Optimal. Leaf size=125 \[ \frac {1}{3} b^4 \text {erfc}(b x)^2+\frac {b e^{-b^2 x^2} \text {erfc}(b x)}{3 \sqrt {\pi } x^3}-\frac {b^2 e^{-2 b^2 x^2}}{3 \pi x^2}-\frac {4 b^4 \text {Ei}\left (-2 b^2 x^2\right )}{3 \pi }-\frac {2 b^3 e^{-b^2 x^2} \text {erfc}(b x)}{3 \sqrt {\pi } x}-\frac {\text {erfc}(b x)^2}{4 x^4} \]
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Rubi [A] time = 0.18, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, 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 30
Rule 2210
Rule 2214
Rule 6365
Rule 6374
Rule 6392
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*}
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Mathematica [A] time = 0.07, size = 97, normalized size = 0.78 \[ \frac {\left (4 b^4 x^4-3\right ) \text {erfc}(b x)^2-\frac {4 b x e^{-b^2 x^2} \left (2 b^2 x^2-1\right ) \text {erfc}(b x)}{\sqrt {\pi }}-\frac {4 b^2 x^2 \left (4 b^2 x^2 \text {Ei}\left (-2 b^2 x^2\right )+e^{-2 b^2 x^2}\right )}{\pi }}{12 x^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 141, normalized size = 1.13 \[ -\frac {3 \, \pi + 8 \, \pi \sqrt {b^{2}} b^{3} x^{4} \operatorname {erf}\left (\sqrt {b^{2}} x\right ) + 16 \, b^{4} x^{4} {\rm Ei}\left (-2 \, b^{2} x^{2}\right ) + 4 \, b^{2} x^{2} e^{\left (-2 \, b^{2} x^{2}\right )} + {\left (3 \, \pi - 4 \, \pi b^{4} x^{4}\right )} \operatorname {erf}\left (b x\right )^{2} + 4 \, \sqrt {\pi } {\left (2 \, b^{3} x^{3} - b x - {\left (2 \, b^{3} x^{3} - b x\right )} \operatorname {erf}\left (b x\right )\right )} e^{\left (-b^{2} x^{2}\right )} - 6 \, \pi \operatorname {erf}\left (b x\right )}{12 \, \pi x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {erfc}\left (b x\right )^{2}}{x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {erfc}\left (b x \right )^{2}}{x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {erfc}\left (b x\right )^{2}}{x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {erfc}\left (b\,x\right )}^2}{x^5} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {erfc}^{2}{\left (b x \right )}}{x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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