Optimal. Leaf size=40 \[ b^2 \text{Erf}(b x)+\frac{b e^{-b^2 x^2}}{\sqrt{\pi } x}-\frac{\text{Erfc}(b x)}{2 x^2} \]
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Rubi [A] time = 0.0378237, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {6362, 2214, 2205} \[ b^2 \text{Erf}(b x)+\frac{b e^{-b^2 x^2}}{\sqrt{\pi } x}-\frac{\text{Erfc}(b x)}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 6362
Rule 2214
Rule 2205
Rubi steps
\begin{align*} \int \frac{\text{erfc}(b x)}{x^3} \, dx &=-\frac{\text{erfc}(b x)}{2 x^2}-\frac{b \int \frac{e^{-b^2 x^2}}{x^2} \, dx}{\sqrt{\pi }}\\ &=\frac{b e^{-b^2 x^2}}{\sqrt{\pi } x}-\frac{\text{erfc}(b x)}{2 x^2}+\frac{\left (2 b^3\right ) \int e^{-b^2 x^2} \, dx}{\sqrt{\pi }}\\ &=\frac{b e^{-b^2 x^2}}{\sqrt{\pi } x}+b^2 \text{erf}(b x)-\frac{\text{erfc}(b x)}{2 x^2}\\ \end{align*}
Mathematica [A] time = 0.0410057, size = 40, normalized size = 1. \[ b^2 \text{Erf}(b x)+\frac{b e^{-b^2 x^2}}{\sqrt{\pi } x}-\frac{\text{Erfc}(b x)}{2 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 51, normalized size = 1.3 \begin{align*}{b}^{2} \left ( -{\frac{{\it erfc} \left ( bx \right ) }{2\,{b}^{2}{x}^{2}}}-{\frac{1}{\sqrt{\pi }} \left ( -{\frac{1}{{{\rm e}^{{b}^{2}{x}^{2}}}bx}}-\sqrt{\pi }{\it Erf} \left ( bx \right ) \right ) } \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.17421, size = 50, normalized size = 1.25 \begin{align*} \frac{\sqrt{b^{2} x^{2}} b \Gamma \left (-\frac{1}{2}, b^{2} x^{2}\right )}{2 \, \sqrt{\pi } x} - \frac{\operatorname{erfc}\left (b x\right )}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.44195, size = 109, normalized size = 2.72 \begin{align*} -\frac{\pi - 2 \, \sqrt{\pi } b x e^{\left (-b^{2} x^{2}\right )} -{\left (\pi + 2 \, \pi b^{2} x^{2}\right )} \operatorname{erf}\left (b x\right )}{2 \, \pi x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.60934, size = 34, normalized size = 0.85 \begin{align*} - b^{2} \operatorname{erfc}{\left (b x \right )} + \frac{b e^{- b^{2} x^{2}}}{\sqrt{\pi } x} - \frac{\operatorname{erfc}{\left (b x \right )}}{2 x^{2}} \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{\operatorname{erfc}\left (b x\right )}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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