3.714 \(\int e^{x^3} (1-e^{4 x^3})^2 x^2 \, dx\)

Optimal. Leaf size=32 \[ \frac{e^{x^3}}{3}-\frac{2 e^{5 x^3}}{15}+\frac{e^{9 x^3}}{27} \]

[Out]

E^x^3/3 - (2*E^(5*x^3))/15 + E^(9*x^3)/27

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Rubi [A]  time = 0.208785, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {6715, 2249, 194} \[ \frac{e^{x^3}}{3}-\frac{2 e^{5 x^3}}{15}+\frac{e^{9 x^3}}{27} \]

Antiderivative was successfully verified.

[In]

Int[E^x^3*(1 - E^(4*x^3))^2*x^2,x]

[Out]

E^x^3/3 - (2*E^(5*x^3))/15 + E^(9*x^3)/27

Rule 6715

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]

Rule 2249

Int[((a_) + (b_.)*(F_)^((e_.)*((c_.) + (d_.)*(x_))))^(p_.)*(G_)^((h_.)*((f_.) + (g_.)*(x_))), x_Symbol] :> Wit
h[{m = FullSimplify[(d*e*Log[F])/(g*h*Log[G])]}, Dist[Denominator[m]/(g*h*Log[G]), Subst[Int[x^(Denominator[m]
- 1)*(a + b*F^(c*e - (d*e*f)/g)*x^Numerator[m])^p, x], x, G^((h*(f + g*x))/Denominator[m])], x] /; LtQ[m, -1]
|| GtQ[m, 1]] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x]

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int e^{x^3} \left (1-e^{4 x^3}\right )^2 x^2 \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int e^x \left (1-e^{4 x}\right )^2 \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (1-x^4\right )^2 \, dx,x,e^{x^3}\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (1-2 x^4+x^8\right ) \, dx,x,e^{x^3}\right )\\ &=\frac{e^{x^3}}{3}-\frac{2 e^{5 x^3}}{15}+\frac{e^{9 x^3}}{27}\\ \end{align*}

Mathematica [A]  time = 0.0238624, size = 29, normalized size = 0.91 \[ \frac{1}{135} e^{x^3} \left (-18 e^{4 x^3}+5 e^{8 x^3}+45\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^x^3*(1 - E^(4*x^3))^2*x^2,x]

[Out]

(E^x^3*(45 - 18*E^(4*x^3) + 5*E^(8*x^3)))/135

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Maple [A]  time = 0.021, size = 24, normalized size = 0.8 \begin{align*}{\frac{ \left ({{\rm e}^{{x}^{3}}} \right ) ^{9}}{27}}-{\frac{2\, \left ({{\rm e}^{{x}^{3}}} \right ) ^{5}}{15}}+{\frac{{{\rm e}^{{x}^{3}}}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x^3)*(1-exp(4*x^3))^2*x^2,x)

[Out]

1/27*exp(x^3)^9-2/15*exp(x^3)^5+1/3*exp(x^3)

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Maxima [A]  time = 0.958564, size = 31, normalized size = 0.97 \begin{align*} \frac{1}{27} \, e^{\left (9 \, x^{3}\right )} - \frac{2}{15} \, e^{\left (5 \, x^{3}\right )} + \frac{1}{3} \, e^{\left (x^{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x^3)*(1-exp(4*x^3))^2*x^2,x, algorithm="maxima")

[Out]

1/27*e^(9*x^3) - 2/15*e^(5*x^3) + 1/3*e^(x^3)

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Fricas [A]  time = 0.869861, size = 63, normalized size = 1.97 \begin{align*} \frac{1}{27} \, e^{\left (9 \, x^{3}\right )} - \frac{2}{15} \, e^{\left (5 \, x^{3}\right )} + \frac{1}{3} \, e^{\left (x^{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x^3)*(1-exp(4*x^3))^2*x^2,x, algorithm="fricas")

[Out]

1/27*e^(9*x^3) - 2/15*e^(5*x^3) + 1/3*e^(x^3)

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Sympy [A]  time = 0.143512, size = 24, normalized size = 0.75 \begin{align*} \frac{e^{9 x^{3}}}{27} - \frac{2 e^{5 x^{3}}}{15} + \frac{e^{x^{3}}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x**3)*(1-exp(4*x**3))**2*x**2,x)

[Out]

exp(9*x**3)/27 - 2*exp(5*x**3)/15 + exp(x**3)/3

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Giac [A]  time = 1.25431, size = 31, normalized size = 0.97 \begin{align*} \frac{1}{27} \, e^{\left (9 \, x^{3}\right )} - \frac{2}{15} \, e^{\left (5 \, x^{3}\right )} + \frac{1}{3} \, e^{\left (x^{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x^3)*(1-exp(4*x^3))^2*x^2,x, algorithm="giac")

[Out]

1/27*e^(9*x^3) - 2/15*e^(5*x^3) + 1/3*e^(x^3)