Optimal. Leaf size=177 \[ \frac{8 b^5 e^{-b^2 x^2} \text{Erfc}(b x)}{45 \sqrt{\pi } x}-\frac{4 b^3 e^{-b^2 x^2} \text{Erfc}(b x)}{45 \sqrt{\pi } x^3}+\frac{2 b e^{-b^2 x^2} \text{Erfc}(b x)}{15 \sqrt{\pi } x^5}-\frac{4}{45} b^6 \text{Erfc}(b x)^2+\frac{28 b^6 \text{ExpIntegralEi}\left (-2 b^2 x^2\right )}{45 \pi }+\frac{2 b^4 e^{-2 b^2 x^2}}{9 \pi x^2}-\frac{b^2 e^{-2 b^2 x^2}}{15 \pi x^4}-\frac{\text{Erfc}(b x)^2}{6 x^6} \]
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Rubi [A] time = 0.281998, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {6365, 6392, 6374, 30, 2210, 2214} \[ \frac{8 b^5 e^{-b^2 x^2} \text{Erfc}(b x)}{45 \sqrt{\pi } x}-\frac{4 b^3 e^{-b^2 x^2} \text{Erfc}(b x)}{45 \sqrt{\pi } x^3}+\frac{2 b e^{-b^2 x^2} \text{Erfc}(b x)}{15 \sqrt{\pi } x^5}-\frac{4}{45} b^6 \text{Erfc}(b x)^2+\frac{28 b^6 \text{Ei}\left (-2 b^2 x^2\right )}{45 \pi }+\frac{2 b^4 e^{-2 b^2 x^2}}{9 \pi x^2}-\frac{b^2 e^{-2 b^2 x^2}}{15 \pi x^4}-\frac{\text{Erfc}(b x)^2}{6 x^6} \]
Antiderivative was successfully verified.
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Rule 6365
Rule 6392
Rule 6374
Rule 30
Rule 2210
Rule 2214
Rubi steps
\begin{align*} \int \frac{\text{erfc}(b x)^2}{x^7} \, dx &=-\frac{\text{erfc}(b x)^2}{6 x^6}-\frac{(2 b) \int \frac{e^{-b^2 x^2} \text{erfc}(b x)}{x^6} \, dx}{3 \sqrt{\pi }}\\ &=\frac{2 b e^{-b^2 x^2} \text{erfc}(b x)}{15 \sqrt{\pi } x^5}-\frac{\text{erfc}(b x)^2}{6 x^6}+\frac{\left (4 b^2\right ) \int \frac{e^{-2 b^2 x^2}}{x^5} \, dx}{15 \pi }+\frac{\left (4 b^3\right ) \int \frac{e^{-b^2 x^2} \text{erfc}(b x)}{x^4} \, dx}{15 \sqrt{\pi }}\\ &=-\frac{b^2 e^{-2 b^2 x^2}}{15 \pi x^4}+\frac{2 b e^{-b^2 x^2} \text{erfc}(b x)}{15 \sqrt{\pi } x^5}-\frac{4 b^3 e^{-b^2 x^2} \text{erfc}(b x)}{45 \sqrt{\pi } x^3}-\frac{\text{erfc}(b x)^2}{6 x^6}-\frac{\left (8 b^4\right ) \int \frac{e^{-2 b^2 x^2}}{x^3} \, dx}{45 \pi }-\frac{\left (4 b^4\right ) \int \frac{e^{-2 b^2 x^2}}{x^3} \, dx}{15 \pi }-\frac{\left (8 b^5\right ) \int \frac{e^{-b^2 x^2} \text{erfc}(b x)}{x^2} \, dx}{45 \sqrt{\pi }}\\ &=-\frac{b^2 e^{-2 b^2 x^2}}{15 \pi x^4}+\frac{2 b^4 e^{-2 b^2 x^2}}{9 \pi x^2}+\frac{2 b e^{-b^2 x^2} \text{erfc}(b x)}{15 \sqrt{\pi } x^5}-\frac{4 b^3 e^{-b^2 x^2} \text{erfc}(b x)}{45 \sqrt{\pi } x^3}+\frac{8 b^5 e^{-b^2 x^2} \text{erfc}(b x)}{45 \sqrt{\pi } x}-\frac{\text{erfc}(b x)^2}{6 x^6}+2 \frac{\left (16 b^6\right ) \int \frac{e^{-2 b^2 x^2}}{x} \, dx}{45 \pi }+\frac{\left (8 b^6\right ) \int \frac{e^{-2 b^2 x^2}}{x} \, dx}{15 \pi }+\frac{\left (16 b^7\right ) \int e^{-b^2 x^2} \text{erfc}(b x) \, dx}{45 \sqrt{\pi }}\\ &=-\frac{b^2 e^{-2 b^2 x^2}}{15 \pi x^4}+\frac{2 b^4 e^{-2 b^2 x^2}}{9 \pi x^2}+\frac{2 b e^{-b^2 x^2} \text{erfc}(b x)}{15 \sqrt{\pi } x^5}-\frac{4 b^3 e^{-b^2 x^2} \text{erfc}(b x)}{45 \sqrt{\pi } x^3}+\frac{8 b^5 e^{-b^2 x^2} \text{erfc}(b x)}{45 \sqrt{\pi } x}-\frac{\text{erfc}(b x)^2}{6 x^6}+\frac{28 b^6 \text{Ei}\left (-2 b^2 x^2\right )}{45 \pi }-\frac{1}{45} \left (8 b^6\right ) \operatorname{Subst}(\int x \, dx,x,\text{erfc}(b x))\\ &=-\frac{b^2 e^{-2 b^2 x^2}}{15 \pi x^4}+\frac{2 b^4 e^{-2 b^2 x^2}}{9 \pi x^2}+\frac{2 b e^{-b^2 x^2} \text{erfc}(b x)}{15 \sqrt{\pi } x^5}-\frac{4 b^3 e^{-b^2 x^2} \text{erfc}(b x)}{45 \sqrt{\pi } x^3}+\frac{8 b^5 e^{-b^2 x^2} \text{erfc}(b x)}{45 \sqrt{\pi } x}-\frac{4}{45} b^6 \text{erfc}(b x)^2-\frac{\text{erfc}(b x)^2}{6 x^6}+\frac{28 b^6 \text{Ei}\left (-2 b^2 x^2\right )}{45 \pi }\\ \end{align*}
Mathematica [A] time = 0.0418421, size = 133, normalized size = 0.75 \[ \frac{e^{-2 b^2 x^2} \left (4 \sqrt{\pi } b x e^{b^2 x^2} \left (4 b^4 x^4-2 b^2 x^2+3\right ) \text{Erfc}(b x)-\pi e^{2 b^2 x^2} \left (8 b^6 x^6+15\right ) \text{Erfc}(b x)^2+56 b^6 x^6 e^{2 b^2 x^2} \text{ExpIntegralEi}\left (-2 b^2 x^2\right )+20 b^4 x^4-6 b^2 x^2\right )}{90 \pi x^6} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.059, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\it erfc} \left ( bx \right ) \right ) ^{2}}{{x}^{7}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erfc}\left (b x\right )^{2}}{x^{7}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.17821, size = 396, normalized size = 2.24 \begin{align*} -\frac{15 \, \pi - 16 \, \pi \sqrt{b^{2}} b^{5} x^{6} \operatorname{erf}\left (\sqrt{b^{2}} x\right ) - 56 \, b^{6} x^{6}{\rm Ei}\left (-2 \, b^{2} x^{2}\right ) +{\left (15 \, \pi + 8 \, \pi b^{6} x^{6}\right )} \operatorname{erf}\left (b x\right )^{2} - 4 \, \sqrt{\pi }{\left (4 \, b^{5} x^{5} - 2 \, b^{3} x^{3} + 3 \, b x -{\left (4 \, b^{5} x^{5} - 2 \, b^{3} x^{3} + 3 \, b x\right )} \operatorname{erf}\left (b x\right )\right )} e^{\left (-b^{2} x^{2}\right )} - 30 \, \pi \operatorname{erf}\left (b x\right ) - 2 \,{\left (10 \, b^{4} x^{4} - 3 \, b^{2} x^{2}\right )} e^{\left (-2 \, b^{2} x^{2}\right )}}{90 \, \pi x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erfc}^{2}{\left (b x \right )}}{x^{7}}\, dx \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 )^{2}}{x^{7}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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