Integrand size = 18, antiderivative size = 65 \[ \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^3} \, dx=-\frac {b}{\sqrt {\pi } x}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{2 x^2}-\frac {2 b^3 x \, _2F_2\left (\frac {1}{2},1;\frac {3}{2},\frac {3}{2};-b^2 x^2\right )}{\sqrt {\pi }} \]
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Time = 0.05 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6528, 6525, 30} \[ \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^3} \, dx=-\frac {2 b^3 x \, _2F_2\left (\frac {1}{2},1;\frac {3}{2},\frac {3}{2};-b^2 x^2\right )}{\sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{2 x^2}-\frac {b}{\sqrt {\pi } x} \]
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Rule 30
Rule 6525
Rule 6528
Rubi steps \begin{align*} \text {integral}& = -\frac {e^{-b^2 x^2} \text {erfi}(b x)}{2 x^2}-b^2 \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x} \, dx+\frac {b \int \frac {1}{x^2} \, dx}{\sqrt {\pi }} \\ & = -\frac {b}{\sqrt {\pi } x}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{2 x^2}-\frac {2 b^3 x \, _2F_2\left (\frac {1}{2},1;\frac {3}{2},\frac {3}{2};-b^2 x^2\right )}{\sqrt {\pi }} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.49 \[ \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^3} \, dx=-\frac {2 b \, _2F_2\left (-\frac {1}{2},1;\frac {1}{2},\frac {3}{2};-b^2 x^2\right )}{\sqrt {\pi } x} \]
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\[\int \frac {\operatorname {erfi}\left (b x \right ) {\mathrm e}^{-b^{2} x^{2}}}{x^{3}}d x\]
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\[ \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^3} \, dx=\int { \frac {\operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{3}} \,d x } \]
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Time = 7.39 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.42 \[ \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^3} \, dx=- \frac {2 b {{}_{2}F_{2}\left (\begin {matrix} - \frac {1}{2}, 1 \\ \frac {1}{2}, \frac {3}{2} \end {matrix}\middle | {- b^{2} x^{2}} \right )}}{\sqrt {\pi } x} \]
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\[ \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^3} \, dx=\int { \frac {\operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{3}} \,d x } \]
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\[ \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^3} \, dx=\int { \frac {\operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^3} \, dx=\int \frac {{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erfi}\left (b\,x\right )}{x^3} \,d x \]
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