*sqrt(pi)*erf(x^3 + 1)*e + 1/2*I*(I*(x^3 + 1)^3*gamma(3/2, (x^3 + 1)^2) 
/((x^3 + 1)^2)^(3/2) - I*sqrt(pi)*(x^3 + 1)*(erf(sqrt((x^3 + 1)^2)) - 1)/s 
qrt((x^3 + 1)^2) - 2*I*e^(-(x^3 + 1)^2))*e + I*(I*sqrt(pi)*(x^3 + 1)*(erf( 
sqrt((x^3 + 1)^2)) - 1)/sqrt((x^3 + 1)^2) + I*e^(-(x^3 + 1)^2))*e
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.85 \[ \int 3 e^{-2 x^3-x^6} x^2 \left (1+x^3\right )^2 \, dx=\frac {1}{4} \, \sqrt {\pi } \operatorname {erf}\left (x^{3} + 1\right ) e - \frac {1}{2} \, {\left (x^{3} + 1\right )} e^{\left (-x^{6} - 2 \, x^{3}\right )} \] Input:

integrate(3*exp(-x^6-2*x^3)*x^2*(x^3+1)^2,x, algorithm="giac")
 

Output:

1/4*sqrt(pi)*erf(x^3 + 1)*e - 1/2*(x^3 + 1)*e^(-x^6 - 2*x^3)
 

Mupad [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.15 \[ \int 3 e^{-2 x^3-x^6} x^2 \left (1+x^3\right )^2 \, dx=\frac {\sqrt {\pi }\,\mathrm {e}\,\mathrm {erf}\left (x^3+1\right )}{4}-\frac {x^3\,{\mathrm {e}}^{-x^6-2\,x^3}}{2}-\frac {{\mathrm {e}}^{-x^6-2\,x^3}}{2} \] Input:

int(3*x^2*exp(- 2*x^3 - x^6)*(x^3 + 1)^2,x)
 

Output:

(pi^(1/2)*exp(1)*erf(x^3 + 1))/4 - (x^3*exp(- 2*x^3 - x^6))/2 - exp(- 2*x^ 
3 - x^6)/2
 

Reduce [F]

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

int(3*exp(-x^6-2*x^3)*x^2*(x^3+1)^2,x)
 

Output:

(3*e**(x**6 + 2*x**3)*int(x**2/e**(x**6 + 2*x**3),x) - x**3 - 1)/(2*e**(x* 
*6 + 2*x**3))