\(\int x^5 \text {erfc}(b x) \, dx\) [104]

Optimal result

Integrand size = 8, antiderivative size = 96 \[ \int x^5 \text {erfc}(b x) \, dx=-\frac {5 e^{-b^2 x^2} x}{8 b^5 \sqrt {\pi }}-\frac {5 e^{-b^2 x^2} x^3}{12 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} x^5}{6 b \sqrt {\pi }}+\frac {5 \text {erf}(b x)}{16 b^6}+\frac {1}{6} x^6 \text {erfc}(b x) \]

[Out]

5/16*erf(b*x)/b^6+1/6*x^6*erfc(b*x)-5/8*x/b^5/exp(b^2*x^2)/Pi^(1/2)-5/12*x^3/b^3/exp(b^2*x^2)/Pi^(1/2)-1/6*x^5
/b/exp(b^2*x^2)/Pi^(1/2)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6497, 2243, 2236} \[ \int x^5 \text {erfc}(b x) \, dx=\frac {5 \text {erf}(b x)}{16 b^6}-\frac {x^5 e^{-b^2 x^2}}{6 \sqrt {\pi } b}-\frac {5 x e^{-b^2 x^2}}{8 \sqrt {\pi } b^5}-\frac {5 x^3 e^{-b^2 x^2}}{12 \sqrt {\pi } b^3}+\frac {1}{6} x^6 \text {erfc}(b x) \]

[In]

Int[x^5*Erfc[b*x],x]

[Out]

(-5*x)/(8*b^5*E^(b^2*x^2)*Sqrt[Pi]) - (5*x^3)/(12*b^3*E^(b^2*x^2)*Sqrt[Pi]) - x^5/(6*b*E^(b^2*x^2)*Sqrt[Pi]) +
(5*Erf[b*x])/(16*b^6) + (x^6*Erfc[b*x])/6

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 6497

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]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} x^6 \text {erfc}(b x)+\frac {b \int e^{-b^2 x^2} x^6 \, dx}{3 \sqrt {\pi }} \\ & = -\frac {e^{-b^2 x^2} x^5}{6 b \sqrt {\pi }}+\frac {1}{6} x^6 \text {erfc}(b x)+\frac {5 \int e^{-b^2 x^2} x^4 \, dx}{6 b \sqrt {\pi }} \\ & = -\frac {5 e^{-b^2 x^2} x^3}{12 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} x^5}{6 b \sqrt {\pi }}+\frac {1}{6} x^6 \text {erfc}(b x)+\frac {5 \int e^{-b^2 x^2} x^2 \, dx}{4 b^3 \sqrt {\pi }} \\ & = -\frac {5 e^{-b^2 x^2} x}{8 b^5 \sqrt {\pi }}-\frac {5 e^{-b^2 x^2} x^3}{12 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} x^5}{6 b \sqrt {\pi }}+\frac {1}{6} x^6 \text {erfc}(b x)+\frac {5 \int e^{-b^2 x^2} \, dx}{8 b^5 \sqrt {\pi }} \\ & = -\frac {5 e^{-b^2 x^2} x}{8 b^5 \sqrt {\pi }}-\frac {5 e^{-b^2 x^2} x^3}{12 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} x^5}{6 b \sqrt {\pi }}+\frac {5 \text {erf}(b x)}{16 b^6}+\frac {1}{6} x^6 \text {erfc}(b x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.65 \[ \int x^5 \text {erfc}(b x) \, dx=\frac {1}{48} \left (-\frac {2 e^{-b^2 x^2} x \left (15+10 b^2 x^2+4 b^4 x^4\right )}{b^5 \sqrt {\pi }}+\frac {15 \text {erf}(b x)}{b^6}+8 x^6 \text {erfc}(b x)\right ) \]

[In]

Integrate[x^5*Erfc[b*x],x]

[Out]

((-2*x*(15 + 10*b^2*x^2 + 4*b^4*x^4))/(b^5*E^(b^2*x^2)*Sqrt[Pi]) + (15*Erf[b*x])/b^6 + 8*x^6*Erfc[b*x])/48

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.84

method	result	size
parallelrisch	\(\frac {8 \,\operatorname {erfc}\left (b x \right ) x^{6} b^{6} \sqrt {\pi }-8 \,{\mathrm e}^{-b^{2} x^{2}} x^{5} b^{5}-20 x^{3} {\mathrm e}^{-b^{2} x^{2}} b^{3}-30 \,{\mathrm e}^{-b^{2} x^{2}} b x -15 \,\operatorname {erfc}\left (b x \right ) \sqrt {\pi }}{48 b^{6} \sqrt {\pi }}\)	\(81\)
derivativedivides	\(\frac {\frac {b^{6} x^{6} \operatorname {erfc}\left (b x \right )}{6}+\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}} x^{5} b^{5}}{2}-\frac {5 x^{3} {\mathrm e}^{-b^{2} x^{2}} b^{3}}{4}-\frac {15 \,{\mathrm e}^{-b^{2} x^{2}} b x}{8}+\frac {15 \,\operatorname {erf}\left (b x \right ) \sqrt {\pi }}{16}}{3 \sqrt {\pi }}}{b^{6}}\)	\(83\)
default	\(\frac {\frac {b^{6} x^{6} \operatorname {erfc}\left (b x \right )}{6}+\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}} x^{5} b^{5}}{2}-\frac {5 x^{3} {\mathrm e}^{-b^{2} x^{2}} b^{3}}{4}-\frac {15 \,{\mathrm e}^{-b^{2} x^{2}} b x}{8}+\frac {15 \,\operatorname {erf}\left (b x \right ) \sqrt {\pi }}{16}}{3 \sqrt {\pi }}}{b^{6}}\)	\(83\)
parts	\(\frac {x^{6} \operatorname {erfc}\left (b x \right )}{6}+\frac {b \left (-\frac {x^{5} {\mathrm e}^{-b^{2} x^{2}}}{2 b^{2}}+\frac {-\frac {5 x^{3} {\mathrm e}^{-b^{2} x^{2}}}{4 b^{2}}+\frac {5 \left (-\frac {3 x \,{\mathrm e}^{-b^{2} x^{2}}}{4 b^{2}}+\frac {3 \sqrt {\pi }\, \operatorname {erf}\left (b x \right )}{8 b^{3}}\right )}{2 b^{2}}}{b^{2}}\right )}{3 \sqrt {\pi }}\)	\(91\)

[In]

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

[Out]

1/48*(8*erfc(b*x)*x^6*b^6*Pi^(1/2)-8*exp(-b^2*x^2)*x^5*b^5-20*x^3*exp(-b^2*x^2)*b^3-30*exp(-b^2*x^2)*b*x-15*er
fc(b*x)*Pi^(1/2))/b^6/Pi^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.74 \[ \int x^5 \text {erfc}(b x) \, dx=\frac {8 \, \pi b^{6} x^{6} - 2 \, \sqrt {\pi } {\left (4 \, b^{5} x^{5} + 10 \, b^{3} x^{3} + 15 \, b x\right )} e^{\left (-b^{2} x^{2}\right )} + {\left (15 \, \pi - 8 \, \pi b^{6} x^{6}\right )} \operatorname {erf}\left (b x\right )}{48 \, \pi b^{6}} \]

[In]

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

[Out]

1/48*(8*pi*b^6*x^6 - 2*sqrt(pi)*(4*b^5*x^5 + 10*b^3*x^3 + 15*b*x)*e^(-b^2*x^2) + (15*pi - 8*pi*b^6*x^6)*erf(b*
x))/(pi*b^6)

Sympy [A] (verification not implemented)

Time = 0.40 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.96 \[ \int x^5 \text {erfc}(b x) \, dx=\begin {cases} \frac {x^{6} \operatorname {erfc}{\left (b x \right )}}{6} - \frac {x^{5} e^{- b^{2} x^{2}}}{6 \sqrt {\pi } b} - \frac {5 x^{3} e^{- b^{2} x^{2}}}{12 \sqrt {\pi } b^{3}} - \frac {5 x e^{- b^{2} x^{2}}}{8 \sqrt {\pi } b^{5}} - \frac {5 \operatorname {erfc}{\left (b x \right )}}{16 b^{6}} & \text {for}\: b \neq 0 \\\frac {x^{6}}{6} & \text {otherwise} \end {cases} \]

[In]

integrate(x**5*erfc(b*x),x)

[Out]

Piecewise((x**6*erfc(b*x)/6 - x**5*exp(-b**2*x**2)/(6*sqrt(pi)*b) - 5*x**3*exp(-b**2*x**2)/(12*sqrt(pi)*b**3)
- 5*x*exp(-b**2*x**2)/(8*sqrt(pi)*b**5) - 5*erfc(b*x)/(16*b**6), Ne(b, 0)), (x**6/6, True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.66 \[ \int x^5 \text {erfc}(b x) \, dx=\frac {1}{6} \, x^{6} \operatorname {erfc}\left (b x\right ) - \frac {b {\left (\frac {2 \, {\left (4 \, b^{4} x^{5} + 10 \, b^{2} x^{3} + 15 \, x\right )} e^{\left (-b^{2} x^{2}\right )}}{b^{6}} - \frac {15 \, \sqrt {\pi } \operatorname {erf}\left (b x\right )}{b^{7}}\right )}}{48 \, \sqrt {\pi }} \]

[In]

integrate(x^5*erfc(b*x),x, algorithm="maxima")

[Out]

1/6*x^6*erfc(b*x) - 1/48*b*(2*(4*b^4*x^5 + 10*b^2*x^3 + 15*x)*e^(-b^2*x^2)/b^6 - 15*sqrt(pi)*erf(b*x)/b^7)/sqr
t(pi)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.72 \[ \int x^5 \text {erfc}(b x) \, dx=-\frac {1}{6} \, x^{6} \operatorname {erf}\left (b x\right ) + \frac {1}{6} \, x^{6} - \frac {b {\left (\frac {2 \, {\left (4 \, b^{4} x^{5} + 10 \, b^{2} x^{3} + 15 \, x\right )} e^{\left (-b^{2} x^{2}\right )}}{b^{6}} + \frac {15 \, \sqrt {\pi } \operatorname {erf}\left (-b x\right )}{b^{7}}\right )}}{48 \, \sqrt {\pi }} \]

[In]

integrate(x^5*erfc(b*x),x, algorithm="giac")

[Out]

-1/6*x^6*erf(b*x) + 1/6*x^6 - 1/48*b*(2*(4*b^4*x^5 + 10*b^2*x^3 + 15*x)*e^(-b^2*x^2)/b^6 + 15*sqrt(pi)*erf(-b*
x)/b^7)/sqrt(pi)

Mupad [B] (verification not implemented)

Time = 4.95 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.81 \[ \int x^5 \text {erfc}(b x) \, dx=\frac {x^6\,\mathrm {erfc}\left (b\,x\right )}{6}-\frac {\frac {5\,\mathrm {erfc}\left (b\,x\right )}{16}+\frac {5\,b^3\,x^3\,{\mathrm {e}}^{-b^2\,x^2}}{12\,\sqrt {\pi }}+\frac {b^5\,x^5\,{\mathrm {e}}^{-b^2\,x^2}}{6\,\sqrt {\pi }}+\frac {5\,b\,x\,{\mathrm {e}}^{-b^2\,x^2}}{8\,\sqrt {\pi }}}{b^6} \]

[In]

int(x^5*erfc(b*x),x)

[Out]

(x^6*erfc(b*x))/6 - ((5*erfc(b*x))/16 + (5*b^3*x^3*exp(-b^2*x^2))/(12*pi^(1/2)) + (b^5*x^5*exp(-b^2*x^2))/(6*p
i^(1/2)) + (5*b*x*exp(-b^2*x^2))/(8*pi^(1/2)))/b^6