Integrand size = 10, antiderivative size = 125 \[ \int \frac {\text {erfc}(b x)^2}{x^5} \, dx=-\frac {b^2 e^{-2 b^2 x^2}}{3 \pi x^2}+\frac {b e^{-b^2 x^2} \text {erfc}(b x)}{3 \sqrt {\pi } x^3}-\frac {2 b^3 e^{-b^2 x^2} \text {erfc}(b x)}{3 \sqrt {\pi } x}+\frac {1}{3} b^4 \text {erfc}(b x)^2-\frac {\text {erfc}(b x)^2}{4 x^4}-\frac {4 b^4 \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{3 \pi } \]
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Time = 0.13 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6500, 6527, 6509, 30, 2241, 2245} \[ \int \frac {\text {erfc}(b x)^2}{x^5} \, dx=\frac {1}{3} b^4 \text {erfc}(b x)^2+\frac {b e^{-b^2 x^2} \text {erfc}(b x)}{3 \sqrt {\pi } x^3}-\frac {b^2 e^{-2 b^2 x^2}}{3 \pi x^2}-\frac {4 b^4 \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{3 \pi }-\frac {2 b^3 e^{-b^2 x^2} \text {erfc}(b x)}{3 \sqrt {\pi } x}-\frac {\text {erfc}(b x)^2}{4 x^4} \]
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Rule 30
Rule 2241
Rule 2245
Rule 6500
Rule 6509
Rule 6527
Rubi steps \begin{align*} \text {integral}& = -\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 \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{3 \pi }+\frac {1}{3} \left (2 b^4\right ) \text {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 \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{3 \pi } \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.78 \[ \int \frac {\text {erfc}(b x)^2}{x^5} \, dx=\frac {-\frac {4 b e^{-b^2 x^2} x \left (-1+2 b^2 x^2\right ) \text {erfc}(b x)}{\sqrt {\pi }}+\left (-3+4 b^4 x^4\right ) \text {erfc}(b x)^2-\frac {4 b^2 x^2 \left (e^{-2 b^2 x^2}+4 b^2 x^2 \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )\right )}{\pi }}{12 x^4} \]
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\[\int \frac {\operatorname {erfc}\left (b x \right )^{2}}{x^{5}}d x\]
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none
Time = 0.26 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.13 \[ \int \frac {\text {erfc}(b x)^2}{x^5} \, dx=-\frac {3 \, \pi + 8 \, \pi \sqrt {b^{2}} b^{3} x^{4} \operatorname {erf}\left (\sqrt {b^{2}} x\right ) + 16 \, b^{4} x^{4} {\rm Ei}\left (-2 \, b^{2} x^{2}\right ) + 4 \, b^{2} x^{2} e^{\left (-2 \, b^{2} x^{2}\right )} + {\left (3 \, \pi - 4 \, \pi b^{4} x^{4}\right )} \operatorname {erf}\left (b x\right )^{2} + 4 \, \sqrt {\pi } {\left (2 \, b^{3} x^{3} - b x - {\left (2 \, b^{3} x^{3} - b x\right )} \operatorname {erf}\left (b x\right )\right )} e^{\left (-b^{2} x^{2}\right )} - 6 \, \pi \operatorname {erf}\left (b x\right )}{12 \, \pi x^{4}} \]
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\[ \int \frac {\text {erfc}(b x)^2}{x^5} \, dx=\int \frac {\operatorname {erfc}^{2}{\left (b x \right )}}{x^{5}}\, dx \]
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\[ \int \frac {\text {erfc}(b x)^2}{x^5} \, dx=\int { \frac {\operatorname {erfc}\left (b x\right )^{2}}{x^{5}} \,d x } \]
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\[ \int \frac {\text {erfc}(b x)^2}{x^5} \, dx=\int { \frac {\operatorname {erfc}\left (b x\right )^{2}}{x^{5}} \,d x } \]
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Timed out. \[ \int \frac {\text {erfc}(b x)^2}{x^5} \, dx=\int \frac {{\mathrm {erfc}\left (b\,x\right )}^2}{x^5} \,d x \]
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