Optimal. Leaf size=56 \[ \frac{b^3 \text{ExpIntegralEi}\left (-b^2 x^2\right )}{3 \sqrt{\pi }}+\frac{b e^{-b^2 x^2}}{3 \sqrt{\pi } x^2}-\frac{\text{Erfc}(b x)}{3 x^3} \]
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Rubi [A] time = 0.0540663, antiderivative size = 56, 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, 2210} \[ \frac{b^3 \text{Ei}\left (-b^2 x^2\right )}{3 \sqrt{\pi }}+\frac{b e^{-b^2 x^2}}{3 \sqrt{\pi } x^2}-\frac{\text{Erfc}(b x)}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 6362
Rule 2214
Rule 2210
Rubi steps
\begin{align*} \int \frac{\text{erfc}(b x)}{x^4} \, dx &=-\frac{\text{erfc}(b x)}{3 x^3}-\frac{(2 b) \int \frac{e^{-b^2 x^2}}{x^3} \, dx}{3 \sqrt{\pi }}\\ &=\frac{b e^{-b^2 x^2}}{3 \sqrt{\pi } x^2}-\frac{\text{erfc}(b x)}{3 x^3}+\frac{\left (2 b^3\right ) \int \frac{e^{-b^2 x^2}}{x} \, dx}{3 \sqrt{\pi }}\\ &=\frac{b e^{-b^2 x^2}}{3 \sqrt{\pi } x^2}-\frac{\text{erfc}(b x)}{3 x^3}+\frac{b^3 \text{Ei}\left (-b^2 x^2\right )}{3 \sqrt{\pi }}\\ \end{align*}
Mathematica [A] time = 0.0395833, size = 49, normalized size = 0.88 \[ \frac{1}{3} \left (\frac{b \left (b^2 \text{ExpIntegralEi}\left (-b^2 x^2\right )+\frac{e^{-b^2 x^2}}{x^2}\right )}{\sqrt{\pi }}-\frac{\text{Erfc}(b x)}{x^3}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 53, normalized size = 1. \begin{align*}{b}^{3} \left ( -{\frac{{\it erfc} \left ( bx \right ) }{3\,{x}^{3}{b}^{3}}}-{\frac{2}{3\,\sqrt{\pi }} \left ( -{\frac{1}{2\,{b}^{2}{x}^{2}{{\rm e}^{{b}^{2}{x}^{2}}}}}+{\frac{{\it Ei} \left ( 1,{b}^{2}{x}^{2} \right ) }{2}} \right ) } \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07609, size = 36, normalized size = 0.64 \begin{align*} \frac{b^{3} \Gamma \left (-1, b^{2} x^{2}\right )}{3 \, \sqrt{\pi }} - \frac{\operatorname{erfc}\left (b x\right )}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.12675, size = 117, normalized size = 2.09 \begin{align*} -\frac{\pi - \pi \operatorname{erf}\left (b x\right ) - \sqrt{\pi }{\left (b^{3} x^{3}{\rm Ei}\left (-b^{2} x^{2}\right ) + b x e^{\left (-b^{2} x^{2}\right )}\right )}}{3 \, \pi x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.50946, size = 48, normalized size = 0.86 \begin{align*} - \frac{b^{3} \operatorname{E}_{1}\left (b^{2} x^{2}\right )}{3 \sqrt{\pi }} + \frac{b e^{- b^{2} x^{2}}}{3 \sqrt{\pi } x^{2}} - \frac{\operatorname{erfc}{\left (b x \right )}}{3 x^{3}} \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^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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