3.178 \(\int e^{-b^2 x^2} x^5 \text{Erfc}(b x) \, dx\)

Optimal. Leaf size=135 \[ -\frac{43 \text{Erf}\left (\sqrt{2} b x\right )}{32 \sqrt{2} b^6}-\frac{x^4 e^{-b^2 x^2} \text{Erfc}(b x)}{2 b^2}-\frac{x^2 e^{-b^2 x^2} \text{Erfc}(b x)}{b^4}-\frac{e^{-b^2 x^2} \text{Erfc}(b x)}{b^6}+\frac{x^3 e^{-2 b^2 x^2}}{4 \sqrt{\pi } b^3}+\frac{11 x e^{-2 b^2 x^2}}{16 \sqrt{\pi } b^5} \]

[Out]

(11*x)/(16*b^5*E^(2*b^2*x^2)*Sqrt[Pi]) + x^3/(4*b^3*E^(2*b^2*x^2)*Sqrt[Pi]) - (43*Erf[Sqrt[2]*b*x])/(32*Sqrt[2
]*b^6) - Erfc[b*x]/(b^6*E^(b^2*x^2)) - (x^2*Erfc[b*x])/(b^4*E^(b^2*x^2)) - (x^4*Erfc[b*x])/(2*b^2*E^(b^2*x^2))

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Rubi [A]  time = 0.194401, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {6386, 6383, 2205, 2212} \[ -\frac{43 \text{Erf}\left (\sqrt{2} b x\right )}{32 \sqrt{2} b^6}-\frac{x^4 e^{-b^2 x^2} \text{Erfc}(b x)}{2 b^2}-\frac{x^2 e^{-b^2 x^2} \text{Erfc}(b x)}{b^4}-\frac{e^{-b^2 x^2} \text{Erfc}(b x)}{b^6}+\frac{x^3 e^{-2 b^2 x^2}}{4 \sqrt{\pi } b^3}+\frac{11 x e^{-2 b^2 x^2}}{16 \sqrt{\pi } b^5} \]

Antiderivative was successfully verified.

[In]

Int[(x^5*Erfc[b*x])/E^(b^2*x^2),x]

[Out]

(11*x)/(16*b^5*E^(2*b^2*x^2)*Sqrt[Pi]) + x^3/(4*b^3*E^(2*b^2*x^2)*Sqrt[Pi]) - (43*Erf[Sqrt[2]*b*x])/(32*Sqrt[2
]*b^6) - Erfc[b*x]/(b^6*E^(b^2*x^2)) - (x^2*Erfc[b*x])/(b^4*E^(b^2*x^2)) - (x^4*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^5 \text{erfc}(b x) \, dx &=-\frac{e^{-b^2 x^2} x^4 \text{erfc}(b x)}{2 b^2}+\frac{2 \int e^{-b^2 x^2} x^3 \text{erfc}(b x) \, dx}{b^2}-\frac{\int e^{-2 b^2 x^2} x^4 \, dx}{b \sqrt{\pi }}\\ &=\frac{e^{-2 b^2 x^2} x^3}{4 b^3 \sqrt{\pi }}-\frac{e^{-b^2 x^2} x^2 \text{erfc}(b x)}{b^4}-\frac{e^{-b^2 x^2} x^4 \text{erfc}(b x)}{2 b^2}+\frac{2 \int e^{-b^2 x^2} x \text{erfc}(b x) \, dx}{b^4}-\frac{3 \int e^{-2 b^2 x^2} x^2 \, dx}{4 b^3 \sqrt{\pi }}-\frac{2 \int e^{-2 b^2 x^2} x^2 \, dx}{b^3 \sqrt{\pi }}\\ &=\frac{11 e^{-2 b^2 x^2} x}{16 b^5 \sqrt{\pi }}+\frac{e^{-2 b^2 x^2} x^3}{4 b^3 \sqrt{\pi }}-\frac{e^{-b^2 x^2} \text{erfc}(b x)}{b^6}-\frac{e^{-b^2 x^2} x^2 \text{erfc}(b x)}{b^4}-\frac{e^{-b^2 x^2} x^4 \text{erfc}(b x)}{2 b^2}-\frac{3 \int e^{-2 b^2 x^2} \, dx}{16 b^5 \sqrt{\pi }}-\frac{\int e^{-2 b^2 x^2} \, dx}{2 b^5 \sqrt{\pi }}-\frac{2 \int e^{-2 b^2 x^2} \, dx}{b^5 \sqrt{\pi }}\\ &=\frac{11 e^{-2 b^2 x^2} x}{16 b^5 \sqrt{\pi }}+\frac{e^{-2 b^2 x^2} x^3}{4 b^3 \sqrt{\pi }}-\frac{43 \text{erf}\left (\sqrt{2} b x\right )}{32 \sqrt{2} b^6}-\frac{e^{-b^2 x^2} \text{erfc}(b x)}{b^6}-\frac{e^{-b^2 x^2} x^2 \text{erfc}(b x)}{b^4}-\frac{e^{-b^2 x^2} x^4 \text{erfc}(b x)}{2 b^2}\\ \end{align*}

Mathematica [A]  time = 0.13191, size = 87, normalized size = 0.64 \[ \frac{4 e^{-2 b^2 x^2} \left (\frac{b x \left (4 b^2 x^2+11\right )}{\sqrt{\pi }}-8 e^{b^2 x^2} \left (b^4 x^4+2 b^2 x^2+2\right ) \text{Erfc}(b x)\right )-43 \sqrt{2} \text{Erf}\left (\sqrt{2} b x\right )}{64 b^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^5*Erfc[b*x])/E^(b^2*x^2),x]

[Out]

(-43*Sqrt[2]*Erf[Sqrt[2]*b*x] + (4*((b*x*(11 + 4*b^2*x^2))/Sqrt[Pi] - 8*E^(b^2*x^2)*(2 + 2*b^2*x^2 + b^4*x^4)*
Erfc[b*x]))/E^(2*b^2*x^2))/(64*b^6)

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Maple [A]  time = 0.127, size = 172, normalized size = 1.3 \begin{align*}{\frac{1}{b} \left ({\frac{1}{{b}^{5}} \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{{\it Erf} \left ( bx \right ) }{{b}^{5}} \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{1}{\sqrt{\pi }{b}^{5}} \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^5*erfc(b*x)/exp(b^2*x^2),x)

[Out]

(1/b^5*(-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))-erf(b*x)/b^5*(-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))+1/Pi^(1/2)/b^5*(-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))/b

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{5} \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^5*erfc(b*x)/exp(b^2*x^2),x, algorithm="maxima")

[Out]

integrate(x^5*erfc(b*x)*e^(-b^2*x^2), x)

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Fricas [A]  time = 2.11923, size = 298, normalized size = 2.21 \begin{align*} -\frac{43 \, \sqrt{2} \pi \sqrt{b^{2}} \operatorname{erf}\left (\sqrt{2} \sqrt{b^{2}} x\right ) - 4 \, \sqrt{\pi }{\left (4 \, b^{4} x^{3} + 11 \, b^{2} x\right )} e^{\left (-2 \, b^{2} x^{2}\right )} + 32 \,{\left (\pi b^{5} x^{4} + 2 \, \pi b^{3} x^{2} + 2 \, \pi b -{\left (\pi b^{5} x^{4} + 2 \, \pi b^{3} x^{2} + 2 \, \pi b\right )} \operatorname{erf}\left (b x\right )\right )} e^{\left (-b^{2} x^{2}\right )}}{64 \, \pi b^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*erfc(b*x)/exp(b^2*x^2),x, algorithm="fricas")

[Out]

-1/64*(43*sqrt(2)*pi*sqrt(b^2)*erf(sqrt(2)*sqrt(b^2)*x) - 4*sqrt(pi)*(4*b^4*x^3 + 11*b^2*x)*e^(-2*b^2*x^2) + 3
2*(pi*b^5*x^4 + 2*pi*b^3*x^2 + 2*pi*b - (pi*b^5*x^4 + 2*pi*b^3*x^2 + 2*pi*b)*erf(b*x))*e^(-b^2*x^2))/(pi*b^7)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*erfc(b*x)/exp(b**2*x**2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{5} \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^5*erfc(b*x)/exp(b^2*x^2),x, algorithm="giac")

[Out]

integrate(x^5*erfc(b*x)*e^(-b^2*x^2), x)