Optimal. Leaf size=96 \[ \frac{4}{45} b^6 \text{Erf}(b x)+\frac{4 b^5 e^{-b^2 x^2}}{45 \sqrt{\pi } x}-\frac{2 b^3 e^{-b^2 x^2}}{45 \sqrt{\pi } x^3}+\frac{b e^{-b^2 x^2}}{15 \sqrt{\pi } x^5}-\frac{\text{Erfc}(b x)}{6 x^6} \]
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Rubi [A] time = 0.0829265, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {6362, 2214, 2205} \[ \frac{4}{45} b^6 \text{Erf}(b x)+\frac{4 b^5 e^{-b^2 x^2}}{45 \sqrt{\pi } x}-\frac{2 b^3 e^{-b^2 x^2}}{45 \sqrt{\pi } x^3}+\frac{b e^{-b^2 x^2}}{15 \sqrt{\pi } x^5}-\frac{\text{Erfc}(b x)}{6 x^6} \]
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^7} \, dx &=-\frac{\text{erfc}(b x)}{6 x^6}-\frac{b \int \frac{e^{-b^2 x^2}}{x^6} \, dx}{3 \sqrt{\pi }}\\ &=\frac{b e^{-b^2 x^2}}{15 \sqrt{\pi } x^5}-\frac{\text{erfc}(b x)}{6 x^6}+\frac{\left (2 b^3\right ) \int \frac{e^{-b^2 x^2}}{x^4} \, dx}{15 \sqrt{\pi }}\\ &=\frac{b e^{-b^2 x^2}}{15 \sqrt{\pi } x^5}-\frac{2 b^3 e^{-b^2 x^2}}{45 \sqrt{\pi } x^3}-\frac{\text{erfc}(b x)}{6 x^6}-\frac{\left (4 b^5\right ) \int \frac{e^{-b^2 x^2}}{x^2} \, dx}{45 \sqrt{\pi }}\\ &=\frac{b e^{-b^2 x^2}}{15 \sqrt{\pi } x^5}-\frac{2 b^3 e^{-b^2 x^2}}{45 \sqrt{\pi } x^3}+\frac{4 b^5 e^{-b^2 x^2}}{45 \sqrt{\pi } x}-\frac{\text{erfc}(b x)}{6 x^6}+\frac{\left (8 b^7\right ) \int e^{-b^2 x^2} \, dx}{45 \sqrt{\pi }}\\ &=\frac{b e^{-b^2 x^2}}{15 \sqrt{\pi } x^5}-\frac{2 b^3 e^{-b^2 x^2}}{45 \sqrt{\pi } x^3}+\frac{4 b^5 e^{-b^2 x^2}}{45 \sqrt{\pi } x}+\frac{4}{45} b^6 \text{erf}(b x)-\frac{\text{erfc}(b x)}{6 x^6}\\ \end{align*}
Mathematica [A] time = 0.0476964, size = 62, normalized size = 0.65 \[ \frac{1}{90} \left (8 b^6 \text{Erf}(b x)+\frac{2 b e^{-b^2 x^2} \left (4 b^4 x^4-2 b^2 x^2+3\right )}{\sqrt{\pi } x^5}-\frac{15 \text{Erfc}(b x)}{x^6}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 87, normalized size = 0.9 \begin{align*}{b}^{6} \left ( -{\frac{{\it erfc} \left ( bx \right ) }{6\,{b}^{6}{x}^{6}}}-{\frac{1}{3\,\sqrt{\pi }} \left ( -{\frac{1}{5\,{{\rm e}^{{b}^{2}{x}^{2}}}{b}^{5}{x}^{5}}}+{\frac{2}{15\,{{\rm e}^{{b}^{2}{x}^{2}}}{b}^{3}{x}^{3}}}-{\frac{4}{15\,{{\rm e}^{{b}^{2}{x}^{2}}}bx}}-{\frac{4\,\sqrt{\pi }{\it Erf} \left ( bx \right ) }{15}} \right ) } \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11779, size = 50, normalized size = 0.52 \begin{align*} \frac{\left (b^{2} x^{2}\right )^{\frac{5}{2}} b \Gamma \left (-\frac{5}{2}, b^{2} x^{2}\right )}{6 \, \sqrt{\pi } x^{5}} - \frac{\operatorname{erfc}\left (b x\right )}{6 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.45557, size = 157, normalized size = 1.64 \begin{align*} -\frac{15 \, \pi - 2 \, \sqrt{\pi }{\left (4 \, b^{5} x^{5} - 2 \, b^{3} x^{3} + 3 \, b x\right )} e^{\left (-b^{2} x^{2}\right )} -{\left (15 \, \pi + 8 \, \pi b^{6} x^{6}\right )} \operatorname{erf}\left (b x\right )}{90 \, \pi x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.63689, size = 87, normalized size = 0.91 \begin{align*} - \frac{4 b^{6} \operatorname{erfc}{\left (b x \right )}}{45} + \frac{4 b^{5} e^{- b^{2} x^{2}}}{45 \sqrt{\pi } x} - \frac{2 b^{3} e^{- b^{2} x^{2}}}{45 \sqrt{\pi } x^{3}} + \frac{b e^{- b^{2} x^{2}}}{15 \sqrt{\pi } x^{5}} - \frac{\operatorname{erfc}{\left (b x \right )}}{6 x^{6}} \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^{7}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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