Integrand size = 8, antiderivative size = 40 \[ \int \frac {\text {erfc}(b x)}{x^3} \, dx=\frac {b e^{-b^2 x^2}}{\sqrt {\pi } x}+b^2 \text {erf}(b x)-\frac {\text {erfc}(b x)}{2 x^2} \]
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Time = 0.03 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6497, 2245, 2236} \[ \int \frac {\text {erfc}(b x)}{x^3} \, dx=b^2 \text {erf}(b x)+\frac {b e^{-b^2 x^2}}{\sqrt {\pi } x}-\frac {\text {erfc}(b x)}{2 x^2} \]
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Rule 2236
Rule 2245
Rule 6497
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {erfc}(b x)}{2 x^2}-\frac {b \int \frac {e^{-b^2 x^2}}{x^2} \, dx}{\sqrt {\pi }} \\ & = \frac {b e^{-b^2 x^2}}{\sqrt {\pi } x}-\frac {\text {erfc}(b x)}{2 x^2}+\frac {\left (2 b^3\right ) \int e^{-b^2 x^2} \, dx}{\sqrt {\pi }} \\ & = \frac {b e^{-b^2 x^2}}{\sqrt {\pi } x}+b^2 \text {erf}(b x)-\frac {\text {erfc}(b x)}{2 x^2} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00 \[ \int \frac {\text {erfc}(b x)}{x^3} \, dx=\frac {b e^{-b^2 x^2}}{\sqrt {\pi } x}+b^2 \text {erf}(b x)-\frac {\text {erfc}(b x)}{2 x^2} \]
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Time = 0.17 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.05
method | result | size |
parts | \(-\frac {\operatorname {erfc}\left (b x \right )}{2 x^{2}}-\frac {b \left (-\frac {{\mathrm e}^{-b^{2} x^{2}}}{x}-b \sqrt {\pi }\, \operatorname {erf}\left (b x \right )\right )}{\sqrt {\pi }}\) | \(42\) |
parallelrisch | \(-\frac {2 x^{2} \operatorname {erfc}\left (b x \right ) \sqrt {\pi }\, b^{2}-2 \,{\mathrm e}^{-b^{2} x^{2}} b x +\operatorname {erfc}\left (b x \right ) \sqrt {\pi }}{2 \sqrt {\pi }\, x^{2}}\) | \(46\) |
derivativedivides | \(b^{2} \left (-\frac {\operatorname {erfc}\left (b x \right )}{2 b^{2} x^{2}}-\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}}}{b x}-\operatorname {erf}\left (b x \right ) \sqrt {\pi }}{\sqrt {\pi }}\right )\) | \(51\) |
default | \(b^{2} \left (-\frac {\operatorname {erfc}\left (b x \right )}{2 b^{2} x^{2}}-\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}}}{b x}-\operatorname {erf}\left (b x \right ) \sqrt {\pi }}{\sqrt {\pi }}\right )\) | \(51\) |
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Time = 0.26 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.08 \[ \int \frac {\text {erfc}(b x)}{x^3} \, dx=-\frac {\pi - 2 \, \sqrt {\pi } b x e^{\left (-b^{2} x^{2}\right )} - {\left (\pi + 2 \, \pi b^{2} x^{2}\right )} \operatorname {erf}\left (b x\right )}{2 \, \pi x^{2}} \]
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Time = 0.19 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.85 \[ \int \frac {\text {erfc}(b x)}{x^3} \, dx=- b^{2} \operatorname {erfc}{\left (b x \right )} + \frac {b e^{- b^{2} x^{2}}}{\sqrt {\pi } x} - \frac {\operatorname {erfc}{\left (b x \right )}}{2 x^{2}} \]
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Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.88 \[ \int \frac {\text {erfc}(b x)}{x^3} \, dx=\frac {b^{2} \sqrt {x^{2}} \Gamma \left (-\frac {1}{2}, b^{2} x^{2}\right )}{2 \, \sqrt {\pi } x} - \frac {\operatorname {erfc}\left (b x\right )}{2 \, x^{2}} \]
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\[ \int \frac {\text {erfc}(b x)}{x^3} \, dx=\int { \frac {\operatorname {erfc}\left (b x\right )}{x^{3}} \,d x } \]
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Time = 4.91 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.95 \[ \int \frac {\text {erfc}(b x)}{x^3} \, dx=-b^2\,\mathrm {erfc}\left (b\,x\right )-\frac {\frac {\mathrm {erfc}\left (b\,x\right )}{2}-\frac {b\,x\,{\mathrm {e}}^{-b^2\,x^2}}{\sqrt {\pi }}}{x^2} \]
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