3.76 \(\int e^{-b^2 x^2} x^3 \text{Erf}(b x) \, dx\)

Optimal. Leaf size=90 \[ -\frac{x^2 e^{-b^2 x^2} \text{Erf}(b x)}{2 b^2}-\frac{e^{-b^2 x^2} \text{Erf}(b x)}{2 b^4}+\frac{5 \text{Erf}\left (\sqrt{2} b x\right )}{8 \sqrt{2} b^4}-\frac{x e^{-2 b^2 x^2}}{4 \sqrt{\pi } b^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.101306, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {6385, 6382, 2205, 2212} \[ -\frac{x^2 e^{-b^2 x^2} \text{Erf}(b x)}{2 b^2}-\frac{e^{-b^2 x^2} \text{Erf}(b x)}{2 b^4}+\frac{5 \text{Erf}\left (\sqrt{2} b x\right )}{8 \sqrt{2} b^4}-\frac{x e^{-2 b^2 x^2}}{4 \sqrt{\pi } b^3} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*Erf[b*x])/E^(b^2*x^2),x]

[Out]

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

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 e^{-b^2 x^2} x^3 \text{erf}(b x) \, dx &=-\frac{e^{-b^2 x^2} x^2 \text{erf}(b x)}{2 b^2}+\frac{\int e^{-b^2 x^2} x \text{erf}(b x) \, dx}{b^2}+\frac{\int e^{-2 b^2 x^2} x^2 \, dx}{b \sqrt{\pi }}\\ &=-\frac{e^{-2 b^2 x^2} x}{4 b^3 \sqrt{\pi }}-\frac{e^{-b^2 x^2} \text{erf}(b x)}{2 b^4}-\frac{e^{-b^2 x^2} x^2 \text{erf}(b x)}{2 b^2}+\frac{\int e^{-2 b^2 x^2} \, dx}{4 b^3 \sqrt{\pi }}+\frac{\int e^{-2 b^2 x^2} \, dx}{b^3 \sqrt{\pi }}\\ &=-\frac{e^{-2 b^2 x^2} x}{4 b^3 \sqrt{\pi }}-\frac{e^{-b^2 x^2} \text{erf}(b x)}{2 b^4}-\frac{e^{-b^2 x^2} x^2 \text{erf}(b x)}{2 b^2}+\frac{5 \text{erf}\left (\sqrt{2} b x\right )}{8 \sqrt{2} b^4}\\ \end{align*}

Mathematica [A]  time = 0.0467127, size = 68, normalized size = 0.76 \[ \frac{-8 e^{-b^2 x^2} \left (b^2 x^2+1\right ) \text{Erf}(b x)-\frac{4 b x e^{-2 b^2 x^2}}{\sqrt{\pi }}+5 \sqrt{2} \text{Erf}\left (\sqrt{2} b x\right )}{16 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^3*Erf[b*x])/E^(b^2*x^2),x]

[Out]

((-4*b*x)/(E^(2*b^2*x^2)*Sqrt[Pi]) - (8*(1 + b^2*x^2)*Erf[b*x])/E^(b^2*x^2) + 5*Sqrt[2]*Erf[Sqrt[2]*b*x])/(16*
b^4)

________________________________________________________________________________________

Maple [A]  time = 0.247, size = 83, normalized size = 0.9 \begin{align*}{\frac{1}{b} \left ({\frac{{\it Erf} \left ( bx \right ) }{{b}^{3}} \left ( -{\frac{{b}^{2}{x}^{2}}{2\,{{\rm e}^{{b}^{2}{x}^{2}}}}}-{\frac{1}{2\,{{\rm e}^{{b}^{2}{x}^{2}}}}} \right ) }-{\frac{1}{{b}^{3}\sqrt{\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^3*erf(b*x)/exp(b^2*x^2),x)

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 3.04953, size = 197, normalized size = 2.19 \begin{align*} -\frac{4 \, \sqrt{\pi } b^{2} x e^{\left (-2 \, b^{2} x^{2}\right )} - 5 \, \sqrt{2} \pi \sqrt{b^{2}} \operatorname{erf}\left (\sqrt{2} \sqrt{b^{2}} x\right ) + 8 \,{\left (\pi b^{3} x^{2} + \pi b\right )} \operatorname{erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{16 \, \pi b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(4*sqrt(pi)*b^2*x*e^(-2*b^2*x^2) - 5*sqrt(2)*pi*sqrt(b^2)*erf(sqrt(2)*sqrt(b^2)*x) + 8*(pi*b^3*x^2 + pi*
b)*erf(b*x)*e^(-b^2*x^2))/(pi*b^5)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**3*exp(-b**2*x**2)*erf(b*x), x)

________________________________________________________________________________________

Giac [A]  time = 1.2589, size = 128, normalized size = 1.42 \begin{align*} -\frac{{\left (b^{2} x^{2} + 1\right )} \operatorname{erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{2 \, 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}}{16 \, \pi b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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