3.105 \(\int x^3 \text{Erfc}(b x) \, dx\)

Optimal. Leaf size=71 \[ \frac{3 \text{Erf}(b x)}{16 b^4}-\frac{x^3 e^{-b^2 x^2}}{4 \sqrt{\pi } b}-\frac{3 x e^{-b^2 x^2}}{8 \sqrt{\pi } b^3}+\frac{1}{4} x^4 \text{Erfc}(b x) \]

[Out]

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

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Rubi [A]  time = 0.0635941, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {6362, 2212, 2205} \[ \frac{3 \text{Erf}(b x)}{16 b^4}-\frac{x^3 e^{-b^2 x^2}}{4 \sqrt{\pi } b}-\frac{3 x e^{-b^2 x^2}}{8 \sqrt{\pi } b^3}+\frac{1}{4} x^4 \text{Erfc}(b x) \]

Antiderivative was successfully verified.

[In]

Int[x^3*Erfc[b*x],x]

[Out]

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

Rule 6362

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

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])

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]

Rubi steps

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

Mathematica [A]  time = 0.0525482, size = 54, normalized size = 0.76 \[ \frac{1}{16} \left (\frac{3 \text{Erf}(b x)}{b^4}-\frac{2 x e^{-b^2 x^2} \left (2 b^2 x^2+3\right )}{\sqrt{\pi } b^3}+4 x^4 \text{Erfc}(b x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*Erfc[b*x],x]

[Out]

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

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Maple [A]  time = 0.042, size = 65, normalized size = 0.9 \begin{align*}{\frac{1}{{b}^{4}} \left ({\frac{{b}^{4}{x}^{4}{\it erfc} \left ( bx \right ) }{4}}+{\frac{1}{2\,\sqrt{\pi }} \left ( -{\frac{{x}^{3}{b}^{3}}{2\,{{\rm e}^{{b}^{2}{x}^{2}}}}}-{\frac{3\,bx}{4\,{{\rm e}^{{b}^{2}{x}^{2}}}}}+{\frac{3\,\sqrt{\pi }{\it Erf} \left ( bx \right ) }{8}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*erfc(b*x),x)

[Out]

1/b^4*(1/4*b^4*x^4*erfc(b*x)+1/2/Pi^(1/2)*(-1/2*b^3*x^3/exp(b^2*x^2)-3/4*b*x/exp(b^2*x^2)+3/8*Pi^(1/2)*erf(b*x
)))

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Maxima [A]  time = 1.00585, size = 74, normalized size = 1.04 \begin{align*} \frac{1}{4} \, x^{4} \operatorname{erfc}\left (b x\right ) - \frac{b{\left (\frac{2 \,{\left (2 \, b^{2} x^{3} + 3 \, x\right )} e^{\left (-b^{2} x^{2}\right )}}{b^{4}} - \frac{3 \, \sqrt{\pi } \operatorname{erf}\left (b x\right )}{b^{5}}\right )}}{16 \, \sqrt{\pi }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.38594, size = 147, normalized size = 2.07 \begin{align*} \frac{4 \, \pi b^{4} x^{4} - 2 \, \sqrt{\pi }{\left (2 \, b^{3} x^{3} + 3 \, 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 )}{16 \, \pi b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.28072, size = 68, normalized size = 0.96 \begin{align*} \begin{cases} \frac{x^{4} \operatorname{erfc}{\left (b x \right )}}{4} - \frac{x^{3} e^{- b^{2} x^{2}}}{4 \sqrt{\pi } b} - \frac{3 x e^{- b^{2} x^{2}}}{8 \sqrt{\pi } b^{3}} - \frac{3 \operatorname{erfc}{\left (b x \right )}}{16 b^{4}} & \text{for}\: b \neq 0 \\\frac{x^{4}}{4} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*erfc(b*x),x)

[Out]

Piecewise((x**4*erfc(b*x)/4 - x**3*exp(-b**2*x**2)/(4*sqrt(pi)*b) - 3*x*exp(-b**2*x**2)/(8*sqrt(pi)*b**3) - 3*
erfc(b*x)/(16*b**4), Ne(b, 0)), (x**4/4, True))

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Giac [A]  time = 1.33988, size = 82, normalized size = 1.15 \begin{align*} -\frac{1}{4} \, x^{4} \operatorname{erf}\left (b x\right ) + \frac{1}{4} \, x^{4} - \frac{b{\left (\frac{2 \,{\left (2 \, b^{2} x^{3} + 3 \, x\right )} e^{\left (-b^{2} x^{2}\right )}}{b^{4}} + \frac{3 \, \sqrt{\pi } \operatorname{erf}\left (-b x\right )}{b^{5}}\right )}}{16 \, \sqrt{\pi }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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