Optimal. Leaf size=60 \[ \sqrt{2} b^2 \text{Erf}\left (\sqrt{2} b x\right )-\frac{e^{-b^2 x^2} \text{Erfc}(b x)}{2 x^2}+\frac{b e^{-2 b^2 x^2}}{\sqrt{\pi } x} \]
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Rubi [A] time = 0.134725, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.075, Rules used = {6392, 2214, 2205} \[ \sqrt{2} b^2 \text{Erf}\left (\sqrt{2} b x\right )-\frac{e^{-b^2 x^2} \text{Erfc}(b x)}{2 x^2}+\frac{b e^{-2 b^2 x^2}}{\sqrt{\pi } x} \]
Antiderivative was successfully verified.
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Rule 6392
Rule 2214
Rule 2205
Rubi steps
\begin{align*} \int \left (\frac{e^{-b^2 x^2} \text{erfc}(b x)}{x^3}+\frac{b^2 e^{-b^2 x^2} \text{erfc}(b x)}{x}\right ) \, dx &=b^2 \int \frac{e^{-b^2 x^2} \text{erfc}(b x)}{x} \, dx+\int \frac{e^{-b^2 x^2} \text{erfc}(b x)}{x^3} \, dx\\ &=-\frac{e^{-b^2 x^2} \text{erfc}(b x)}{2 x^2}-\frac{b \int \frac{e^{-2 b^2 x^2}}{x^2} \, dx}{\sqrt{\pi }}\\ &=\frac{b e^{-2 b^2 x^2}}{\sqrt{\pi } x}-\frac{e^{-b^2 x^2} \text{erfc}(b x)}{2 x^2}+\frac{\left (4 b^3\right ) \int e^{-2 b^2 x^2} \, dx}{\sqrt{\pi }}\\ &=\frac{b e^{-2 b^2 x^2}}{\sqrt{\pi } x}+\sqrt{2} b^2 \text{erf}\left (\sqrt{2} b x\right )-\frac{e^{-b^2 x^2} \text{erfc}(b x)}{2 x^2}\\ \end{align*}
Mathematica [A] time = 0.0774352, size = 60, normalized size = 1. \[ \sqrt{2} b^2 \text{Erf}\left (\sqrt{2} b x\right )-\frac{e^{-b^2 x^2} \text{Erfc}(b x)}{2 x^2}+\frac{b e^{-2 b^2 x^2}}{\sqrt{\pi } x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.302, size = 84, normalized size = 1.4 \begin{align*}{\frac{1}{b} \left ( -{\frac{b}{2\,{{\rm e}^{{b}^{2}{x}^{2}}}{x}^{2}}}+{\frac{{\it Erf} \left ( bx \right ) b}{2\,{{\rm e}^{{b}^{2}{x}^{2}}}{x}^{2}}}-{\frac{{b}^{3}}{\sqrt{\pi }} \left ( -{\frac{1}{ \left ({{\rm e}^{{b}^{2}{x}^{2}}} \right ) ^{2}bx}}-\sqrt{2}\sqrt{\pi }{\it Erf} \left ( bx\sqrt{2} \right ) \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b^{2} \operatorname{erfc}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x} + \frac{\operatorname{erfc}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.29337, size = 184, normalized size = 3.07 \begin{align*} \frac{2 \, \sqrt{2} \pi \sqrt{b^{2}} b x^{2} \operatorname{erf}\left (\sqrt{2} \sqrt{b^{2}} x\right ) + 2 \, \sqrt{\pi } b x e^{\left (-2 \, b^{2} x^{2}\right )} -{\left (\pi - \pi \operatorname{erf}\left (b x\right )\right )} e^{\left (-b^{2} x^{2}\right )}}{2 \, \pi x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (b^{2} x^{2} + 1\right ) e^{- b^{2} x^{2}} \operatorname{erfc}{\left (b x \right )}}{x^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b^{2} \operatorname{erfc}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x} + \frac{\operatorname{erfc}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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