3.2.78 \(\int e^{-b^2 x^2} x^5 \text {Erfc}(b x) \, dx\) [178]

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

[Out]

-erfc(b*x)/b^6/exp(b^2*x^2)-x^2*erfc(b*x)/b^4/exp(b^2*x^2)-1/2*x^4*erfc(b*x)/b^2/exp(b^2*x^2)-43/64*erf(b*x*2^
(1/2))/b^6*2^(1/2)+11/16*x/b^5/exp(2*b^2*x^2)/Pi^(1/2)+1/4*x^3/b^3/exp(2*b^2*x^2)/Pi^(1/2)

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Rubi [A]
time = 0.13, antiderivative size = 135, normalized size of antiderivative = 1.00, 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 = {6521, 6518, 2236, 2243} \begin {gather*} -\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 {e^{-b^2 x^2} \text {Erfc}(b x)}{b^6}+\frac {11 x e^{-2 b^2 x^2}}{16 \sqrt {\pi } b^5}-\frac {x^2 e^{-b^2 x^2} \text {Erfc}(b x)}{b^4}+\frac {x^3 e^{-2 b^2 x^2}}{4 \sqrt {\pi } b^3} \end {gather*}

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 2236

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 2243

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 6518

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 6521

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
]

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*}

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Mathematica [A]
time = 0.09, size = 87, normalized size = 0.64 \begin {gather*} \frac {-43 \sqrt {2} \text {Erf}\left (\sqrt {2} b x\right )+4 e^{-2 b^2 x^2} \left (\frac {b x \left (11+4 b^2 x^2\right )}{\sqrt {\pi }}-8 e^{b^2 x^2} \left (2+2 b^2 x^2+b^4 x^4\right ) \text {Erfc}(b x)\right )}{64 b^6} \end {gather*}

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 = 0.25, size = 172, normalized size = 1.27

method	result	size
default	\(\frac {\frac {-\frac {b^{4} x^{4} {\mathrm e}^{-b^{2} x^{2}}}{2}-b^{2} x^{2} {\mathrm e}^{-b^{2} x^{2}}-{\mathrm e}^{-b^{2} x^{2}}}{b^{5}}-\frac {\erf \left (b x \right ) \left (-\frac {b^{4} x^{4} {\mathrm e}^{-b^{2} x^{2}}}{2}-b^{2} x^{2} {\mathrm e}^{-b^{2} x^{2}}-{\mathrm e}^{-b^{2} x^{2}}\right )}{b^{5}}+\frac {-\frac {43 \sqrt {2}\, \sqrt {\pi }\, \erf \left (b x \sqrt {2}\right )}{64}+\frac {{\mathrm e}^{-2 b^{2} x^{2}} b^{3} x^{3}}{4}+\frac {11 \,{\mathrm e}^{-2 b^{2} x^{2}} b x}{16}}{\sqrt {\pi }\, b^{5}}}{b}\)	\(172\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*erfc(b*x)/exp(b^2*x^2),x,method=_RETURNVERBOSE)

[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))+1/4/exp(b^2*x^2
)^2*b^3*x^3+11/16/exp(b^2*x^2)^2*b*x))/b

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.35, size = 121, normalized size = 0.90 \begin {gather*} -\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 {gather*}

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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{5} e^{- b^{2} x^{2}} \operatorname {erfc}{\left (b x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**5*exp(-b**2*x**2)*erfc(b*x), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^5\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erfc}\left (b\,x\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*exp(-b^2*x^2)*erfc(b*x),x)

[Out]

int(x^5*exp(-b^2*x^2)*erfc(b*x), x)

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