3.115 \(\int \frac{\text{Erfc}(b x)}{x^2} \, dx\)

Optimal. Leaf size=27 \[ -\frac{b \text{ExpIntegralEi}\left (-b^2 x^2\right )}{\sqrt{\pi }}-\frac{\text{Erfc}(b x)}{x} \]

[Out]

-(Erfc[b*x]/x) - (b*ExpIntegralEi[-(b^2*x^2)])/Sqrt[Pi]

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Rubi [A]  time = 0.0312142, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6362, 2210} \[ -\frac{b \text{Ei}\left (-b^2 x^2\right )}{\sqrt{\pi }}-\frac{\text{Erfc}(b x)}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Erfc[b*x]/x) - (b*ExpIntegralEi[-(b^2*x^2)])/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 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^2} \, dx &=-\frac{\text{erfc}(b x)}{x}-\frac{(2 b) \int \frac{e^{-b^2 x^2}}{x} \, dx}{\sqrt{\pi }}\\ &=-\frac{\text{erfc}(b x)}{x}-\frac{b \text{Ei}\left (-b^2 x^2\right )}{\sqrt{\pi }}\\ \end{align*}

Mathematica [A]  time = 0.0152504, size = 27, normalized size = 1. \[ -\frac{b \text{ExpIntegralEi}\left (-b^2 x^2\right )}{\sqrt{\pi }}-\frac{\text{Erfc}(b x)}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Erfc[b*x]/x) - (b*ExpIntegralEi[-(b^2*x^2)])/Sqrt[Pi]

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Maple [A]  time = 0.043, size = 29, normalized size = 1.1 \begin{align*} b \left ( -{\frac{{\it erfc} \left ( bx \right ) }{bx}}+{\frac{{\it Ei} \left ( 1,{b}^{2}{x}^{2} \right ) }{\sqrt{\pi }}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.11359, size = 34, normalized size = 1.26 \begin{align*} -\frac{b{\rm Ei}\left (-b^{2} x^{2}\right )}{\sqrt{\pi }} - \frac{\operatorname{erfc}\left (b x\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b*Ei(-b^2*x^2)/sqrt(pi) - erfc(b*x)/x

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Fricas [A]  time = 2.35477, size = 76, normalized size = 2.81 \begin{align*} -\frac{\pi + \sqrt{\pi } b x{\rm Ei}\left (-b^{2} x^{2}\right ) - \pi \operatorname{erf}\left (b x\right )}{\pi x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(pi + sqrt(pi)*b*x*Ei(-b^2*x^2) - pi*erf(b*x))/(pi*x)

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Sympy [A]  time = 1.27443, size = 20, normalized size = 0.74 \begin{align*} \frac{b \operatorname{E}_{1}\left (b^{2} x^{2}\right )}{\sqrt{\pi }} - \frac{\operatorname{erfc}{\left (b x \right )}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*expint(1, b**2*x**2)/sqrt(pi) - erfc(b*x)/x

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erfc}\left (b x\right )}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(erfc(b*x)/x^2, x)