3.183 \(\int \frac{e^{-b^2 x^2} \text{Erfc}(b x)}{x^5} \, dx\)

Optimal. Leaf size=161 \[ \frac{1}{2} b^4 \text{Unintegrable}\left (\frac{e^{-b^2 x^2} \text{Erfc}(b x)}{x},x\right )-\frac{2}{3} \sqrt{2} b^4 \text{Erf}\left (\sqrt{2} b x\right )-\frac{b^4 \text{Erf}\left (\sqrt{2} b x\right )}{\sqrt{2}}+\frac{b^2 e^{-b^2 x^2} \text{Erfc}(b x)}{4 x^2}-\frac{e^{-b^2 x^2} \text{Erfc}(b x)}{4 x^4}-\frac{7 b^3 e^{-2 b^2 x^2}}{6 \sqrt{\pi } x}+\frac{b e^{-2 b^2 x^2}}{6 \sqrt{\pi } x^3} \]

[Out]

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

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Rubi [A]  time = 0.182261, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{e^{-b^2 x^2} \text{Erfc}(b x)}{x^5} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Erfc[b*x]/(E^(b^2*x^2)*x^5),x]

[Out]

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

Rubi steps

\begin{align*} \int \frac{e^{-b^2 x^2} \text{erfc}(b x)}{x^5} \, dx &=-\frac{e^{-b^2 x^2} \text{erfc}(b x)}{4 x^4}-\frac{1}{2} b^2 \int \frac{e^{-b^2 x^2} \text{erfc}(b x)}{x^3} \, dx-\frac{b \int \frac{e^{-2 b^2 x^2}}{x^4} \, dx}{2 \sqrt{\pi }}\\ &=\frac{b e^{-2 b^2 x^2}}{6 \sqrt{\pi } x^3}-\frac{e^{-b^2 x^2} \text{erfc}(b x)}{4 x^4}+\frac{b^2 e^{-b^2 x^2} \text{erfc}(b x)}{4 x^2}+\frac{1}{2} b^4 \int \frac{e^{-b^2 x^2} \text{erfc}(b x)}{x} \, dx+\frac{b^3 \int \frac{e^{-2 b^2 x^2}}{x^2} \, dx}{2 \sqrt{\pi }}+\frac{\left (2 b^3\right ) \int \frac{e^{-2 b^2 x^2}}{x^2} \, dx}{3 \sqrt{\pi }}\\ &=\frac{b e^{-2 b^2 x^2}}{6 \sqrt{\pi } x^3}-\frac{7 b^3 e^{-2 b^2 x^2}}{6 \sqrt{\pi } x}-\frac{e^{-b^2 x^2} \text{erfc}(b x)}{4 x^4}+\frac{b^2 e^{-b^2 x^2} \text{erfc}(b x)}{4 x^2}+\frac{1}{2} b^4 \int \frac{e^{-b^2 x^2} \text{erfc}(b x)}{x} \, dx-\frac{\left (2 b^5\right ) \int e^{-2 b^2 x^2} \, dx}{\sqrt{\pi }}-\frac{\left (8 b^5\right ) \int e^{-2 b^2 x^2} \, dx}{3 \sqrt{\pi }}\\ &=\frac{b e^{-2 b^2 x^2}}{6 \sqrt{\pi } x^3}-\frac{7 b^3 e^{-2 b^2 x^2}}{6 \sqrt{\pi } x}-\frac{b^4 \text{erf}\left (\sqrt{2} b x\right )}{\sqrt{2}}-\frac{2}{3} \sqrt{2} b^4 \text{erf}\left (\sqrt{2} b x\right )-\frac{e^{-b^2 x^2} \text{erfc}(b x)}{4 x^4}+\frac{b^2 e^{-b^2 x^2} \text{erfc}(b x)}{4 x^2}+\frac{1}{2} b^4 \int \frac{e^{-b^2 x^2} \text{erfc}(b x)}{x} \, dx\\ \end{align*}

Mathematica [A]  time = 0.271161, size = 0, normalized size = 0. \[ \int \frac{e^{-b^2 x^2} \text{Erfc}(b x)}{x^5} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Erfc[b*x]/(E^(b^2*x^2)*x^5),x]

[Out]

Integrate[Erfc[b*x]/(E^(b^2*x^2)*x^5), x]

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Maple [A]  time = 0.33, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it erfc} \left ( bx \right ) }{{{\rm e}^{{b}^{2}{x}^{2}}}{x}^{5}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erfc}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(erfc(b*x)*e^(-b^2*x^2)/x^5, x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (\operatorname{erf}\left (b x\right ) - 1\right )} e^{\left (-b^{2} x^{2}\right )}}{x^{5}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(erf(b*x) - 1)*e^(-b^2*x^2)/x^5, x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{- b^{2} x^{2}} \operatorname{erfc}{\left (b x \right )}}{x^{5}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(exp(-b**2*x**2)*erfc(b*x)/x**5, x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erfc}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(erfc(b*x)*e^(-b^2*x^2)/x^5, x)