Optimal. Leaf size=71 \[ -\frac{1}{3} b^4 \text{Erf}(b x)-\frac{b^3 e^{-b^2 x^2}}{3 \sqrt{\pi } x}+\frac{b e^{-b^2 x^2}}{6 \sqrt{\pi } x^3}-\frac{\text{Erfc}(b x)}{4 x^4} \]
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Rubi [A] time = 0.059194, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {6362, 2214, 2205} \[ -\frac{1}{3} b^4 \text{Erf}(b x)-\frac{b^3 e^{-b^2 x^2}}{3 \sqrt{\pi } x}+\frac{b e^{-b^2 x^2}}{6 \sqrt{\pi } x^3}-\frac{\text{Erfc}(b x)}{4 x^4} \]
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^5} \, dx &=-\frac{\text{erfc}(b x)}{4 x^4}-\frac{b \int \frac{e^{-b^2 x^2}}{x^4} \, dx}{2 \sqrt{\pi }}\\ &=\frac{b e^{-b^2 x^2}}{6 \sqrt{\pi } x^3}-\frac{\text{erfc}(b x)}{4 x^4}+\frac{b^3 \int \frac{e^{-b^2 x^2}}{x^2} \, dx}{3 \sqrt{\pi }}\\ &=\frac{b e^{-b^2 x^2}}{6 \sqrt{\pi } x^3}-\frac{b^3 e^{-b^2 x^2}}{3 \sqrt{\pi } x}-\frac{\text{erfc}(b x)}{4 x^4}-\frac{\left (2 b^5\right ) \int e^{-b^2 x^2} \, dx}{3 \sqrt{\pi }}\\ &=\frac{b e^{-b^2 x^2}}{6 \sqrt{\pi } x^3}-\frac{b^3 e^{-b^2 x^2}}{3 \sqrt{\pi } x}-\frac{1}{3} b^4 \text{erf}(b x)-\frac{\text{erfc}(b x)}{4 x^4}\\ \end{align*}
Mathematica [A] time = 0.0351263, size = 53, normalized size = 0.75 \[ \frac{1}{12} \left (-4 b^4 \text{Erf}(b x)+\frac{2 e^{-b^2 x^2} \left (b-2 b^3 x^2\right )}{\sqrt{\pi } x^3}-\frac{3 \text{Erfc}(b x)}{x^4}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 69, normalized size = 1. \begin{align*}{b}^{4} \left ( -{\frac{{\it erfc} \left ( bx \right ) }{4\,{b}^{4}{x}^{4}}}-{\frac{1}{2\,\sqrt{\pi }} \left ( -{\frac{1}{3\,{{\rm e}^{{b}^{2}{x}^{2}}}{b}^{3}{x}^{3}}}+{\frac{2}{3\,{{\rm e}^{{b}^{2}{x}^{2}}}bx}}+{\frac{2\,\sqrt{\pi }{\it Erf} \left ( bx \right ) }{3}} \right ) } \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09469, size = 50, normalized size = 0.7 \begin{align*} \frac{\left (b^{2} x^{2}\right )^{\frac{3}{2}} b \Gamma \left (-\frac{3}{2}, b^{2} x^{2}\right )}{4 \, \sqrt{\pi } x^{3}} - \frac{\operatorname{erfc}\left (b x\right )}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.42292, size = 135, normalized size = 1.9 \begin{align*} -\frac{3 \, \pi + 2 \, \sqrt{\pi }{\left (2 \, b^{3} x^{3} - b x\right )} e^{\left (-b^{2} x^{2}\right )} -{\left (3 \, \pi - 4 \, \pi b^{4} x^{4}\right )} \operatorname{erf}\left (b x\right )}{12 \, \pi x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.58735, size = 60, normalized size = 0.85 \begin{align*} \frac{b^{4} \operatorname{erfc}{\left (b x \right )}}{3} - \frac{b^{3} e^{- b^{2} x^{2}}}{3 \sqrt{\pi } x} + \frac{b e^{- b^{2} x^{2}}}{6 \sqrt{\pi } x^{3}} - \frac{\operatorname{erfc}{\left (b x \right )}}{4 x^{4}} \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^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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