\(\int x^5 \text {erfc}(b x)^2 \, dx\) [125]

Optimal result

Integrand size = 10, antiderivative size = 178 \[ \int x^5 \text {erfc}(b x)^2 \, dx=\frac {11 e^{-2 b^2 x^2}}{12 b^6 \pi }+\frac {7 e^{-2 b^2 x^2} x^2}{12 b^4 \pi }+\frac {e^{-2 b^2 x^2} x^4}{6 b^2 \pi }-\frac {5 e^{-b^2 x^2} x \text {erfc}(b x)}{4 b^5 \sqrt {\pi }}-\frac {5 e^{-b^2 x^2} x^3 \text {erfc}(b x)}{6 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} x^5 \text {erfc}(b x)}{3 b \sqrt {\pi }}-\frac {5 \text {erfc}(b x)^2}{16 b^6}+\frac {1}{6} x^6 \text {erfc}(b x)^2 \]

[Out]

11/12/b^6/exp(2*b^2*x^2)/Pi+7/12*x^2/b^4/exp(2*b^2*x^2)/Pi+1/6*x^4/b^2/exp(2*b^2*x^2)/Pi-5/16*erfc(b*x)^2/b^6+
1/6*x^6*erfc(b*x)^2-5/4*x*erfc(b*x)/b^5/exp(b^2*x^2)/Pi^(1/2)-5/6*x^3*erfc(b*x)/b^3/exp(b^2*x^2)/Pi^(1/2)-1/3*
x^5*erfc(b*x)/b/exp(b^2*x^2)/Pi^(1/2)

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6500, 6521, 6509, 30, 2240, 2243} \[ \int x^5 \text {erfc}(b x)^2 \, dx=-\frac {5 \text {erfc}(b x)^2}{16 b^6}-\frac {x^5 e^{-b^2 x^2} \text {erfc}(b x)}{3 \sqrt {\pi } b}+\frac {x^4 e^{-2 b^2 x^2}}{6 \pi b^2}+\frac {11 e^{-2 b^2 x^2}}{12 \pi b^6}-\frac {5 x e^{-b^2 x^2} \text {erfc}(b x)}{4 \sqrt {\pi } b^5}+\frac {7 x^2 e^{-2 b^2 x^2}}{12 \pi b^4}-\frac {5 x^3 e^{-b^2 x^2} \text {erfc}(b x)}{6 \sqrt {\pi } b^3}+\frac {1}{6} x^6 \text {erfc}(b x)^2 \]

[In]

Int[x^5*Erfc[b*x]^2,x]

[Out]

11/(12*b^6*E^(2*b^2*x^2)*Pi) + (7*x^2)/(12*b^4*E^(2*b^2*x^2)*Pi) + x^4/(6*b^2*E^(2*b^2*x^2)*Pi) - (5*x*Erfc[b*
x])/(4*b^5*E^(b^2*x^2)*Sqrt[Pi]) - (5*x^3*Erfc[b*x])/(6*b^3*E^(b^2*x^2)*Sqrt[Pi]) - (x^5*Erfc[b*x])/(3*b*E^(b^
2*x^2)*Sqrt[Pi]) - (5*Erfc[b*x]^2)/(16*b^6) + (x^6*Erfc[b*x]^2)/6

Rule 30

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

Rule 2240

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 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 6500

Int[Erfc[(b_.)*(x_)]^2*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*(Erfc[b*x]^2/(m + 1)), x] + Dist[4*(b/(Sqrt[Pi]
*(m + 1))), Int[(x^(m + 1)*Erfc[b*x])/E^(b^2*x^2), x], x] /; FreeQ[b, x] && (IGtQ[m, 0] || ILtQ[(m + 1)/2, 0])

Rule 6509

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 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*} \text {integral}& = \frac {1}{6} x^6 \text {erfc}(b x)^2+\frac {(2 b) \int e^{-b^2 x^2} x^6 \text {erfc}(b x) \, dx}{3 \sqrt {\pi }} \\ & = -\frac {e^{-b^2 x^2} x^5 \text {erfc}(b x)}{3 b \sqrt {\pi }}+\frac {1}{6} x^6 \text {erfc}(b x)^2-\frac {2 \int e^{-2 b^2 x^2} x^5 \, dx}{3 \pi }+\frac {5 \int e^{-b^2 x^2} x^4 \text {erfc}(b x) \, dx}{3 b \sqrt {\pi }} \\ & = \frac {e^{-2 b^2 x^2} x^4}{6 b^2 \pi }-\frac {5 e^{-b^2 x^2} x^3 \text {erfc}(b x)}{6 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} x^5 \text {erfc}(b x)}{3 b \sqrt {\pi }}+\frac {1}{6} x^6 \text {erfc}(b x)^2-\frac {2 \int e^{-2 b^2 x^2} x^3 \, dx}{3 b^2 \pi }-\frac {5 \int e^{-2 b^2 x^2} x^3 \, dx}{3 b^2 \pi }+\frac {5 \int e^{-b^2 x^2} x^2 \text {erfc}(b x) \, dx}{2 b^3 \sqrt {\pi }} \\ & = \frac {7 e^{-2 b^2 x^2} x^2}{12 b^4 \pi }+\frac {e^{-2 b^2 x^2} x^4}{6 b^2 \pi }-\frac {5 e^{-b^2 x^2} x \text {erfc}(b x)}{4 b^5 \sqrt {\pi }}-\frac {5 e^{-b^2 x^2} x^3 \text {erfc}(b x)}{6 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} x^5 \text {erfc}(b x)}{3 b \sqrt {\pi }}+\frac {1}{6} x^6 \text {erfc}(b x)^2-\frac {\int e^{-2 b^2 x^2} x \, dx}{3 b^4 \pi }-\frac {5 \int e^{-2 b^2 x^2} x \, dx}{6 b^4 \pi }-\frac {5 \int e^{-2 b^2 x^2} x \, dx}{2 b^4 \pi }+\frac {5 \int e^{-b^2 x^2} \text {erfc}(b x) \, dx}{4 b^5 \sqrt {\pi }} \\ & = \frac {11 e^{-2 b^2 x^2}}{12 b^6 \pi }+\frac {7 e^{-2 b^2 x^2} x^2}{12 b^4 \pi }+\frac {e^{-2 b^2 x^2} x^4}{6 b^2 \pi }-\frac {5 e^{-b^2 x^2} x \text {erfc}(b x)}{4 b^5 \sqrt {\pi }}-\frac {5 e^{-b^2 x^2} x^3 \text {erfc}(b x)}{6 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} x^5 \text {erfc}(b x)}{3 b \sqrt {\pi }}+\frac {1}{6} x^6 \text {erfc}(b x)^2-\frac {5 \text {Subst}(\int x \, dx,x,\text {erfc}(b x))}{8 b^6} \\ & = \frac {11 e^{-2 b^2 x^2}}{12 b^6 \pi }+\frac {7 e^{-2 b^2 x^2} x^2}{12 b^4 \pi }+\frac {e^{-2 b^2 x^2} x^4}{6 b^2 \pi }-\frac {5 e^{-b^2 x^2} x \text {erfc}(b x)}{4 b^5 \sqrt {\pi }}-\frac {5 e^{-b^2 x^2} x^3 \text {erfc}(b x)}{6 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} x^5 \text {erfc}(b x)}{3 b \sqrt {\pi }}-\frac {5 \text {erfc}(b x)^2}{16 b^6}+\frac {1}{6} x^6 \text {erfc}(b x)^2 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.87 \[ \int x^5 \text {erfc}(b x)^2 \, dx=\frac {1}{48} \left (8 x^6+8 x^6 \text {erf}(b x)^2+\frac {e^{-2 b^2 x^2} \left (44+28 b^2 x^2+8 b^4 x^4+4 b e^{b^2 x^2} \sqrt {\pi } x \left (15+10 b^2 x^2+4 b^4 x^4\right ) \text {erf}(b x)-15 e^{2 b^2 x^2} \pi \text {erf}(b x)^2\right )}{b^6 \pi }-16 x^6 \left (\text {erf}(b x)+\frac {b x \Gamma \left (\frac {7}{2},b^2 x^2\right )}{\sqrt {\pi } \left (b^2 x^2\right )^{7/2}}\right )\right ) \]

[In]

Integrate[x^5*Erfc[b*x]^2,x]

[Out]

(8*x^6 + 8*x^6*Erf[b*x]^2 + (44 + 28*b^2*x^2 + 8*b^4*x^4 + 4*b*E^(b^2*x^2)*Sqrt[Pi]*x*(15 + 10*b^2*x^2 + 4*b^4
*x^4)*Erf[b*x] - 15*E^(2*b^2*x^2)*Pi*Erf[b*x]^2)/(b^6*E^(2*b^2*x^2)*Pi) - 16*x^6*(Erf[b*x] + (b*x*Gamma[7/2, b
^2*x^2])/(Sqrt[Pi]*(b^2*x^2)^(7/2))))/48

Maple [A] (verified)

Time = 0.58 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.90

method	result	size
parallelrisch	\(\frac {8 \operatorname {erfc}\left (b x \right )^{2} x^{6} b^{6} \pi ^{\frac {3}{2}}-16 \,{\mathrm e}^{-b^{2} x^{2}} \operatorname {erfc}\left (b x \right ) x^{5} b^{5} \pi +8 \,{\mathrm e}^{-2 b^{2} x^{2}} x^{4} b^{4} \sqrt {\pi }-40 \,{\mathrm e}^{-b^{2} x^{2}} \operatorname {erfc}\left (b x \right ) x^{3} b^{3} \pi +28 x^{2} {\mathrm e}^{-2 b^{2} x^{2}} b^{2} \sqrt {\pi }-60 \,{\mathrm e}^{-b^{2} x^{2}} x \,\operatorname {erfc}\left (b x \right ) b \pi -15 \operatorname {erfc}\left (b x \right )^{2} \pi ^{\frac {3}{2}}+44 \,{\mathrm e}^{-2 b^{2} x^{2}} \sqrt {\pi }}{48 b^{6} \pi ^{\frac {3}{2}}}\)	\(160\)

[In]

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

[Out]

1/48*(8*erfc(b*x)^2*x^6*b^6*Pi^(3/2)-16*exp(-b^2*x^2)*erfc(b*x)*x^5*b^5*Pi+8*exp(-b^2*x^2)^2*x^4*b^4*Pi^(1/2)-
40*exp(-b^2*x^2)*erfc(b*x)*x^3*b^3*Pi+28*x^2*exp(-b^2*x^2)^2*b^2*Pi^(1/2)-60*exp(-b^2*x^2)*x*erfc(b*x)*b*Pi-15
*erfc(b*x)^2*Pi^(3/2)+44*exp(-b^2*x^2)^2*Pi^(1/2))/b^6/Pi^(3/2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.84 \[ \int x^5 \text {erfc}(b x)^2 \, dx=\frac {8 \, \pi b^{6} x^{6} - {\left (15 \, \pi - 8 \, \pi b^{6} x^{6}\right )} \operatorname {erf}\left (b x\right )^{2} - 4 \, \sqrt {\pi } {\left (4 \, b^{5} x^{5} + 10 \, b^{3} x^{3} + 15 \, b x - {\left (4 \, b^{5} x^{5} + 10 \, b^{3} x^{3} + 15 \, b x\right )} \operatorname {erf}\left (b x\right )\right )} e^{\left (-b^{2} x^{2}\right )} + 2 \, {\left (15 \, \pi - 8 \, \pi b^{6} x^{6}\right )} \operatorname {erf}\left (b x\right ) + 4 \, {\left (2 \, b^{4} x^{4} + 7 \, b^{2} x^{2} + 11\right )} e^{\left (-2 \, b^{2} x^{2}\right )}}{48 \, \pi b^{6}} \]

[In]

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

[Out]

1/48*(8*pi*b^6*x^6 - (15*pi - 8*pi*b^6*x^6)*erf(b*x)^2 - 4*sqrt(pi)*(4*b^5*x^5 + 10*b^3*x^3 + 15*b*x - (4*b^5*
x^5 + 10*b^3*x^3 + 15*b*x)*erf(b*x))*e^(-b^2*x^2) + 2*(15*pi - 8*pi*b^6*x^6)*erf(b*x) + 4*(2*b^4*x^4 + 7*b^2*x
^2 + 11)*e^(-2*b^2*x^2))/(pi*b^6)

Sympy [A] (verification not implemented)

Time = 0.55 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.97 \[ \int x^5 \text {erfc}(b x)^2 \, dx=\begin {cases} \frac {x^{6} \operatorname {erfc}^{2}{\left (b x \right )}}{6} - \frac {x^{5} e^{- b^{2} x^{2}} \operatorname {erfc}{\left (b x \right )}}{3 \sqrt {\pi } b} + \frac {x^{4} e^{- 2 b^{2} x^{2}}}{6 \pi b^{2}} - \frac {5 x^{3} e^{- b^{2} x^{2}} \operatorname {erfc}{\left (b x \right )}}{6 \sqrt {\pi } b^{3}} + \frac {7 x^{2} e^{- 2 b^{2} x^{2}}}{12 \pi b^{4}} - \frac {5 x e^{- b^{2} x^{2}} \operatorname {erfc}{\left (b x \right )}}{4 \sqrt {\pi } b^{5}} - \frac {5 \operatorname {erfc}^{2}{\left (b x \right )}}{16 b^{6}} + \frac {11 e^{- 2 b^{2} x^{2}}}{12 \pi b^{6}} & \text {for}\: b \neq 0 \\\frac {x^{6}}{6} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((x**6*erfc(b*x)**2/6 - x**5*exp(-b**2*x**2)*erfc(b*x)/(3*sqrt(pi)*b) + x**4*exp(-2*b**2*x**2)/(6*pi*
b**2) - 5*x**3*exp(-b**2*x**2)*erfc(b*x)/(6*sqrt(pi)*b**3) + 7*x**2*exp(-2*b**2*x**2)/(12*pi*b**4) - 5*x*exp(-
b**2*x**2)*erfc(b*x)/(4*sqrt(pi)*b**5) - 5*erfc(b*x)**2/(16*b**6) + 11*exp(-2*b**2*x**2)/(12*pi*b**6), Ne(b, 0
)), (x**6/6, True))

Maxima [F]

\[ \int x^5 \text {erfc}(b x)^2 \, dx=\int { x^{5} \operatorname {erfc}\left (b x\right )^{2} \,d x } \]

[In]

integrate(x^5*erfc(b*x)^2,x, algorithm="maxima")

[Out]

integrate(x^5*erfc(b*x)^2, x)

Giac [F]

\[ \int x^5 \text {erfc}(b x)^2 \, dx=\int { x^{5} \operatorname {erfc}\left (b x\right )^{2} \,d x } \]

[In]

integrate(x^5*erfc(b*x)^2,x, algorithm="giac")

[Out]

integrate(x^5*erfc(b*x)^2, x)

Mupad [B] (verification not implemented)

Time = 4.96 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.80 \[ \int x^5 \text {erfc}(b x)^2 \, dx=\frac {x^6\,{\mathrm {erfc}\left (b\,x\right )}^2}{6}-\frac {\frac {5\,\pi \,{\mathrm {erfc}\left (b\,x\right )}^2}{16}-\frac {11\,{\mathrm {e}}^{-2\,b^2\,x^2}}{12}-\frac {7\,b^2\,x^2\,{\mathrm {e}}^{-2\,b^2\,x^2}}{12}-\frac {b^4\,x^4\,{\mathrm {e}}^{-2\,b^2\,x^2}}{6}+\frac {5\,b^3\,x^3\,\sqrt {\pi }\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erfc}\left (b\,x\right )}{6}+\frac {b^5\,x^5\,\sqrt {\pi }\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erfc}\left (b\,x\right )}{3}+\frac {5\,b\,x\,\sqrt {\pi }\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erfc}\left (b\,x\right )}{4}}{b^6\,\pi } \]

[In]

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

[Out]

(x^6*erfc(b*x)^2)/6 - ((5*pi*erfc(b*x)^2)/16 - (11*exp(-2*b^2*x^2))/12 - (7*b^2*x^2*exp(-2*b^2*x^2))/12 - (b^4
*x^4*exp(-2*b^2*x^2))/6 + (5*b^3*x^3*pi^(1/2)*exp(-b^2*x^2)*erfc(b*x))/6 + (b^5*x^5*pi^(1/2)*exp(-b^2*x^2)*erf
c(b*x))/3 + (5*b*x*pi^(1/2)*exp(-b^2*x^2)*erfc(b*x))/4)/(b^6*pi)