3.117 \(\int \frac{\text{Erfc}(b x)}{x^6} \, dx\)

Optimal. Leaf size=81 \[ -\frac{b^5 \text{ExpIntegralEi}\left (-b^2 x^2\right )}{10 \sqrt{\pi }}-\frac{b^3 e^{-b^2 x^2}}{10 \sqrt{\pi } x^2}+\frac{b e^{-b^2 x^2}}{10 \sqrt{\pi } x^4}-\frac{\text{Erfc}(b x)}{5 x^5} \]

[Out]

b/(10*E^(b^2*x^2)*Sqrt[Pi]*x^4) - b^3/(10*E^(b^2*x^2)*Sqrt[Pi]*x^2) - Erfc[b*x]/(5*x^5) - (b^5*ExpIntegralEi[-
(b^2*x^2)])/(10*Sqrt[Pi])

________________________________________________________________________________________

Rubi [A]  time = 0.0740645, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {6362, 2214, 2210} \[ -\frac{b^5 \text{Ei}\left (-b^2 x^2\right )}{10 \sqrt{\pi }}-\frac{b^3 e^{-b^2 x^2}}{10 \sqrt{\pi } x^2}+\frac{b e^{-b^2 x^2}}{10 \sqrt{\pi } x^4}-\frac{\text{Erfc}(b x)}{5 x^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

b/(10*E^(b^2*x^2)*Sqrt[Pi]*x^4) - b^3/(10*E^(b^2*x^2)*Sqrt[Pi]*x^2) - Erfc[b*x]/(5*x^5) - (b^5*ExpIntegralEi[-
(b^2*x^2)])/(10*Sqrt[Pi])

Rule 6362

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]

Rule 2214

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*F^(a + b*(c + d*x)^n))/(d*(m + 1)), x] - Dist[(b*n*Log[F])/(m + 1), 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[-4, (m + 1)/n, 5] && IntegerQ[n
] && ((GtQ[n, 0] && LtQ[m, -1]) || (GtQ[-n, 0] && LeQ[-n, m + 1]))

Rule 2210

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Simp[(F^a*ExpIntegralEi[
b*(c + d*x)^n*Log[F]])/(f*n), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[d*e - c*f, 0]

Rubi steps

\begin{align*} \int \frac{\text{erfc}(b x)}{x^6} \, dx &=-\frac{\text{erfc}(b x)}{5 x^5}-\frac{(2 b) \int \frac{e^{-b^2 x^2}}{x^5} \, dx}{5 \sqrt{\pi }}\\ &=\frac{b e^{-b^2 x^2}}{10 \sqrt{\pi } x^4}-\frac{\text{erfc}(b x)}{5 x^5}+\frac{b^3 \int \frac{e^{-b^2 x^2}}{x^3} \, dx}{5 \sqrt{\pi }}\\ &=\frac{b e^{-b^2 x^2}}{10 \sqrt{\pi } x^4}-\frac{b^3 e^{-b^2 x^2}}{10 \sqrt{\pi } x^2}-\frac{\text{erfc}(b x)}{5 x^5}-\frac{b^5 \int \frac{e^{-b^2 x^2}}{x} \, dx}{5 \sqrt{\pi }}\\ &=\frac{b e^{-b^2 x^2}}{10 \sqrt{\pi } x^4}-\frac{b^3 e^{-b^2 x^2}}{10 \sqrt{\pi } x^2}-\frac{\text{erfc}(b x)}{5 x^5}-\frac{b^5 \text{Ei}\left (-b^2 x^2\right )}{10 \sqrt{\pi }}\\ \end{align*}

Mathematica [A]  time = 0.0239886, size = 73, normalized size = 0.9 \[ -\frac{b^5 \text{ExpIntegralEi}\left (-b^2 x^2\right )}{10 \sqrt{\pi }}+e^{-b^2 x^2} \left (\frac{b}{10 \sqrt{\pi } x^4}-\frac{b^3}{10 \sqrt{\pi } x^2}\right )-\frac{\text{Erfc}(b x)}{5 x^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b/(10*Sqrt[Pi]*x^4) - b^3/(10*Sqrt[Pi]*x^2))/E^(b^2*x^2) - Erfc[b*x]/(5*x^5) - (b^5*ExpIntegralEi[-(b^2*x^2)]
)/(10*Sqrt[Pi])

________________________________________________________________________________________

Maple [A]  time = 0.042, size = 71, normalized size = 0.9 \begin{align*}{b}^{5} \left ( -{\frac{{\it erfc} \left ( bx \right ) }{5\,{b}^{5}{x}^{5}}}-{\frac{2}{5\,\sqrt{\pi }} \left ( -{\frac{1}{4\,{{\rm e}^{{b}^{2}{x}^{2}}}{b}^{4}{x}^{4}}}+{\frac{1}{4\,{b}^{2}{x}^{2}{{\rm e}^{{b}^{2}{x}^{2}}}}}-{\frac{{\it Ei} \left ( 1,{b}^{2}{x}^{2} \right ) }{4}} \right ) } \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^5*(-1/5/b^5/x^5*erfc(b*x)-2/5/Pi^(1/2)*(-1/4/exp(b^2*x^2)/b^4/x^4+1/4/exp(b^2*x^2)/b^2/x^2-1/4*Ei(1,b^2*x^2)
))

________________________________________________________________________________________

Maxima [A]  time = 1.09173, size = 36, normalized size = 0.44 \begin{align*} \frac{b^{5} \Gamma \left (-2, b^{2} x^{2}\right )}{5 \, \sqrt{\pi }} - \frac{\operatorname{erfc}\left (b x\right )}{5 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*b^5*gamma(-2, b^2*x^2)/sqrt(pi) - 1/5*erfc(b*x)/x^5

________________________________________________________________________________________

Fricas [A]  time = 2.13838, size = 140, normalized size = 1.73 \begin{align*} -\frac{2 \, \pi - 2 \, \pi \operatorname{erf}\left (b x\right ) + \sqrt{\pi }{\left (b^{5} x^{5}{\rm Ei}\left (-b^{2} x^{2}\right ) +{\left (b^{3} x^{3} - b x\right )} e^{\left (-b^{2} x^{2}\right )}\right )}}{10 \, \pi x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10*(2*pi - 2*pi*erf(b*x) + sqrt(pi)*(b^5*x^5*Ei(-b^2*x^2) + (b^3*x^3 - b*x)*e^(-b^2*x^2)))/(pi*x^5)

________________________________________________________________________________________

Sympy [A]  time = 5.2556, size = 70, normalized size = 0.86 \begin{align*} \frac{b^{5} \operatorname{E}_{1}\left (b^{2} x^{2}\right )}{10 \sqrt{\pi }} - \frac{b^{3} e^{- b^{2} x^{2}}}{10 \sqrt{\pi } x^{2}} + \frac{b e^{- b^{2} x^{2}}}{10 \sqrt{\pi } x^{4}} - \frac{\operatorname{erfc}{\left (b x \right )}}{5 x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b**5*expint(1, b**2*x**2)/(10*sqrt(pi)) - b**3*exp(-b**2*x**2)/(10*sqrt(pi)*x**2) + b*exp(-b**2*x**2)/(10*sqrt
(pi)*x**4) - erfc(b*x)/(5*x**5)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erfc}\left (b x\right )}{x^{6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(erfc(b*x)/x^6, x)