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

Optimal. Leaf size=113 \[ \frac{2 x^2 e^{-b^2 x^2} \text{Erf}(b x)}{3 \sqrt{\pi } b}+\frac{2 e^{-b^2 x^2} \text{Erf}(b x)}{3 \sqrt{\pi } b^3}-\frac{5 \text{Erf}\left (\sqrt{2} b x\right )}{6 \sqrt{2 \pi } b^3}+\frac{x e^{-2 b^2 x^2}}{3 \pi b^2}+\frac{1}{3} x^3 \text{Erf}(b x)^2 \]

[Out]

x/(3*b^2*E^(2*b^2*x^2)*Pi) + (2*Erf[b*x])/(3*b^3*E^(b^2*x^2)*Sqrt[Pi]) + (2*x^2*Erf[b*x])/(3*b*E^(b^2*x^2)*Sqr
t[Pi]) + (x^3*Erf[b*x]^2)/3 - (5*Erf[Sqrt[2]*b*x])/(6*b^3*Sqrt[2*Pi])

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Rubi [A]  time = 0.135931, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6364, 6385, 6382, 2205, 2212} \[ \frac{2 x^2 e^{-b^2 x^2} \text{Erf}(b x)}{3 \sqrt{\pi } b}+\frac{2 e^{-b^2 x^2} \text{Erf}(b x)}{3 \sqrt{\pi } b^3}-\frac{5 \text{Erf}\left (\sqrt{2} b x\right )}{6 \sqrt{2 \pi } b^3}+\frac{x e^{-2 b^2 x^2}}{3 \pi b^2}+\frac{1}{3} x^3 \text{Erf}(b x)^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/(3*b^2*E^(2*b^2*x^2)*Pi) + (2*Erf[b*x])/(3*b^3*E^(b^2*x^2)*Sqrt[Pi]) + (2*x^2*Erf[b*x])/(3*b*E^(b^2*x^2)*Sqr
t[Pi]) + (x^3*Erf[b*x]^2)/3 - (5*Erf[Sqrt[2]*b*x])/(6*b^3*Sqrt[2*Pi])

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 6382

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

Rule 2205

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 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^2 \text{erf}(b x)^2 \, dx &=\frac{1}{3} x^3 \text{erf}(b x)^2-\frac{(4 b) \int e^{-b^2 x^2} x^3 \text{erf}(b x) \, dx}{3 \sqrt{\pi }}\\ &=\frac{2 e^{-b^2 x^2} x^2 \text{erf}(b x)}{3 b \sqrt{\pi }}+\frac{1}{3} x^3 \text{erf}(b x)^2-\frac{4 \int e^{-2 b^2 x^2} x^2 \, dx}{3 \pi }-\frac{4 \int e^{-b^2 x^2} x \text{erf}(b x) \, dx}{3 b \sqrt{\pi }}\\ &=\frac{e^{-2 b^2 x^2} x}{3 b^2 \pi }+\frac{2 e^{-b^2 x^2} \text{erf}(b x)}{3 b^3 \sqrt{\pi }}+\frac{2 e^{-b^2 x^2} x^2 \text{erf}(b x)}{3 b \sqrt{\pi }}+\frac{1}{3} x^3 \text{erf}(b x)^2-\frac{\int e^{-2 b^2 x^2} \, dx}{3 b^2 \pi }-\frac{4 \int e^{-2 b^2 x^2} \, dx}{3 b^2 \pi }\\ &=\frac{e^{-2 b^2 x^2} x}{3 b^2 \pi }+\frac{2 e^{-b^2 x^2} \text{erf}(b x)}{3 b^3 \sqrt{\pi }}+\frac{2 e^{-b^2 x^2} x^2 \text{erf}(b x)}{3 b \sqrt{\pi }}+\frac{1}{3} x^3 \text{erf}(b x)^2-\frac{\sqrt{\frac{2}{\pi }} \text{erf}\left (\sqrt{2} b x\right )}{3 b^3}-\frac{\text{erf}\left (\sqrt{2} b x\right )}{6 b^3 \sqrt{2 \pi }}\\ \end{align*}

Mathematica [A]  time = 0.073654, size = 88, normalized size = 0.78 \[ \frac{4 \pi b^3 x^3 \text{Erf}(b x)^2+8 \sqrt{\pi } e^{-b^2 x^2} \left (b^2 x^2+1\right ) \text{Erf}(b x)+4 b x e^{-2 b^2 x^2}-5 \sqrt{2 \pi } \text{Erf}\left (\sqrt{2} b x\right )}{12 \pi b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((4*b*x)/E^(2*b^2*x^2) + (8*Sqrt[Pi]*(1 + b^2*x^2)*Erf[b*x])/E^(b^2*x^2) + 4*b^3*Pi*x^3*Erf[b*x]^2 - 5*Sqrt[2*
Pi]*Erf[Sqrt[2]*b*x])/(12*b^3*Pi)

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Maple [A]  time = 0.046, size = 95, normalized size = 0.8 \begin{align*}{\frac{1}{{b}^{3}} \left ({\frac{ \left ({\it Erf} \left ( bx \right ) \right ) ^{2}{b}^{3}{x}^{3}}{3}}-{\frac{4\,{\it Erf} \left ( bx \right ) }{3\,\sqrt{\pi }} \left ( -{\frac{{b}^{2}{x}^{2}}{2\,{{\rm e}^{{b}^{2}{x}^{2}}}}}-{\frac{1}{2\,{{\rm e}^{{b}^{2}{x}^{2}}}}} \right ) }+{\frac{4}{3\,\pi } \left ( -{\frac{5\,\sqrt{2}\sqrt{\pi }{\it Erf} \left ( bx\sqrt{2} \right ) }{16}}+{\frac{bx}{4\, \left ({{\rm e}^{{b}^{2}{x}^{2}}} \right ) ^{2}}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b^3*(1/3*erf(b*x)^2*b^3*x^3-4/3*erf(b*x)/Pi^(1/2)*(-1/2*b^2*x^2/exp(b^2*x^2)-1/2/exp(b^2*x^2))+4/3/Pi*(-5/16
*2^(1/2)*Pi^(1/2)*erf(b*x*2^(1/2))+1/4/exp(b^2*x^2)^2*b*x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.60602, size = 231, normalized size = 2.04 \begin{align*} \frac{4 \, \pi b^{4} x^{3} \operatorname{erf}\left (b x\right )^{2} + 4 \, b^{2} x e^{\left (-2 \, b^{2} x^{2}\right )} + 8 \, \sqrt{\pi }{\left (b^{3} x^{2} + b\right )} \operatorname{erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} - 5 \, \sqrt{2} \sqrt{\pi } \sqrt{b^{2}} \operatorname{erf}\left (\sqrt{2} \sqrt{b^{2}} x\right )}{12 \, \pi b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(4*pi*b^4*x^3*erf(b*x)^2 + 4*b^2*x*e^(-2*b^2*x^2) + 8*sqrt(pi)*(b^3*x^2 + b)*erf(b*x)*e^(-b^2*x^2) - 5*sq
rt(2)*sqrt(pi)*sqrt(b^2)*erf(sqrt(2)*sqrt(b^2)*x))/(pi*b^4)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2*erf(b*x)**2, x)

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Giac [A]  time = 1.45898, size = 151, normalized size = 1.34 \begin{align*} \frac{1}{3} \, x^{3} \operatorname{erf}\left (b x\right )^{2} + \frac{b{\left (\frac{8 \,{\left (b^{2} x^{2} + 1\right )} \operatorname{erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{b^{4}} + \frac{\sqrt{\pi } b^{2}{\left (\frac{4 \, x e^{\left (-2 \, b^{2} x^{2}\right )}}{b^{2}} + \frac{\sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\sqrt{2} b x\right )}{b^{3}}\right )} + \frac{4 \, \sqrt{2} \pi \operatorname{erf}\left (-\sqrt{2} b x\right )}{b}}{\pi b^{3}}\right )}}{12 \, \sqrt{\pi }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3*erf(b*x)^2 + 1/12*b*(8*(b^2*x^2 + 1)*erf(b*x)*e^(-b^2*x^2)/b^4 + (sqrt(pi)*b^2*(4*x*e^(-2*b^2*x^2)/b^2
+ sqrt(2)*sqrt(pi)*erf(-sqrt(2)*b*x)/b^3) + 4*sqrt(2)*pi*erf(-sqrt(2)*b*x)/b)/(pi*b^3))/sqrt(pi)