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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.10145, 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 = {6386, 6383, 2205, 2212} \[ -\frac{5 \text{Erf}\left (\sqrt{2} b x\right )}{8 \sqrt{2} b^4}-\frac{x^2 e^{-b^2 x^2} \text{Erfc}(b x)}{2 b^2}-\frac{e^{-b^2 x^2} \text{Erfc}(b x)}{2 b^4}+\frac{x e^{-2 b^2 x^2}}{4 \sqrt{\pi } b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6386

Int[E^((c_.) + (d_.)*(x_)^2)*Erfc[(a_.) + (b_.)*(x_)]*(x_)^(m_), x_Symbol] :> Simp[(x^(m - 1)*E^(c + d*x^2)*Er
fc[a + b*x])/(2*d), x] + (-Dist[(m - 1)/(2*d), Int[x^(m - 2)*E^(c + d*x^2)*Erfc[a + b*x], x], x] + Dist[b/(d*S
qrt[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 6383

Int[E^((c_.) + (d_.)*(x_)^2)*Erfc[(a_.) + (b_.)*(x_)]*(x_), x_Symbol] :> Simp[(E^(c + d*x^2)*Erfc[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{erfc}(b x) \, dx &=-\frac{e^{-b^2 x^2} x^2 \text{erfc}(b x)}{2 b^2}+\frac{\int e^{-b^2 x^2} x \text{erfc}(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{erfc}(b x)}{2 b^4}-\frac{e^{-b^2 x^2} x^2 \text{erfc}(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{5 \text{erf}\left (\sqrt{2} b x\right )}{8 \sqrt{2} b^4}-\frac{e^{-b^2 x^2} \text{erfc}(b x)}{2 b^4}-\frac{e^{-b^2 x^2} x^2 \text{erfc}(b x)}{2 b^2}\\ \end{align*}

Mathematica [A]  time = 0.0924486, size = 69, normalized size = 0.77 \[ \frac{4 e^{-2 b^2 x^2} \left (\frac{b x}{\sqrt{\pi }}-2 e^{b^2 x^2} \left (b^2 x^2+1\right ) \text{Erfc}(b x)\right )-5 \sqrt{2} \text{Erf}\left (\sqrt{2} b x\right )}{16 b^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.245, size = 118, normalized size = 1.3 \begin{align*}{\frac{1}{b} \left ({\frac{1}{{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{{\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*erfc(b*x)/exp(b^2*x^2),x)

[Out]

(1/b^3*(-1/2*b^2*x^2/exp(b^2*x^2)-1/2/exp(b^2*x^2))-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*} \int x^{3} \operatorname{erfc}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^3*erfc(b*x)*e^(-b^2*x^2), x)

________________________________________________________________________________________

Fricas [A]  time = 2.28048, size = 225, normalized size = 2.5 \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 -{\left (\pi b^{3} x^{2} + \pi b\right )} \operatorname{erf}\left (b x\right )\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*erfc(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
- (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{erfc}{\left (b x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^3*erfc(b*x)*e^(-b^2*x^2), x)