Optimal. Leaf size=27 \[ -\frac{b \text{ExpIntegralEi}\left (-b^2 x^2\right )}{\sqrt{\pi }}-\frac{\text{Erfc}(b x)}{x} \]
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Rubi [A] time = 0.0312142, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6362, 2210} \[ -\frac{b \text{Ei}\left (-b^2 x^2\right )}{\sqrt{\pi }}-\frac{\text{Erfc}(b x)}{x} \]
Antiderivative was successfully verified.
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Rule 6362
Rule 2210
Rubi steps
\begin{align*} \int \frac{\text{erfc}(b x)}{x^2} \, dx &=-\frac{\text{erfc}(b x)}{x}-\frac{(2 b) \int \frac{e^{-b^2 x^2}}{x} \, dx}{\sqrt{\pi }}\\ &=-\frac{\text{erfc}(b x)}{x}-\frac{b \text{Ei}\left (-b^2 x^2\right )}{\sqrt{\pi }}\\ \end{align*}
Mathematica [A] time = 0.0152504, size = 27, normalized size = 1. \[ -\frac{b \text{ExpIntegralEi}\left (-b^2 x^2\right )}{\sqrt{\pi }}-\frac{\text{Erfc}(b x)}{x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 29, normalized size = 1.1 \begin{align*} b \left ( -{\frac{{\it erfc} \left ( bx \right ) }{bx}}+{\frac{{\it Ei} \left ( 1,{b}^{2}{x}^{2} \right ) }{\sqrt{\pi }}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11359, size = 34, normalized size = 1.26 \begin{align*} -\frac{b{\rm Ei}\left (-b^{2} x^{2}\right )}{\sqrt{\pi }} - \frac{\operatorname{erfc}\left (b x\right )}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.35477, size = 76, normalized size = 2.81 \begin{align*} -\frac{\pi + \sqrt{\pi } b x{\rm Ei}\left (-b^{2} x^{2}\right ) - \pi \operatorname{erf}\left (b x\right )}{\pi x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.27443, size = 20, normalized size = 0.74 \begin{align*} \frac{b \operatorname{E}_{1}\left (b^{2} x^{2}\right )}{\sqrt{\pi }} - \frac{\operatorname{erfc}{\left (b x \right )}}{x} \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^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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