Integrand size = 8, antiderivative size = 81 \[ \int \frac {\text {erfc}(b x)}{x^6} \, dx=\frac {b e^{-b^2 x^2}}{10 \sqrt {\pi } x^4}-\frac {b^3 e^{-b^2 x^2}}{10 \sqrt {\pi } x^2}-\frac {\text {erfc}(b x)}{5 x^5}-\frac {b^5 \operatorname {ExpIntegralEi}\left (-b^2 x^2\right )}{10 \sqrt {\pi }} \]
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Time = 0.05 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6497, 2245, 2241} \[ \int \frac {\text {erfc}(b x)}{x^6} \, dx=\frac {b e^{-b^2 x^2}}{10 \sqrt {\pi } x^4}-\frac {b^5 \operatorname {ExpIntegralEi}\left (-b^2 x^2\right )}{10 \sqrt {\pi }}-\frac {b^3 e^{-b^2 x^2}}{10 \sqrt {\pi } x^2}-\frac {\text {erfc}(b x)}{5 x^5} \]
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Rule 2241
Rule 2245
Rule 6497
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {erfc}(b x)}{5 x^5}-\frac {(2 b) \int \frac {e^{-b^2 x^2}}{x^5} \, dx}{5 \sqrt {\pi }} \\ & = \frac {b e^{-b^2 x^2}}{10 \sqrt {\pi } x^4}-\frac {\text {erfc}(b x)}{5 x^5}+\frac {b^3 \int \frac {e^{-b^2 x^2}}{x^3} \, dx}{5 \sqrt {\pi }} \\ & = \frac {b e^{-b^2 x^2}}{10 \sqrt {\pi } x^4}-\frac {b^3 e^{-b^2 x^2}}{10 \sqrt {\pi } x^2}-\frac {\text {erfc}(b x)}{5 x^5}-\frac {b^5 \int \frac {e^{-b^2 x^2}}{x} \, dx}{5 \sqrt {\pi }} \\ & = \frac {b e^{-b^2 x^2}}{10 \sqrt {\pi } x^4}-\frac {b^3 e^{-b^2 x^2}}{10 \sqrt {\pi } x^2}-\frac {\text {erfc}(b x)}{5 x^5}-\frac {b^5 \operatorname {ExpIntegralEi}\left (-b^2 x^2\right )}{10 \sqrt {\pi }} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.90 \[ \int \frac {\text {erfc}(b x)}{x^6} \, dx=e^{-b^2 x^2} \left (\frac {b}{10 \sqrt {\pi } x^4}-\frac {b^3}{10 \sqrt {\pi } x^2}\right )-\frac {\text {erfc}(b x)}{5 x^5}-\frac {b^5 \operatorname {ExpIntegralEi}\left (-b^2 x^2\right )}{10 \sqrt {\pi }} \]
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Time = 0.66 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.81
method | result | size |
parts | \(-\frac {\operatorname {erfc}\left (b x \right )}{5 x^{5}}-\frac {2 b \left (-\frac {{\mathrm e}^{-b^{2} x^{2}}}{4 x^{4}}-\frac {b^{2} \left (-\frac {{\mathrm e}^{-b^{2} x^{2}}}{2 x^{2}}+\frac {b^{2} \operatorname {Ei}_{1}\left (b^{2} x^{2}\right )}{2}\right )}{2}\right )}{5 \sqrt {\pi }}\) | \(66\) |
derivativedivides | \(b^{5} \left (-\frac {\operatorname {erfc}\left (b x \right )}{5 b^{5} x^{5}}-\frac {2 \left (-\frac {{\mathrm e}^{-b^{2} x^{2}}}{4 b^{4} x^{4}}+\frac {{\mathrm e}^{-b^{2} x^{2}}}{4 x^{2} b^{2}}-\frac {\operatorname {Ei}_{1}\left (b^{2} x^{2}\right )}{4}\right )}{5 \sqrt {\pi }}\right )\) | \(71\) |
default | \(b^{5} \left (-\frac {\operatorname {erfc}\left (b x \right )}{5 b^{5} x^{5}}-\frac {2 \left (-\frac {{\mathrm e}^{-b^{2} x^{2}}}{4 b^{4} x^{4}}+\frac {{\mathrm e}^{-b^{2} x^{2}}}{4 x^{2} b^{2}}-\frac {\operatorname {Ei}_{1}\left (b^{2} x^{2}\right )}{4}\right )}{5 \sqrt {\pi }}\right )\) | \(71\) |
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Time = 0.25 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.77 \[ \int \frac {\text {erfc}(b x)}{x^6} \, dx=-\frac {2 \, \pi - 2 \, \pi \operatorname {erf}\left (b x\right ) + \sqrt {\pi } {\left (b^{5} x^{5} {\rm Ei}\left (-b^{2} x^{2}\right ) + {\left (b^{3} x^{3} - b x\right )} e^{\left (-b^{2} x^{2}\right )}\right )}}{10 \, \pi x^{5}} \]
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Time = 1.84 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.86 \[ \int \frac {\text {erfc}(b x)}{x^6} \, dx=\frac {b^{5} \operatorname {E}_{1}\left (b^{2} x^{2}\right )}{10 \sqrt {\pi }} - \frac {b^{3} e^{- b^{2} x^{2}}}{10 \sqrt {\pi } x^{2}} + \frac {b e^{- b^{2} x^{2}}}{10 \sqrt {\pi } x^{4}} - \frac {\operatorname {erfc}{\left (b x \right )}}{5 x^{5}} \]
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Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.33 \[ \int \frac {\text {erfc}(b x)}{x^6} \, dx=\frac {b^{5} \Gamma \left (-2, b^{2} x^{2}\right )}{5 \, \sqrt {\pi }} - \frac {\operatorname {erfc}\left (b x\right )}{5 \, x^{5}} \]
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\[ \int \frac {\text {erfc}(b x)}{x^6} \, dx=\int { \frac {\operatorname {erfc}\left (b x\right )}{x^{6}} \,d x } \]
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Time = 4.73 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.81 \[ \int \frac {\text {erfc}(b x)}{x^6} \, dx=-\frac {\frac {\mathrm {erfc}\left (b\,x\right )}{5}+\frac {b^3\,x^3\,{\mathrm {e}}^{-b^2\,x^2}}{10\,\sqrt {\pi }}-\frac {b\,x\,{\mathrm {e}}^{-b^2\,x^2}}{10\,\sqrt {\pi }}}{x^5}-\frac {b^5\,\mathrm {ei}\left (-b^2\,x^2\right )}{10\,\sqrt {\pi }} \]
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