3.23 \(\int x^3 \text{Erf}(b x)^2 \, dx\)

Optimal. Leaf size=126 \[ \frac{x^3 e^{-b^2 x^2} \text{Erf}(b x)}{2 \sqrt{\pi } b}+\frac{3 x e^{-b^2 x^2} \text{Erf}(b x)}{4 \sqrt{\pi } b^3}-\frac{3 \text{Erf}(b x)^2}{16 b^4}+\frac{x^2 e^{-2 b^2 x^2}}{4 \pi b^2}+\frac{e^{-2 b^2 x^2}}{2 \pi b^4}+\frac{1}{4} x^4 \text{Erf}(b x)^2 \]

[Out]

1/(2*b^4*E^(2*b^2*x^2)*Pi) + x^2/(4*b^2*E^(2*b^2*x^2)*Pi) + (3*x*Erf[b*x])/(4*b^3*E^(b^2*x^2)*Sqrt[Pi]) + (x^3
*Erf[b*x])/(2*b*E^(b^2*x^2)*Sqrt[Pi]) - (3*Erf[b*x]^2)/(16*b^4) + (x^4*Erf[b*x]^2)/4

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Rubi [A]  time = 0.181377, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {6364, 6385, 6373, 30, 2209, 2212} \[ \frac{x^3 e^{-b^2 x^2} \text{Erf}(b x)}{2 \sqrt{\pi } b}+\frac{3 x e^{-b^2 x^2} \text{Erf}(b x)}{4 \sqrt{\pi } b^3}-\frac{3 \text{Erf}(b x)^2}{16 b^4}+\frac{x^2 e^{-2 b^2 x^2}}{4 \pi b^2}+\frac{e^{-2 b^2 x^2}}{2 \pi b^4}+\frac{1}{4} x^4 \text{Erf}(b x)^2 \]

Antiderivative was successfully verified.

[In]

Int[x^3*Erf[b*x]^2,x]

[Out]

1/(2*b^4*E^(2*b^2*x^2)*Pi) + x^2/(4*b^2*E^(2*b^2*x^2)*Pi) + (3*x*Erf[b*x])/(4*b^3*E^(b^2*x^2)*Sqrt[Pi]) + (x^3
*Erf[b*x])/(2*b*E^(b^2*x^2)*Sqrt[Pi]) - (3*Erf[b*x]^2)/(16*b^4) + (x^4*Erf[b*x]^2)/4

Rule 6364

Int[Erf[(b_.)*(x_)]^2*(x_)^(m_.), x_Symbol] :> Simp[(x^(m + 1)*Erf[b*x]^2)/(m + 1), x] - Dist[(4*b)/(Sqrt[Pi]*
(m + 1)), Int[(x^(m + 1)*Erf[b*x])/E^(b^2*x^2), x], x] /; FreeQ[b, x] && (IGtQ[m, 0] || ILtQ[(m + 1)/2, 0])

Rule 6385

Int[E^((c_.) + (d_.)*(x_)^2)*Erf[(a_.) + (b_.)*(x_)]*(x_)^(m_), x_Symbol] :> Simp[(x^(m - 1)*E^(c + d*x^2)*Erf
[a + b*x])/(2*d), x] + (-Dist[(m - 1)/(2*d), Int[x^(m - 2)*E^(c + d*x^2)*Erf[a + b*x], x], x] - Dist[b/(d*Sqrt
[Pi]), Int[x^(m - 1)*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2), x], x]) /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 1]

Rule 6373

Int[E^((c_.) + (d_.)*(x_)^2)*Erf[(b_.)*(x_)]^(n_.), x_Symbol] :> Dist[(E^c*Sqrt[Pi])/(2*b), Subst[Int[x^n, x],
x, Erf[b*x]], x] /; FreeQ[{b, c, d, n}, x] && EqQ[d, -b^2]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2209

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

Rule 2212

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
- n + 1)*F^(a + b*(c + d*x)^n))/(b*d*n*Log[F]), x] - Dist[(m - n + 1)/(b*n*Log[F]), Int[(c + d*x)^(m - n)*F^(
a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[(2*(m + 1))/n] && LtQ[0, (m + 1)/n, 5] &&
IntegerQ[n] && (LtQ[0, n, m + 1] || LtQ[m, n, 0])

Rubi steps

\begin{align*} \int x^3 \text{erf}(b x)^2 \, dx &=\frac{1}{4} x^4 \text{erf}(b x)^2-\frac{b \int e^{-b^2 x^2} x^4 \text{erf}(b x) \, dx}{\sqrt{\pi }}\\ &=\frac{e^{-b^2 x^2} x^3 \text{erf}(b x)}{2 b \sqrt{\pi }}+\frac{1}{4} x^4 \text{erf}(b x)^2-\frac{\int e^{-2 b^2 x^2} x^3 \, dx}{\pi }-\frac{3 \int e^{-b^2 x^2} x^2 \text{erf}(b x) \, dx}{2 b \sqrt{\pi }}\\ &=\frac{e^{-2 b^2 x^2} x^2}{4 b^2 \pi }+\frac{3 e^{-b^2 x^2} x \text{erf}(b x)}{4 b^3 \sqrt{\pi }}+\frac{e^{-b^2 x^2} x^3 \text{erf}(b x)}{2 b \sqrt{\pi }}+\frac{1}{4} x^4 \text{erf}(b x)^2-\frac{\int e^{-2 b^2 x^2} x \, dx}{2 b^2 \pi }-\frac{3 \int e^{-2 b^2 x^2} x \, dx}{2 b^2 \pi }-\frac{3 \int e^{-b^2 x^2} \text{erf}(b x) \, dx}{4 b^3 \sqrt{\pi }}\\ &=\frac{e^{-2 b^2 x^2}}{2 b^4 \pi }+\frac{e^{-2 b^2 x^2} x^2}{4 b^2 \pi }+\frac{3 e^{-b^2 x^2} x \text{erf}(b x)}{4 b^3 \sqrt{\pi }}+\frac{e^{-b^2 x^2} x^3 \text{erf}(b x)}{2 b \sqrt{\pi }}+\frac{1}{4} x^4 \text{erf}(b x)^2-\frac{3 \operatorname{Subst}(\int x \, dx,x,\text{erf}(b x))}{8 b^4}\\ &=\frac{e^{-2 b^2 x^2}}{2 b^4 \pi }+\frac{e^{-2 b^2 x^2} x^2}{4 b^2 \pi }+\frac{3 e^{-b^2 x^2} x \text{erf}(b x)}{4 b^3 \sqrt{\pi }}+\frac{e^{-b^2 x^2} x^3 \text{erf}(b x)}{2 b \sqrt{\pi }}-\frac{3 \text{erf}(b x)^2}{16 b^4}+\frac{1}{4} x^4 \text{erf}(b x)^2\\ \end{align*}

Mathematica [A]  time = 0.0350976, size = 90, normalized size = 0.71 \[ \frac{e^{-2 b^2 x^2} \left (4 \sqrt{\pi } b x e^{b^2 x^2} \left (2 b^2 x^2+3\right ) \text{Erf}(b x)+\pi e^{2 b^2 x^2} \left (4 b^4 x^4-3\right ) \text{Erf}(b x)^2+4 b^2 x^2+8\right )}{16 \pi b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*Erf[b*x]^2,x]

[Out]

(8 + 4*b^2*x^2 + 4*b*E^(b^2*x^2)*Sqrt[Pi]*x*(3 + 2*b^2*x^2)*Erf[b*x] + E^(2*b^2*x^2)*Pi*(-3 + 4*b^4*x^4)*Erf[b
*x]^2)/(16*b^4*E^(2*b^2*x^2)*Pi)

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Maple [F]  time = 0.053, size = 0, normalized size = 0. \begin{align*} \int{x}^{3} \left ({\it Erf} \left ( bx \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*erf(b*x)^2,x)

[Out]

int(x^3*erf(b*x)^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{-\frac{{\left (2 \, b^{2} x^{2} + 1\right )} e^{\left (-2 \, b^{2} x^{2}\right )}}{4 \, b^{2}} - \frac{3 \, e^{\left (-2 \, b^{2} x^{2}\right )}}{4 \, b^{2}}}{2 \, \pi b^{2}} - \frac{{\left (3 \, \pi - 4 \, \pi b^{4} x^{4}\right )} \operatorname{erf}\left (b x\right )^{2} - 4 \,{\left (2 \, \sqrt{\pi } b^{3} x^{3} + 3 \, \sqrt{\pi } b x\right )} \operatorname{erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{16 \, \pi b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*erf(b*x)^2,x, algorithm="maxima")

[Out]

-1/2*integrate((2*b^2*x^3 + 3*x)*e^(-2*b^2*x^2), x)/(pi*b^2) - 1/16*((3*pi - 4*pi*b^4*x^4)*erf(b*x)^2 - 4*(2*s
qrt(pi)*b^3*x^3 + 3*sqrt(pi)*b*x)*erf(b*x)*e^(-b^2*x^2))/(pi*b^4)

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Fricas [A]  time = 2.96057, size = 186, normalized size = 1.48 \begin{align*} \frac{4 \, \sqrt{\pi }{\left (2 \, b^{3} x^{3} + 3 \, b x\right )} \operatorname{erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} -{\left (3 \, \pi - 4 \, \pi b^{4} x^{4}\right )} \operatorname{erf}\left (b x\right )^{2} + 4 \,{\left (b^{2} x^{2} + 2\right )} e^{\left (-2 \, b^{2} x^{2}\right )}}{16 \, \pi b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*erf(b*x)^2,x, algorithm="fricas")

[Out]

1/16*(4*sqrt(pi)*(2*b^3*x^3 + 3*b*x)*erf(b*x)*e^(-b^2*x^2) - (3*pi - 4*pi*b^4*x^4)*erf(b*x)^2 + 4*(b^2*x^2 + 2
)*e^(-2*b^2*x^2))/(pi*b^4)

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Sympy [A]  time = 5.43399, size = 117, normalized size = 0.93 \begin{align*} \begin{cases} \frac{x^{4} \operatorname{erf}^{2}{\left (b x \right )}}{4} + \frac{x^{3} e^{- b^{2} x^{2}} \operatorname{erf}{\left (b x \right )}}{2 \sqrt{\pi } b} + \frac{x^{2} e^{- 2 b^{2} x^{2}}}{4 \pi b^{2}} + \frac{3 x e^{- b^{2} x^{2}} \operatorname{erf}{\left (b x \right )}}{4 \sqrt{\pi } b^{3}} - \frac{3 \operatorname{erf}^{2}{\left (b x \right )}}{16 b^{4}} + \frac{e^{- 2 b^{2} x^{2}}}{2 \pi b^{4}} & \text{for}\: b \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*erf(b*x)**2,x)

[Out]

Piecewise((x**4*erf(b*x)**2/4 + x**3*exp(-b**2*x**2)*erf(b*x)/(2*sqrt(pi)*b) + x**2*exp(-2*b**2*x**2)/(4*pi*b*
*2) + 3*x*exp(-b**2*x**2)*erf(b*x)/(4*sqrt(pi)*b**3) - 3*erf(b*x)**2/(16*b**4) + exp(-2*b**2*x**2)/(2*pi*b**4)
, Ne(b, 0)), (0, True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \operatorname{erf}\left (b x\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*erf(b*x)^2,x, algorithm="giac")

[Out]

integrate(x^3*erf(b*x)^2, x)