Optimal. Leaf size=125 \[ -\frac{2 b^3 e^{-b^2 x^2} \text{Erfc}(b x)}{3 \sqrt{\pi } x}+\frac{b e^{-b^2 x^2} \text{Erfc}(b x)}{3 \sqrt{\pi } x^3}+\frac{1}{3} b^4 \text{Erfc}(b x)^2-\frac{4 b^4 \text{ExpIntegralEi}\left (-2 b^2 x^2\right )}{3 \pi }-\frac{b^2 e^{-2 b^2 x^2}}{3 \pi x^2}-\frac{\text{Erfc}(b x)^2}{4 x^4} \]
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Rubi [A] time = 0.177593, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ -\frac{2 b^3 e^{-b^2 x^2} \text{Erfc}(b x)}{3 \sqrt{\pi } x}+\frac{b e^{-b^2 x^2} \text{Erfc}(b x)}{3 \sqrt{\pi } x^3}+\frac{1}{3} b^4 \text{Erfc}(b x)^2-\frac{4 b^4 \text{Ei}\left (-2 b^2 x^2\right )}{3 \pi }-\frac{b^2 e^{-2 b^2 x^2}}{3 \pi x^2}-\frac{\text{Erfc}(b x)^2}{4 x^4} \]
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^5} \, dx &=-\frac{\text{erfc}(b x)^2}{4 x^4}-\frac{b \int \frac{e^{-b^2 x^2} \text{erfc}(b x)}{x^4} \, dx}{\sqrt{\pi }}\\ &=\frac{b e^{-b^2 x^2} \text{erfc}(b x)}{3 \sqrt{\pi } x^3}-\frac{\text{erfc}(b x)^2}{4 x^4}+\frac{\left (2 b^2\right ) \int \frac{e^{-2 b^2 x^2}}{x^3} \, dx}{3 \pi }+\frac{\left (2 b^3\right ) \int \frac{e^{-b^2 x^2} \text{erfc}(b x)}{x^2} \, dx}{3 \sqrt{\pi }}\\ &=-\frac{b^2 e^{-2 b^2 x^2}}{3 \pi x^2}+\frac{b e^{-b^2 x^2} \text{erfc}(b x)}{3 \sqrt{\pi } x^3}-\frac{2 b^3 e^{-b^2 x^2} \text{erfc}(b x)}{3 \sqrt{\pi } x}-\frac{\text{erfc}(b x)^2}{4 x^4}-2 \frac{\left (4 b^4\right ) \int \frac{e^{-2 b^2 x^2}}{x} \, dx}{3 \pi }-\frac{\left (4 b^5\right ) \int e^{-b^2 x^2} \text{erfc}(b x) \, dx}{3 \sqrt{\pi }}\\ &=-\frac{b^2 e^{-2 b^2 x^2}}{3 \pi x^2}+\frac{b e^{-b^2 x^2} \text{erfc}(b x)}{3 \sqrt{\pi } x^3}-\frac{2 b^3 e^{-b^2 x^2} \text{erfc}(b x)}{3 \sqrt{\pi } x}-\frac{\text{erfc}(b x)^2}{4 x^4}-\frac{4 b^4 \text{Ei}\left (-2 b^2 x^2\right )}{3 \pi }+\frac{1}{3} \left (2 b^4\right ) \operatorname{Subst}(\int x \, dx,x,\text{erfc}(b x))\\ &=-\frac{b^2 e^{-2 b^2 x^2}}{3 \pi x^2}+\frac{b e^{-b^2 x^2} \text{erfc}(b x)}{3 \sqrt{\pi } x^3}-\frac{2 b^3 e^{-b^2 x^2} \text{erfc}(b x)}{3 \sqrt{\pi } x}+\frac{1}{3} b^4 \text{erfc}(b x)^2-\frac{\text{erfc}(b x)^2}{4 x^4}-\frac{4 b^4 \text{Ei}\left (-2 b^2 x^2\right )}{3 \pi }\\ \end{align*}
Mathematica [A] time = 0.0819507, size = 97, normalized size = 0.78 \[ \frac{-\frac{4 b x e^{-b^2 x^2} \left (2 b^2 x^2-1\right ) \text{Erfc}(b x)}{\sqrt{\pi }}+\left (4 b^4 x^4-3\right ) \text{Erfc}(b x)^2-\frac{4 b^2 x^2 \left (4 b^2 x^2 \text{ExpIntegralEi}\left (-2 b^2 x^2\right )+e^{-2 b^2 x^2}\right )}{\pi }}{12 x^4} \]
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}^{5}}}\, 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^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.14996, size = 329, normalized size = 2.63 \begin{align*} -\frac{3 \, \pi + 8 \, \pi \sqrt{b^{2}} b^{3} x^{4} \operatorname{erf}\left (\sqrt{b^{2}} x\right ) + 16 \, b^{4} x^{4}{\rm Ei}\left (-2 \, b^{2} x^{2}\right ) + 4 \, b^{2} x^{2} e^{\left (-2 \, b^{2} x^{2}\right )} +{\left (3 \, \pi - 4 \, \pi b^{4} x^{4}\right )} \operatorname{erf}\left (b x\right )^{2} + 4 \, \sqrt{\pi }{\left (2 \, b^{3} x^{3} - b x -{\left (2 \, b^{3} x^{3} - b x\right )} \operatorname{erf}\left (b x\right )\right )} e^{\left (-b^{2} x^{2}\right )} - 6 \, \pi \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erfc}^{2}{\left (b x \right )}}{x^{5}}\, 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^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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