\(\int x^6 \text {erfc}(b x) \, dx\) [111]

Optimal result

Integrand size = 8, antiderivative size = 109 \[ \int x^6 \text {erfc}(b x) \, dx=-\frac {6 e^{-b^2 x^2}}{7 b^7 \sqrt {\pi }}-\frac {6 e^{-b^2 x^2} x^2}{7 b^5 \sqrt {\pi }}-\frac {3 e^{-b^2 x^2} x^4}{7 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} x^6}{7 b \sqrt {\pi }}+\frac {1}{7} x^7 \text {erfc}(b x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 109, 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, 2240} \[ \int x^6 \text {erfc}(b x) \, dx=-\frac {x^6 e^{-b^2 x^2}}{7 \sqrt {\pi } b}-\frac {6 e^{-b^2 x^2}}{7 \sqrt {\pi } b^7}-\frac {6 x^2 e^{-b^2 x^2}}{7 \sqrt {\pi } b^5}-\frac {3 x^4 e^{-b^2 x^2}}{7 \sqrt {\pi } b^3}+\frac {1}{7} x^7 \text {erfc}(b x) \]

[In]

Int[x^6*Erfc[b*x],x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.67 \[ \int x^6 \text {erfc}(b x) \, dx=\frac {e^{-b^2 x^2} \left (-6-6 b^2 x^2-3 b^4 x^4-b^6 x^6+b^7 e^{b^2 x^2} \sqrt {\pi } x^7 \text {erfc}(b x)\right )}{7 b^7 \sqrt {\pi }} \]

[In]

Integrate[x^6*Erfc[b*x],x]

[Out]

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

Maple [A] (verified)

Time = 0.39 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.79

method	result	size
parallelrisch	\(\frac {\operatorname {erfc}\left (b x \right ) x^{7} b^{7} \sqrt {\pi }-{\mathrm e}^{-b^{2} x^{2}} x^{6} b^{6}-3 \,{\mathrm e}^{-b^{2} x^{2}} x^{4} b^{4}-6 x^{2} {\mathrm e}^{-b^{2} x^{2}} b^{2}-6 \,{\mathrm e}^{-b^{2} x^{2}}}{7 b^{7} \sqrt {\pi }}\)	\(86\)
derivativedivides	\(\frac {\frac {b^{7} x^{7} \operatorname {erfc}\left (b x \right )}{7}+\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}} x^{6} b^{6}}{7}-\frac {3 \,{\mathrm e}^{-b^{2} x^{2}} x^{4} b^{4}}{7}-\frac {6 x^{2} {\mathrm e}^{-b^{2} x^{2}} b^{2}}{7}-\frac {6 \,{\mathrm e}^{-b^{2} x^{2}}}{7}}{\sqrt {\pi }}}{b^{7}}\)	\(90\)
default	\(\frac {\frac {b^{7} x^{7} \operatorname {erfc}\left (b x \right )}{7}+\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}} x^{6} b^{6}}{7}-\frac {3 \,{\mathrm e}^{-b^{2} x^{2}} x^{4} b^{4}}{7}-\frac {6 x^{2} {\mathrm e}^{-b^{2} x^{2}} b^{2}}{7}-\frac {6 \,{\mathrm e}^{-b^{2} x^{2}}}{7}}{\sqrt {\pi }}}{b^{7}}\)	\(90\)
parts	\(\frac {x^{7} \operatorname {erfc}\left (b x \right )}{7}+\frac {2 b \left (-\frac {x^{6} {\mathrm e}^{-b^{2} x^{2}}}{2 b^{2}}+\frac {-\frac {3 x^{4} {\mathrm e}^{-b^{2} x^{2}}}{2 b^{2}}+\frac {3 \left (-\frac {x^{2} {\mathrm e}^{-b^{2} x^{2}}}{b^{2}}-\frac {{\mathrm e}^{-b^{2} x^{2}}}{b^{4}}\right )}{b^{2}}}{b^{2}}\right )}{7 \sqrt {\pi }}\)	\(95\)

[In]

int(x^6*erfc(b*x),x,method=_RETURNVERBOSE)

[Out]

1/7*(erfc(b*x)*x^7*b^7*Pi^(1/2)-exp(-b^2*x^2)*x^6*b^6-3*exp(-b^2*x^2)*x^4*b^4-6*x^2*exp(-b^2*x^2)*b^2-6*exp(-b
^2*x^2))/b^7/Pi^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.62 \[ \int x^6 \text {erfc}(b x) \, dx=-\frac {\pi b^{7} x^{7} \operatorname {erf}\left (b x\right ) - \pi b^{7} x^{7} + \sqrt {\pi } {\left (b^{6} x^{6} + 3 \, b^{4} x^{4} + 6 \, b^{2} x^{2} + 6\right )} e^{\left (-b^{2} x^{2}\right )}}{7 \, \pi b^{7}} \]

[In]

integrate(x^6*erfc(b*x),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.52 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.94 \[ \int x^6 \text {erfc}(b x) \, dx=\begin {cases} \frac {x^{7} \operatorname {erfc}{\left (b x \right )}}{7} - \frac {x^{6} e^{- b^{2} x^{2}}}{7 \sqrt {\pi } b} - \frac {3 x^{4} e^{- b^{2} x^{2}}}{7 \sqrt {\pi } b^{3}} - \frac {6 x^{2} e^{- b^{2} x^{2}}}{7 \sqrt {\pi } b^{5}} - \frac {6 e^{- b^{2} x^{2}}}{7 \sqrt {\pi } b^{7}} & \text {for}\: b \neq 0 \\\frac {x^{7}}{7} & \text {otherwise} \end {cases} \]

[In]

integrate(x**6*erfc(b*x),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.48 \[ \int x^6 \text {erfc}(b x) \, dx=\frac {1}{7} \, x^{7} \operatorname {erfc}\left (b x\right ) - \frac {{\left (b^{6} x^{6} + 3 \, b^{4} x^{4} + 6 \, b^{2} x^{2} + 6\right )} e^{\left (-b^{2} x^{2}\right )}}{7 \, \sqrt {\pi } b^{7}} \]

[In]

integrate(x^6*erfc(b*x),x, algorithm="maxima")

[Out]

1/7*x^7*erfc(b*x) - 1/7*(b^6*x^6 + 3*b^4*x^4 + 6*b^2*x^2 + 6)*e^(-b^2*x^2)/(sqrt(pi)*b^7)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.52 \[ \int x^6 \text {erfc}(b x) \, dx=-\frac {1}{7} \, x^{7} \operatorname {erf}\left (b x\right ) + \frac {1}{7} \, x^{7} - \frac {{\left (b^{6} x^{6} + 3 \, b^{4} x^{4} + 6 \, b^{2} x^{2} + 6\right )} e^{\left (-b^{2} x^{2}\right )}}{7 \, \sqrt {\pi } b^{7}} \]

[In]

integrate(x^6*erfc(b*x),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 4.98 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.83 \[ \int x^6 \text {erfc}(b x) \, dx=\frac {x^7\,\mathrm {erfc}\left (b\,x\right )}{7}-\frac {\frac {6\,{\mathrm {e}}^{-b^2\,x^2}}{7\,\sqrt {\pi }}+\frac {6\,b^2\,x^2\,{\mathrm {e}}^{-b^2\,x^2}}{7\,\sqrt {\pi }}+\frac {3\,b^4\,x^4\,{\mathrm {e}}^{-b^2\,x^2}}{7\,\sqrt {\pi }}+\frac {b^6\,x^6\,{\mathrm {e}}^{-b^2\,x^2}}{7\,\sqrt {\pi }}}{b^7} \]

[In]

int(x^6*erfc(b*x),x)

[Out]

(x^7*erfc(b*x))/7 - ((6*exp(-b^2*x^2))/(7*pi^(1/2)) + (6*b^2*x^2*exp(-b^2*x^2))/(7*pi^(1/2)) + (3*b^4*x^4*exp(
-b^2*x^2))/(7*pi^(1/2)) + (b^6*x^6*exp(-b^2*x^2))/(7*pi^(1/2)))/b^7