Integrand size = 40, antiderivative size = 62 \[ \int \left (\frac {e^{-b^2 x^2} \text {erf}(b x)}{x^3}+\frac {b^2 e^{-b^2 x^2} \text {erf}(b x)}{x}\right ) \, dx=-\frac {b e^{-2 b^2 x^2}}{\sqrt {\pi } x}-\frac {e^{-b^2 x^2} \text {erf}(b x)}{2 x^2}-\sqrt {2} b^2 \text {erf}\left (\sqrt {2} b x\right ) \]
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Time = 0.11 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {6526, 2245, 2236} \[ \int \left (\frac {e^{-b^2 x^2} \text {erf}(b x)}{x^3}+\frac {b^2 e^{-b^2 x^2} \text {erf}(b x)}{x}\right ) \, dx=-\frac {e^{-b^2 x^2} \text {erf}(b x)}{2 x^2}-\sqrt {2} b^2 \text {erf}\left (\sqrt {2} b x\right )-\frac {b e^{-2 b^2 x^2}}{\sqrt {\pi } x} \]
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Rule 2236
Rule 2245
Rule 6526
Rubi steps \begin{align*} \text {integral}& = b^2 \int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x} \, dx+\int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x^3} \, dx \\ & = -\frac {e^{-b^2 x^2} \text {erf}(b x)}{2 x^2}+\frac {b \int \frac {e^{-2 b^2 x^2}}{x^2} \, dx}{\sqrt {\pi }} \\ & = -\frac {b e^{-2 b^2 x^2}}{\sqrt {\pi } x}-\frac {e^{-b^2 x^2} \text {erf}(b x)}{2 x^2}-\frac {\left (4 b^3\right ) \int e^{-2 b^2 x^2} \, dx}{\sqrt {\pi }} \\ & = -\frac {b e^{-2 b^2 x^2}}{\sqrt {\pi } x}-\frac {e^{-b^2 x^2} \text {erf}(b x)}{2 x^2}-\sqrt {2} b^2 \text {erf}\left (\sqrt {2} b x\right ) \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00 \[ \int \left (\frac {e^{-b^2 x^2} \text {erf}(b x)}{x^3}+\frac {b^2 e^{-b^2 x^2} \text {erf}(b x)}{x}\right ) \, dx=-\frac {b e^{-2 b^2 x^2}}{\sqrt {\pi } x}-\frac {e^{-b^2 x^2} \text {erf}(b x)}{2 x^2}-\sqrt {2} b^2 \text {erf}\left (\sqrt {2} b x\right ) \]
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Time = 1.70 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.08
method | result | size |
default | \(\frac {-\frac {\operatorname {erf}\left (b x \right ) b \,{\mathrm e}^{-b^{2} x^{2}}}{2 x^{2}}+\frac {b^{3} \left (-\frac {{\mathrm e}^{-2 b^{2} x^{2}}}{b x}-\sqrt {2}\, \sqrt {\pi }\, \operatorname {erf}\left (b x \sqrt {2}\right )\right )}{\sqrt {\pi }}}{b}\) | \(67\) |
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none
Time = 0.26 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.06 \[ \int \left (\frac {e^{-b^2 x^2} \text {erf}(b x)}{x^3}+\frac {b^2 e^{-b^2 x^2} \text {erf}(b x)}{x}\right ) \, dx=-\frac {2 \, \sqrt {2} \pi \sqrt {b^{2}} b x^{2} \operatorname {erf}\left (\sqrt {2} \sqrt {b^{2}} x\right ) + 2 \, \sqrt {\pi } b x e^{\left (-2 \, b^{2} x^{2}\right )} + \pi \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{2 \, \pi x^{2}} \]
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\[ \int \left (\frac {e^{-b^2 x^2} \text {erf}(b x)}{x^3}+\frac {b^2 e^{-b^2 x^2} \text {erf}(b x)}{x}\right ) \, dx=\int \frac {\left (b^{2} x^{2} + 1\right ) e^{- b^{2} x^{2}} \operatorname {erf}{\left (b x \right )}}{x^{3}}\, dx \]
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\[ \int \left (\frac {e^{-b^2 x^2} \text {erf}(b x)}{x^3}+\frac {b^2 e^{-b^2 x^2} \text {erf}(b x)}{x}\right ) \, dx=\int { \frac {b^{2} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x} + \frac {\operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{3}} \,d x } \]
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\[ \int \left (\frac {e^{-b^2 x^2} \text {erf}(b x)}{x^3}+\frac {b^2 e^{-b^2 x^2} \text {erf}(b x)}{x}\right ) \, dx=\int { \frac {b^{2} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x} + \frac {\operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{3}} \,d x } \]
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Time = 5.00 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.84 \[ \int \left (\frac {e^{-b^2 x^2} \text {erf}(b x)}{x^3}+\frac {b^2 e^{-b^2 x^2} \text {erf}(b x)}{x}\right ) \, dx=-\frac {\frac {{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erf}\left (b\,x\right )}{2}+\frac {b\,x\,{\mathrm {e}}^{-2\,b^2\,x^2}}{\sqrt {\pi }}}{x^2}-\sqrt {2}\,b^2\,\mathrm {erf}\left (\sqrt {2}\,b\,x\right ) \]
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