Integrand size = 40, antiderivative size = 33 \[ \int \left (\frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^3}+\frac {b^2 e^{-b^2 x^2} \text {erfi}(b x)}{x}\right ) \, dx=-\frac {b}{\sqrt {\pi } x}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{2 x^2} \]
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Time = 0.08 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {6528, 6525, 30} \[ \int \left (\frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^3}+\frac {b^2 e^{-b^2 x^2} \text {erfi}(b x)}{x}\right ) \, dx=-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{2 x^2}-\frac {b}{\sqrt {\pi } x} \]
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Rule 30
Rule 6525
Rule 6528
Rubi steps \begin{align*} \text {integral}& = b^2 \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x} \, dx+\int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^3} \, dx \\ & = -\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 }}-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} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \left (\frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^3}+\frac {b^2 e^{-b^2 x^2} \text {erfi}(b x)}{x}\right ) \, dx=-\frac {b}{\sqrt {\pi } x}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{2 x^2} \]
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Time = 0.28 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.24
method | result | size |
default | \(\frac {\left (-2 \,{\mathrm e}^{b^{2} x^{2}} b x -\operatorname {erfi}\left (b x \right ) \sqrt {\pi }\right ) {\mathrm e}^{-b^{2} x^{2}}}{2 \sqrt {\pi }\, x^{2}}\) | \(41\) |
parallelrisch | \(\frac {\left (-2 \,{\mathrm e}^{b^{2} x^{2}} b x -\operatorname {erfi}\left (b x \right ) \sqrt {\pi }\right ) {\mathrm e}^{-b^{2} x^{2}}}{2 \sqrt {\pi }\, x^{2}}\) | \(41\) |
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Time = 0.26 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.18 \[ \int \left (\frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^3}+\frac {b^2 e^{-b^2 x^2} \text {erfi}(b x)}{x}\right ) \, dx=-\frac {{\left (2 \, \sqrt {\pi } b x e^{\left (b^{2} x^{2}\right )} + \pi \operatorname {erfi}\left (b x\right )\right )} e^{\left (-b^{2} x^{2}\right )}}{2 \, \pi x^{2}} \]
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Time = 10.51 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.61 \[ \int \left (\frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^3}+\frac {b^2 e^{-b^2 x^2} \text {erfi}(b x)}{x}\right ) \, dx=\frac {2 b^{3} x {{}_{2}F_{2}\left (\begin {matrix} \frac {1}{2}, 1 \\ \frac {3}{2}, \frac {3}{2} \end {matrix}\middle | {- b^{2} x^{2}} \right )}}{\sqrt {\pi }} - \frac {2 b {{}_{2}F_{2}\left (\begin {matrix} - \frac {1}{2}, 1 \\ \frac {1}{2}, \frac {3}{2} \end {matrix}\middle | {- b^{2} x^{2}} \right )}}{\sqrt {\pi } x} \]
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\[ \int \left (\frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^3}+\frac {b^2 e^{-b^2 x^2} \text {erfi}(b x)}{x}\right ) \, dx=\int { \frac {b^{2} \operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x} + \frac {\operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{3}} \,d x } \]
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\[ \int \left (\frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^3}+\frac {b^2 e^{-b^2 x^2} \text {erfi}(b x)}{x}\right ) \, dx=\int { \frac {b^{2} \operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x} + \frac {\operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{3}} \,d x } \]
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Time = 0.21 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85 \[ \int \left (\frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^3}+\frac {b^2 e^{-b^2 x^2} \text {erfi}(b x)}{x}\right ) \, dx=-\frac {{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erfi}\left (b\,x\right )}{2\,x^2}-\frac {b}{x\,\sqrt {\pi }} \]
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