3.29 \(\int x^4 \text{Erf}(b x)^2 \, dx\)

Optimal. Leaf size=165 \[ \frac{2 x^4 e^{-b^2 x^2} \text{Erf}(b x)}{5 \sqrt{\pi } b}+\frac{4 x^2 e^{-b^2 x^2} \text{Erf}(b x)}{5 \sqrt{\pi } b^3}+\frac{4 e^{-b^2 x^2} \text{Erf}(b x)}{5 \sqrt{\pi } b^5}-\frac{43 \text{Erf}\left (\sqrt{2} b x\right )}{40 \sqrt{2 \pi } b^5}+\frac{x^3 e^{-2 b^2 x^2}}{5 \pi b^2}+\frac{11 x e^{-2 b^2 x^2}}{20 \pi b^4}+\frac{1}{5} x^5 \text{Erf}(b x)^2 \]

[Out]

(11*x)/(20*b^4*E^(2*b^2*x^2)*Pi) + x^3/(5*b^2*E^(2*b^2*x^2)*Pi) + (4*Erf[b*x])/(5*b^5*E^(b^2*x^2)*Sqrt[Pi]) +
(4*x^2*Erf[b*x])/(5*b^3*E^(b^2*x^2)*Sqrt[Pi]) + (2*x^4*Erf[b*x])/(5*b*E^(b^2*x^2)*Sqrt[Pi]) + (x^5*Erf[b*x]^2)
/5 - (43*Erf[Sqrt[2]*b*x])/(40*b^5*Sqrt[2*Pi])

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Rubi [A]  time = 0.247006, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 10, 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} \[ \frac{2 x^4 e^{-b^2 x^2} \text{Erf}(b x)}{5 \sqrt{\pi } b}+\frac{4 x^2 e^{-b^2 x^2} \text{Erf}(b x)}{5 \sqrt{\pi } b^3}+\frac{4 e^{-b^2 x^2} \text{Erf}(b x)}{5 \sqrt{\pi } b^5}-\frac{43 \text{Erf}\left (\sqrt{2} b x\right )}{40 \sqrt{2 \pi } b^5}+\frac{x^3 e^{-2 b^2 x^2}}{5 \pi b^2}+\frac{11 x e^{-2 b^2 x^2}}{20 \pi b^4}+\frac{1}{5} x^5 \text{Erf}(b x)^2 \]

Antiderivative was successfully verified.

[In]

Int[x^4*Erf[b*x]^2,x]

[Out]

(11*x)/(20*b^4*E^(2*b^2*x^2)*Pi) + x^3/(5*b^2*E^(2*b^2*x^2)*Pi) + (4*Erf[b*x])/(5*b^5*E^(b^2*x^2)*Sqrt[Pi]) +
(4*x^2*Erf[b*x])/(5*b^3*E^(b^2*x^2)*Sqrt[Pi]) + (2*x^4*Erf[b*x])/(5*b*E^(b^2*x^2)*Sqrt[Pi]) + (x^5*Erf[b*x]^2)
/5 - (43*Erf[Sqrt[2]*b*x])/(40*b^5*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^4 \text{erf}(b x)^2 \, dx &=\frac{1}{5} x^5 \text{erf}(b x)^2-\frac{(4 b) \int e^{-b^2 x^2} x^5 \text{erf}(b x) \, dx}{5 \sqrt{\pi }}\\ &=\frac{2 e^{-b^2 x^2} x^4 \text{erf}(b x)}{5 b \sqrt{\pi }}+\frac{1}{5} x^5 \text{erf}(b x)^2-\frac{4 \int e^{-2 b^2 x^2} x^4 \, dx}{5 \pi }-\frac{8 \int e^{-b^2 x^2} x^3 \text{erf}(b x) \, dx}{5 b \sqrt{\pi }}\\ &=\frac{e^{-2 b^2 x^2} x^3}{5 b^2 \pi }+\frac{4 e^{-b^2 x^2} x^2 \text{erf}(b x)}{5 b^3 \sqrt{\pi }}+\frac{2 e^{-b^2 x^2} x^4 \text{erf}(b x)}{5 b \sqrt{\pi }}+\frac{1}{5} x^5 \text{erf}(b x)^2-\frac{3 \int e^{-2 b^2 x^2} x^2 \, dx}{5 b^2 \pi }-\frac{8 \int e^{-2 b^2 x^2} x^2 \, dx}{5 b^2 \pi }-\frac{8 \int e^{-b^2 x^2} x \text{erf}(b x) \, dx}{5 b^3 \sqrt{\pi }}\\ &=\frac{11 e^{-2 b^2 x^2} x}{20 b^4 \pi }+\frac{e^{-2 b^2 x^2} x^3}{5 b^2 \pi }+\frac{4 e^{-b^2 x^2} \text{erf}(b x)}{5 b^5 \sqrt{\pi }}+\frac{4 e^{-b^2 x^2} x^2 \text{erf}(b x)}{5 b^3 \sqrt{\pi }}+\frac{2 e^{-b^2 x^2} x^4 \text{erf}(b x)}{5 b \sqrt{\pi }}+\frac{1}{5} x^5 \text{erf}(b x)^2-\frac{3 \int e^{-2 b^2 x^2} \, dx}{20 b^4 \pi }-\frac{2 \int e^{-2 b^2 x^2} \, dx}{5 b^4 \pi }-\frac{8 \int e^{-2 b^2 x^2} \, dx}{5 b^4 \pi }\\ &=\frac{11 e^{-2 b^2 x^2} x}{20 b^4 \pi }+\frac{e^{-2 b^2 x^2} x^3}{5 b^2 \pi }+\frac{4 e^{-b^2 x^2} \text{erf}(b x)}{5 b^5 \sqrt{\pi }}+\frac{4 e^{-b^2 x^2} x^2 \text{erf}(b x)}{5 b^3 \sqrt{\pi }}+\frac{2 e^{-b^2 x^2} x^4 \text{erf}(b x)}{5 b \sqrt{\pi }}+\frac{1}{5} x^5 \text{erf}(b x)^2-\frac{2 \sqrt{\frac{2}{\pi }} \text{erf}\left (\sqrt{2} b x\right )}{5 b^5}-\frac{11 \text{erf}\left (\sqrt{2} b x\right )}{40 b^5 \sqrt{2 \pi }}\\ \end{align*}

Mathematica [A]  time = 0.0903972, size = 106, normalized size = 0.64 \[ \frac{16 \pi b^5 x^5 \text{Erf}(b x)^2+32 \sqrt{\pi } e^{-b^2 x^2} \left (b^4 x^4+2 b^2 x^2+2\right ) \text{Erf}(b x)+4 b x e^{-2 b^2 x^2} \left (4 b^2 x^2+11\right )-43 \sqrt{2 \pi } \text{Erf}\left (\sqrt{2} b x\right )}{80 \pi b^5} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*Erf[b*x]^2,x]

[Out]

((4*b*x*(11 + 4*b^2*x^2))/E^(2*b^2*x^2) + (32*Sqrt[Pi]*(2 + 2*b^2*x^2 + b^4*x^4)*Erf[b*x])/E^(b^2*x^2) + 16*b^
5*Pi*x^5*Erf[b*x]^2 - 43*Sqrt[2*Pi]*Erf[Sqrt[2]*b*x])/(80*b^5*Pi)

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Maple [A]  time = 0.052, size = 131, normalized size = 0.8 \begin{align*}{\frac{1}{{b}^{5}} \left ({\frac{ \left ({\it Erf} \left ( bx \right ) \right ) ^{2}{b}^{5}{x}^{5}}{5}}-{\frac{4\,{\it Erf} \left ( bx \right ) }{5\,\sqrt{\pi }} \left ( -{\frac{{b}^{4}{x}^{4}}{2\,{{\rm e}^{{b}^{2}{x}^{2}}}}}-{\frac{{b}^{2}{x}^{2}}{{{\rm e}^{{b}^{2}{x}^{2}}}}}- \left ({{\rm e}^{{b}^{2}{x}^{2}}} \right ) ^{-1} \right ) }+{\frac{4}{5\,\pi } \left ( -{\frac{43\,\sqrt{2}\sqrt{\pi }{\it Erf} \left ( bx\sqrt{2} \right ) }{64}}+{\frac{11\,bx}{16\, \left ({{\rm e}^{{b}^{2}{x}^{2}}} \right ) ^{2}}}+{\frac{{x}^{3}{b}^{3}}{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^4*erf(b*x)^2,x)

[Out]

1/b^5*(1/5*erf(b*x)^2*b^5*x^5-4/5*erf(b*x)/Pi^(1/2)*(-1/2*b^4*x^4/exp(b^2*x^2)-b^2*x^2/exp(b^2*x^2)-1/exp(b^2*
x^2))+4/5/Pi*(-43/64*2^(1/2)*Pi^(1/2)*erf(b*x*2^(1/2))+11/16/exp(b^2*x^2)^2*b*x+1/4/exp(b^2*x^2)^2*b^3*x^3))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{-\frac{1}{16} \, b^{4}{\left (\frac{4 \,{\left (4 \, b^{2} x^{3} + 3 \, x\right )} e^{\left (-2 \, b^{2} x^{2}\right )}}{b^{4}} - \frac{3 \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (\sqrt{2} b x\right )}{b^{5}}\right )} - \frac{1}{2} \, 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{2 \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (\sqrt{2} b x\right )}{b}}{5 \, \pi b^{4}} + \frac{\sqrt{\pi } b^{5} x^{5} \operatorname{erf}\left (b x\right )^{2} + 2 \,{\left (b^{4} x^{4} + 2 \, b^{2} x^{2} + 2\right )} \operatorname{erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{5 \, \sqrt{\pi } b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*erf(b*x)^2,x, algorithm="maxima")

[Out]

-1/5*integrate(4*(b^4*x^4 + 2*b^2*x^2 + 2)*e^(-2*b^2*x^2), x)/(pi*b^4) + 1/5*(sqrt(pi)*b^5*x^5*erf(b*x)^2 + 2*
(b^4*x^4 + 2*b^2*x^2 + 2)*erf(b*x)*e^(-b^2*x^2))/(sqrt(pi)*b^5)

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Fricas [A]  time = 2.54934, size = 277, normalized size = 1.68 \begin{align*} \frac{16 \, \pi b^{6} x^{5} \operatorname{erf}\left (b x\right )^{2} + 32 \, \sqrt{\pi }{\left (b^{5} x^{4} + 2 \, b^{3} x^{2} + 2 \, b\right )} \operatorname{erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} - 43 \, \sqrt{2} \sqrt{\pi } \sqrt{b^{2}} \operatorname{erf}\left (\sqrt{2} \sqrt{b^{2}} x\right ) + 4 \,{\left (4 \, b^{4} x^{3} + 11 \, b^{2} x\right )} e^{\left (-2 \, b^{2} x^{2}\right )}}{80 \, \pi b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*erf(b*x)^2,x, algorithm="fricas")

[Out]

1/80*(16*pi*b^6*x^5*erf(b*x)^2 + 32*sqrt(pi)*(b^5*x^4 + 2*b^3*x^2 + 2*b)*erf(b*x)*e^(-b^2*x^2) - 43*sqrt(2)*sq
rt(pi)*sqrt(b^2)*erf(sqrt(2)*sqrt(b^2)*x) + 4*(4*b^4*x^3 + 11*b^2*x)*e^(-2*b^2*x^2))/(pi*b^6)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{4} \operatorname{erf}^{2}{\left (b x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*erf(b*x)**2,x)

[Out]

Integral(x**4*erf(b*x)**2, x)

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Giac [A]  time = 1.39101, size = 235, normalized size = 1.42 \begin{align*} \frac{1}{5} \, x^{5} \operatorname{erf}\left (b x\right )^{2} + \frac{b{\left (\frac{32 \,{\left (b^{4} x^{4} + 2 \, b^{2} x^{2} + 2\right )} \operatorname{erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{b^{6}} + \frac{\sqrt{\pi } b^{4}{\left (\frac{4 \,{\left (4 \, b^{2} x^{3} + 3 \, x\right )} e^{\left (-2 \, b^{2} x^{2}\right )}}{b^{4}} + \frac{3 \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\sqrt{2} b x\right )}{b^{5}}\right )} + 8 \, \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{32 \, \sqrt{2} \pi \operatorname{erf}\left (-\sqrt{2} b x\right )}{b}}{\pi b^{5}}\right )}}{80 \, \sqrt{\pi }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*erf(b*x)^2,x, algorithm="giac")

[Out]

1/5*x^5*erf(b*x)^2 + 1/80*b*(32*(b^4*x^4 + 2*b^2*x^2 + 2)*erf(b*x)*e^(-b^2*x^2)/b^6 + (sqrt(pi)*b^4*(4*(4*b^2*
x^3 + 3*x)*e^(-2*b^2*x^2)/b^4 + 3*sqrt(2)*sqrt(pi)*erf(-sqrt(2)*b*x)/b^5) + 8*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) + 32*sqrt(2)*pi*erf(-sqrt(2)*b*x)/b)/(pi*b^5))/sqrt(pi)