3.129 \(\int \frac{\text{Erfc}(b x)^2}{x^3} \, dx\)

Optimal. Leaf size=67 \[ \frac{2 b e^{-b^2 x^2} \text{Erfc}(b x)}{\sqrt{\pi } x}+b^2 \left (-\text{Erfc}(b x)^2\right )+\frac{2 b^2 \text{ExpIntegralEi}\left (-2 b^2 x^2\right )}{\pi }-\frac{\text{Erfc}(b x)^2}{2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.0946859, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6365, 6392, 6374, 30, 2210} \[ \frac{2 b e^{-b^2 x^2} \text{Erfc}(b x)}{\sqrt{\pi } x}+b^2 \left (-\text{Erfc}(b x)^2\right )+\frac{2 b^2 \text{Ei}\left (-2 b^2 x^2\right )}{\pi }-\frac{\text{Erfc}(b x)^2}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[Erfc[b*x]^2/x^3,x]

[Out]

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

Rule 6365

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 6392

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])/(m + 1), x] + (-Dist[(2*d)/(m + 1), Int[x^(m + 2)*E^(c + d*x^2)*Erfc[a + b*x], x], x] + Dist[(2*b
)/((m + 1)*Sqrt[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]
&& ILtQ[m, -1]

Rule 6374

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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[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)^2}{x^3} \, dx &=-\frac{\text{erfc}(b x)^2}{2 x^2}-\frac{(2 b) \int \frac{e^{-b^2 x^2} \text{erfc}(b x)}{x^2} \, dx}{\sqrt{\pi }}\\ &=\frac{2 b e^{-b^2 x^2} \text{erfc}(b x)}{\sqrt{\pi } x}-\frac{\text{erfc}(b x)^2}{2 x^2}+\frac{\left (4 b^2\right ) \int \frac{e^{-2 b^2 x^2}}{x} \, dx}{\pi }+\frac{\left (4 b^3\right ) \int e^{-b^2 x^2} \text{erfc}(b x) \, dx}{\sqrt{\pi }}\\ &=\frac{2 b e^{-b^2 x^2} \text{erfc}(b x)}{\sqrt{\pi } x}-\frac{\text{erfc}(b x)^2}{2 x^2}+\frac{2 b^2 \text{Ei}\left (-2 b^2 x^2\right )}{\pi }-\left (2 b^2\right ) \operatorname{Subst}(\int x \, dx,x,\text{erfc}(b x))\\ &=\frac{2 b e^{-b^2 x^2} \text{erfc}(b x)}{\sqrt{\pi } x}-b^2 \text{erfc}(b x)^2-\frac{\text{erfc}(b x)^2}{2 x^2}+\frac{2 b^2 \text{Ei}\left (-2 b^2 x^2\right )}{\pi }\\ \end{align*}

Mathematica [A]  time = 0.0343234, size = 63, normalized size = 0.94 \[ \frac{2 b e^{-b^2 x^2} \text{Erfc}(b x)}{\sqrt{\pi } x}+\left (-b^2-\frac{1}{2 x^2}\right ) \text{Erfc}(b x)^2+\frac{2 b^2 \text{ExpIntegralEi}\left (-2 b^2 x^2\right )}{\pi } \]

Antiderivative was successfully verified.

[In]

Integrate[Erfc[b*x]^2/x^3,x]

[Out]

(2*b*Erfc[b*x])/(E^(b^2*x^2)*Sqrt[Pi]*x) + (-b^2 - 1/(2*x^2))*Erfc[b*x]^2 + (2*b^2*ExpIntegralEi[-2*b^2*x^2])/
Pi

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Maple [F]  time = 0.059, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\it erfc} \left ( bx \right ) \right ) ^{2}}{{x}^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erfc(b*x)^2/x^3,x)

[Out]

int(erfc(b*x)^2/x^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erfc}\left (b x\right )^{2}}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfc(b*x)^2/x^3,x, algorithm="maxima")

[Out]

integrate(erfc(b*x)^2/x^3, x)

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Fricas [A]  time = 2.06378, size = 247, normalized size = 3.69 \begin{align*} -\frac{\pi - 4 \, \pi \sqrt{b^{2}} b x^{2} \operatorname{erf}\left (\sqrt{b^{2}} x\right ) - 4 \, b^{2} x^{2}{\rm Ei}\left (-2 \, b^{2} x^{2}\right ) +{\left (\pi + 2 \, \pi b^{2} x^{2}\right )} \operatorname{erf}\left (b x\right )^{2} + 4 \, \sqrt{\pi }{\left (b x \operatorname{erf}\left (b x\right ) - b x\right )} e^{\left (-b^{2} x^{2}\right )} - 2 \, \pi \operatorname{erf}\left (b x\right )}{2 \, \pi x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfc(b*x)^2/x^3,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erfc}^{2}{\left (b x \right )}}{x^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfc(b*x)**2/x**3,x)

[Out]

Integral(erfc(b*x)**2/x**3, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erfc}\left (b x\right )^{2}}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfc(b*x)^2/x^3,x, algorithm="giac")

[Out]

integrate(erfc(b*x)^2/x^3, x)