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

Optimal. Leaf size=39 \[ -\frac {1}{2} e^{-2 x^3-x^6} \left (1+x^3\right )+\frac {1}{4} e \sqrt {\pi } \text {erf}\left (1+x^3\right ) \]

[Out]

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

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Rubi [A]
time = 0.12, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {12, 6847, 2269, 2266, 2236} \begin {gather*} \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 ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2236

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] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 2266

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

Rule 2269

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_))^(m_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*
(F^(a + b*x + c*x^2)/(2*c*Log[F])), x] - Dist[(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 6847

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

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=3 \int e^{-2 x^3-x^6} x^2 \left (1+x^3\right )^2 \, dx\\ &=\text {Subst}\left (\int e^{-2 x-x^2} (1+x)^2 \, dx,x,x^3\right )\\ &=-\frac {1}{2} e^{-2 x^3-x^6} \left (1+x^3\right )+\frac {1}{2} \text {Subst}\left (\int e^{-2 x-x^2} \, dx,x,x^3\right )\\ &=-\frac {1}{2} e^{-2 x^3-x^6} \left (1+x^3\right )+\frac {1}{2} e \text {Subst}\left (\int e^{-\frac {1}{4} (-2-2 x)^2} \, dx,x,x^3\right )\\ &=-\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 )\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.33, size = 37, normalized size = 0.95 \begin {gather*} \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 ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int 3 x^{2} \left (x^{3}+1\right )^{2} {\mathrm e}^{-x^{6}-2 x^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [C] Result contains complex when optimal does not.
time = 0.47, size = 137, normalized size = 3.51 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)/sqrt((x^3 + 1)^2) - 2*I*e^(-(x^3 + 1)^2))*e + I*(I*sqrt(pi)*(x^3 + 1)*(er
f(sqrt((x^3 + 1)^2)) - 1)/sqrt((x^3 + 1)^2) + I*e^(-(x^3 + 1)^2))*e

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Fricas [A]
time = 0.58, size = 33, normalized size = 0.85 \begin {gather*} \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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} 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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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))

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Giac [A]
time = 0.46, size = 33, normalized size = 0.85 \begin {gather*} \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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.31, size = 45, normalized size = 1.15 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(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

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Chatgpt [F] Failed to verify
time = 1.00, size = 36, normalized size = 0.92 \begin {gather*} -\frac {\left (x^{3}+1\right )^{2} {\mathrm e}^{-x^{6}-2 x^{3}}}{2}-{\mathrm e}^{-x^{6}-2 x^{3}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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