3.184 \(\int e^{-b^2 x^2} x^4 \text{Erfc}(b x) \, dx\)

Optimal. Leaf size=112 \[ -\frac{x^3 e^{-b^2 x^2} \text{Erfc}(b x)}{2 b^2}-\frac{3 x e^{-b^2 x^2} \text{Erfc}(b x)}{4 b^4}-\frac{3 \sqrt{\pi } \text{Erfc}(b x)^2}{16 b^5}+\frac{x^2 e^{-2 b^2 x^2}}{4 \sqrt{\pi } b^3}+\frac{e^{-2 b^2 x^2}}{2 \sqrt{\pi } b^5} \]

[Out]

1/(2*b^5*E^(2*b^2*x^2)*Sqrt[Pi]) + x^2/(4*b^3*E^(2*b^2*x^2)*Sqrt[Pi]) - (3*x*Erfc[b*x])/(4*b^4*E^(b^2*x^2)) -
(x^3*Erfc[b*x])/(2*b^2*E^(b^2*x^2)) - (3*Sqrt[Pi]*Erfc[b*x]^2)/(16*b^5)

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Rubi [A]  time = 0.166182, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {6386, 6374, 30, 2209, 2212} \[ -\frac{x^3 e^{-b^2 x^2} \text{Erfc}(b x)}{2 b^2}-\frac{3 x e^{-b^2 x^2} \text{Erfc}(b x)}{4 b^4}-\frac{3 \sqrt{\pi } \text{Erfc}(b x)^2}{16 b^5}+\frac{x^2 e^{-2 b^2 x^2}}{4 \sqrt{\pi } b^3}+\frac{e^{-2 b^2 x^2}}{2 \sqrt{\pi } b^5} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*Erfc[b*x])/E^(b^2*x^2),x]

[Out]

1/(2*b^5*E^(2*b^2*x^2)*Sqrt[Pi]) + x^2/(4*b^3*E^(2*b^2*x^2)*Sqrt[Pi]) - (3*x*Erfc[b*x])/(4*b^4*E^(b^2*x^2)) -
(x^3*Erfc[b*x])/(2*b^2*E^(b^2*x^2)) - (3*Sqrt[Pi]*Erfc[b*x]^2)/(16*b^5)

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 6374

Int[E^((c_.) + (d_.)*(x_)^2)*Erfc[(b_.)*(x_)]^(n_.), x_Symbol] :> -Dist[(E^c*Sqrt[Pi])/(2*b), Subst[Int[x^n, x
], x, Erfc[b*x]], x] /; FreeQ[{b, c, d, n}, x] && EqQ[d, -b^2]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

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^4 \text{erfc}(b x) \, dx &=-\frac{e^{-b^2 x^2} x^3 \text{erfc}(b x)}{2 b^2}+\frac{3 \int e^{-b^2 x^2} x^2 \text{erfc}(b x) \, dx}{2 b^2}-\frac{\int e^{-2 b^2 x^2} x^3 \, dx}{b \sqrt{\pi }}\\ &=\frac{e^{-2 b^2 x^2} x^2}{4 b^3 \sqrt{\pi }}-\frac{3 e^{-b^2 x^2} x \text{erfc}(b x)}{4 b^4}-\frac{e^{-b^2 x^2} x^3 \text{erfc}(b x)}{2 b^2}+\frac{3 \int e^{-b^2 x^2} \text{erfc}(b x) \, dx}{4 b^4}-\frac{\int e^{-2 b^2 x^2} x \, dx}{2 b^3 \sqrt{\pi }}-\frac{3 \int e^{-2 b^2 x^2} x \, dx}{2 b^3 \sqrt{\pi }}\\ &=\frac{e^{-2 b^2 x^2}}{2 b^5 \sqrt{\pi }}+\frac{e^{-2 b^2 x^2} x^2}{4 b^3 \sqrt{\pi }}-\frac{3 e^{-b^2 x^2} x \text{erfc}(b x)}{4 b^4}-\frac{e^{-b^2 x^2} x^3 \text{erfc}(b x)}{2 b^2}-\frac{\left (3 \sqrt{\pi }\right ) \operatorname{Subst}(\int x \, dx,x,\text{erfc}(b x))}{8 b^5}\\ &=\frac{e^{-2 b^2 x^2}}{2 b^5 \sqrt{\pi }}+\frac{e^{-2 b^2 x^2} x^2}{4 b^3 \sqrt{\pi }}-\frac{3 e^{-b^2 x^2} x \text{erfc}(b x)}{4 b^4}-\frac{e^{-b^2 x^2} x^3 \text{erfc}(b x)}{2 b^2}-\frac{3 \sqrt{\pi } \text{erfc}(b x)^2}{16 b^5}\\ \end{align*}

Mathematica [A]  time = 0.136288, size = 112, normalized size = 1. \[ -\frac{-4 \sqrt{\pi } b x e^{-b^2 x^2} \left (2 b^2 x^2+3\right ) \text{Erf}(b x)-4 e^{-2 b^2 x^2} \left (b^2 x^2+2\right )+4 \sqrt{\pi } b x e^{-b^2 x^2} \left (2 b^2 x^2+3\right )+3 \pi \text{Erf}(b x)^2-6 \pi \text{Erf}(b x)}{16 \sqrt{\pi } b^5} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(x^4*Erfc[b*x])/E^(b^2*x^2),x]

[Out]

-((-4*(2 + b^2*x^2))/E^(2*b^2*x^2) + (4*b*Sqrt[Pi]*x*(3 + 2*b^2*x^2))/E^(b^2*x^2) - 6*Pi*Erf[b*x] - (4*b*Sqrt[
Pi]*x*(3 + 2*b^2*x^2)*Erf[b*x])/E^(b^2*x^2) + 3*Pi*Erf[b*x]^2)/(16*b^5*Sqrt[Pi])

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Maple [F]  time = 0.253, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{4}{\it erfc} \left ( bx \right ) }{{{\rm e}^{{b}^{2}{x}^{2}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*erfc(b*x)/exp(b^2*x^2),x)

[Out]

int(x^4*erfc(b*x)/exp(b^2*x^2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{4} \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^4*erfc(b*x)/exp(b^2*x^2),x, algorithm="maxima")

[Out]

integrate(x^4*erfc(b*x)*e^(-b^2*x^2), x)

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Fricas [A]  time = 2.08305, size = 235, normalized size = 2.1 \begin{align*} -\frac{4 \,{\left (2 \, \pi b^{3} x^{3} + 3 \, \pi b x -{\left (2 \, \pi b^{3} x^{3} + 3 \, \pi b x\right )} \operatorname{erf}\left (b x\right )\right )} e^{\left (-b^{2} x^{2}\right )} + \sqrt{\pi }{\left (3 \, \pi \operatorname{erf}\left (b x\right )^{2} - 6 \, \pi \operatorname{erf}\left (b x\right ) - 4 \,{\left (b^{2} x^{2} + 2\right )} e^{\left (-2 \, b^{2} x^{2}\right )}\right )}}{16 \, \pi b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*erfc(b*x)/exp(b^2*x^2),x, algorithm="fricas")

[Out]

-1/16*(4*(2*pi*b^3*x^3 + 3*pi*b*x - (2*pi*b^3*x^3 + 3*pi*b*x)*erf(b*x))*e^(-b^2*x^2) + sqrt(pi)*(3*pi*erf(b*x)
^2 - 6*pi*erf(b*x) - 4*(b^2*x^2 + 2)*e^(-2*b^2*x^2)))/(pi*b^5)

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Sympy [A]  time = 116.695, size = 151, normalized size = 1.35 \begin{align*} \begin{cases} - \frac{x^{3} e^{- b^{2} x^{2}} \operatorname{erfc}{\left (b x \right )}}{2 b^{2}} + \frac{x^{2} e^{- 2 b^{2} x^{2}}}{4 \sqrt{\pi } b^{3}} - \frac{3 x e^{- b^{2} x^{2}} \operatorname{erfc}{\left (b x \right )}}{4 b^{4}} + \frac{3 \sqrt{\pi } \operatorname{erf}^{2}{\left (x \sqrt{b^{2}} \right )}}{16 b^{5}} + \frac{e^{- 2 b^{2} x^{2}}}{2 \sqrt{\pi } b^{5}} + \frac{3 \sqrt{\pi } \sqrt{b^{2}} \operatorname{erf}{\left (x \sqrt{b^{2}} \right )} \operatorname{erfc}{\left (b x \right )}}{8 b^{6}} & \text{for}\: b \neq 0 \\\frac{x^{5}}{5} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*erfc(b*x)/exp(b**2*x**2),x)

[Out]

Piecewise((-x**3*exp(-b**2*x**2)*erfc(b*x)/(2*b**2) + x**2*exp(-2*b**2*x**2)/(4*sqrt(pi)*b**3) - 3*x*exp(-b**2
*x**2)*erfc(b*x)/(4*b**4) + 3*sqrt(pi)*erf(x*sqrt(b**2))**2/(16*b**5) + exp(-2*b**2*x**2)/(2*sqrt(pi)*b**5) +
3*sqrt(pi)*sqrt(b**2)*erf(x*sqrt(b**2))*erfc(b*x)/(8*b**6), Ne(b, 0)), (x**5/5, True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{4} \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^4*erfc(b*x)/exp(b^2*x^2),x, algorithm="giac")

[Out]

integrate(x^4*erfc(b*x)*e^(-b^2*x^2), x)