Integrand size = 18, antiderivative size = 53 \[ \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^2} \, dx=-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{x}+\frac {1}{2} b \sqrt {\pi } \text {erfc}(b x)^2-\frac {b \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{\sqrt {\pi }} \]
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Time = 0.05 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6527, 6509, 30, 2241} \[ \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^2} \, dx=-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{x}-\frac {b \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{\sqrt {\pi }}+\frac {1}{2} \sqrt {\pi } b \text {erfc}(b x)^2 \]
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Rule 30
Rule 2241
Rule 6509
Rule 6527
Rubi steps \begin{align*} \text {integral}& = -\frac {e^{-b^2 x^2} \text {erfc}(b x)}{x}-\left (2 b^2\right ) \int e^{-b^2 x^2} \text {erfc}(b x) \, dx-\frac {(2 b) \int \frac {e^{-2 b^2 x^2}}{x} \, dx}{\sqrt {\pi }} \\ & = -\frac {e^{-b^2 x^2} \text {erfc}(b x)}{x}-\frac {b \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{\sqrt {\pi }}+\left (b \sqrt {\pi }\right ) \text {Subst}(\int x \, dx,x,\text {erfc}(b x)) \\ & = -\frac {e^{-b^2 x^2} \text {erfc}(b x)}{x}+\frac {1}{2} b \sqrt {\pi } \text {erfc}(b x)^2-\frac {b \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{\sqrt {\pi }} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^2} \, dx=-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{x}+\frac {1}{2} b \sqrt {\pi } \text {erfc}(b x)^2-\frac {b \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{\sqrt {\pi }} \]
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\[\int \frac {\operatorname {erfc}\left (b x \right ) {\mathrm e}^{-b^{2} x^{2}}}{x^{2}}d x\]
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none
Time = 0.26 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.45 \[ \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^2} \, dx=-\frac {2 \, \pi ^{\frac {3}{2}} \sqrt {b^{2}} x \operatorname {erf}\left (\sqrt {b^{2}} x\right ) + 2 \, {\left (\pi - \pi \operatorname {erf}\left (b x\right )\right )} e^{\left (-b^{2} x^{2}\right )} - \sqrt {\pi } {\left (\pi b x \operatorname {erf}\left (b x\right )^{2} - 2 \, b x {\rm Ei}\left (-2 \, b^{2} x^{2}\right )\right )}}{2 \, \pi x} \]
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\[ \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^2} \, dx=\int \frac {e^{- b^{2} x^{2}} \operatorname {erfc}{\left (b x \right )}}{x^{2}}\, dx \]
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\[ \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^2} \, dx=\int { \frac {\operatorname {erfc}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{2}} \,d x } \]
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\[ \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^2} \, dx=\int { \frac {\operatorname {erfc}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^2} \, dx=\int \frac {{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erfc}\left (b\,x\right )}{x^2} \,d x \]
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