Optimal. Leaf size=67 \[ \frac {2 b e^{-b^2 x^2} \text {erfc}(b x)}{\sqrt {\pi } x}+b^2 \left (-\text {erfc}(b x)^2\right )+\frac {2 b^2 \text {Ei}\left (-2 b^2 x^2\right )}{\pi }-\frac {\text {erfc}(b x)^2}{2 x^2} \]
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Rubi [A] time = 0.09, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6365, 6392, 6374, 30, 2210} \[ \frac {2 b e^{-b^2 x^2} \text {Erfc}(b x)}{\sqrt {\pi } x}+b^2 \left (-\text {Erfc}(b x)^2\right )+\frac {2 b^2 \text {Ei}\left (-2 b^2 x^2\right )}{\pi }-\frac {\text {Erfc}(b x)^2}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2210
Rule 6365
Rule 6374
Rule 6392
Rubi steps
\begin {align*} \int \frac {\text {erfc}(b x)^2}{x^3} \, dx &=-\frac {\text {erfc}(b x)^2}{2 x^2}-\frac {(2 b) \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^2} \, dx}{\sqrt {\pi }}\\ &=\frac {2 b e^{-b^2 x^2} \text {erfc}(b x)}{\sqrt {\pi } x}-\frac {\text {erfc}(b x)^2}{2 x^2}+\frac {\left (4 b^2\right ) \int \frac {e^{-2 b^2 x^2}}{x} \, dx}{\pi }+\frac {\left (4 b^3\right ) \int e^{-b^2 x^2} \text {erfc}(b x) \, dx}{\sqrt {\pi }}\\ &=\frac {2 b e^{-b^2 x^2} \text {erfc}(b x)}{\sqrt {\pi } x}-\frac {\text {erfc}(b x)^2}{2 x^2}+\frac {2 b^2 \text {Ei}\left (-2 b^2 x^2\right )}{\pi }-\left (2 b^2\right ) \operatorname {Subst}(\int x \, dx,x,\text {erfc}(b x))\\ &=\frac {2 b e^{-b^2 x^2} \text {erfc}(b x)}{\sqrt {\pi } x}-b^2 \text {erfc}(b x)^2-\frac {\text {erfc}(b x)^2}{2 x^2}+\frac {2 b^2 \text {Ei}\left (-2 b^2 x^2\right )}{\pi }\\ \end {align*}
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Mathematica [A] time = 0.04, size = 63, normalized size = 0.94 \[ \frac {2 b e^{-b^2 x^2} \text {erfc}(b x)}{\sqrt {\pi } x}+\left (-b^2-\frac {1}{2 x^2}\right ) \text {erfc}(b x)^2+\frac {2 b^2 \text {Ei}\left (-2 b^2 x^2\right )}{\pi } \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 98, normalized size = 1.46 \[ -\frac {\pi - 4 \, \pi \sqrt {b^{2}} b x^{2} \operatorname {erf}\left (\sqrt {b^{2}} x\right ) - 4 \, b^{2} x^{2} {\rm Ei}\left (-2 \, b^{2} x^{2}\right ) + {\left (\pi + 2 \, \pi b^{2} x^{2}\right )} \operatorname {erf}\left (b x\right )^{2} + 4 \, \sqrt {\pi } {\left (b x \operatorname {erf}\left (b x\right ) - b x\right )} e^{\left (-b^{2} x^{2}\right )} - 2 \, \pi \operatorname {erf}\left (b x\right )}{2 \, \pi x^{2}} \]
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 \[ \int \frac {\operatorname {erfc}\left (b x\right )^{2}}{x^{3}}\,{d x} \]
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 \[ \int \frac {\mathrm {erfc}\left (b x \right )^{2}}{x^{3}}\, dx \]
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 \[ \int \frac {\operatorname {erfc}\left (b x\right )^{2}}{x^{3}}\,{d x} \]
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 \[ \int \frac {{\mathrm {erfc}\left (b\,x\right )}^2}{x^3} \,d x \]
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 \[ \int \frac {\operatorname {erfc}^{2}{\left (b x \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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