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