Integrand size = 16, antiderivative size = 43 \[ \int e^{-b^2 x^2} x \text {erf}(b x) \, dx=-\frac {e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}+\frac {\text {erf}\left (\sqrt {2} b x\right )}{2 \sqrt {2} b^2} \]
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Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6517, 2236} \[ \int e^{-b^2 x^2} x \text {erf}(b x) \, dx=\frac {\text {erf}\left (\sqrt {2} b x\right )}{2 \sqrt {2} b^2}-\frac {e^{-b^2 x^2} \text {erf}(b x)}{2 b^2} \]
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Rule 2236
Rule 6517
Rubi steps \begin{align*} \text {integral}& = -\frac {e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}+\frac {\int e^{-2 b^2 x^2} \, dx}{b \sqrt {\pi }} \\ & = -\frac {e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}+\frac {\text {erf}\left (\sqrt {2} b x\right )}{2 \sqrt {2} b^2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.91 \[ \int e^{-b^2 x^2} x \text {erf}(b x) \, dx=\frac {-2 e^{-b^2 x^2} \text {erf}(b x)+\sqrt {2} \text {erf}\left (\sqrt {2} b x\right )}{4 b^2} \]
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Time = 0.44 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.91
method | result | size |
default | \(\frac {-\frac {\operatorname {erf}\left (b x \right ) {\mathrm e}^{-b^{2} x^{2}}}{2 b}+\frac {\sqrt {2}\, \operatorname {erf}\left (b x \sqrt {2}\right )}{4 b}}{b}\) | \(39\) |
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Time = 0.25 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00 \[ \int e^{-b^2 x^2} x \text {erf}(b x) \, dx=-\frac {2 \, b \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} - \sqrt {2} \sqrt {b^{2}} \operatorname {erf}\left (\sqrt {2} \sqrt {b^{2}} x\right )}{4 \, b^{3}} \]
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\[ \int e^{-b^2 x^2} x \text {erf}(b x) \, dx=\int x e^{- b^{2} x^{2}} \operatorname {erf}{\left (b x \right )}\, dx \]
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Time = 0.30 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.79 \[ \int e^{-b^2 x^2} x \text {erf}(b x) \, dx=-\frac {\operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{2 \, b^{2}} + \frac {\sqrt {2} \operatorname {erf}\left (\sqrt {2} b x\right )}{4 \, b^{2}} \]
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Time = 0.28 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.81 \[ \int e^{-b^2 x^2} x \text {erf}(b x) \, dx=-\frac {\operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{2 \, b^{2}} - \frac {\sqrt {2} \operatorname {erf}\left (-\sqrt {2} b x\right )}{4 \, b^{2}} \]
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Time = 5.05 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00 \[ \int e^{-b^2 x^2} x \text {erf}(b x) \, dx=\frac {\sqrt {2}\,\mathrm {erf}\left (\sqrt {2}\,x\,\sqrt {b^2}\right )}{4\,b\,\sqrt {b^2}}-\frac {{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erf}\left (b\,x\right )}{2\,b^2} \]
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