Integrand size = 18, antiderivative size = 90 \[ \int e^{-b^2 x^2} x^3 \text {erf}(b x) \, dx=-\frac {e^{-2 b^2 x^2} x}{4 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erf}(b x)}{2 b^4}-\frac {e^{-b^2 x^2} x^2 \text {erf}(b x)}{2 b^2}+\frac {5 \text {erf}\left (\sqrt {2} b x\right )}{8 \sqrt {2} b^4} \]
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Time = 0.07 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6520, 6517, 2236, 2243} \[ \int e^{-b^2 x^2} x^3 \text {erf}(b x) \, dx=\frac {5 \text {erf}\left (\sqrt {2} b x\right )}{8 \sqrt {2} b^4}-\frac {x^2 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}-\frac {e^{-b^2 x^2} \text {erf}(b x)}{2 b^4}-\frac {x e^{-2 b^2 x^2}}{4 \sqrt {\pi } b^3} \]
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Rule 2236
Rule 2243
Rule 6517
Rule 6520
Rubi steps \begin{align*} \text {integral}& = -\frac {e^{-b^2 x^2} x^2 \text {erf}(b x)}{2 b^2}+\frac {\int e^{-b^2 x^2} x \text {erf}(b x) \, dx}{b^2}+\frac {\int e^{-2 b^2 x^2} x^2 \, dx}{b \sqrt {\pi }} \\ & = -\frac {e^{-2 b^2 x^2} x}{4 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erf}(b x)}{2 b^4}-\frac {e^{-b^2 x^2} x^2 \text {erf}(b x)}{2 b^2}+\frac {\int e^{-2 b^2 x^2} \, dx}{4 b^3 \sqrt {\pi }}+\frac {\int e^{-2 b^2 x^2} \, dx}{b^3 \sqrt {\pi }} \\ & = -\frac {e^{-2 b^2 x^2} x}{4 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erf}(b x)}{2 b^4}-\frac {e^{-b^2 x^2} x^2 \text {erf}(b x)}{2 b^2}+\frac {5 \text {erf}\left (\sqrt {2} b x\right )}{8 \sqrt {2} b^4} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.76 \[ \int e^{-b^2 x^2} x^3 \text {erf}(b x) \, dx=\frac {-\frac {4 b e^{-2 b^2 x^2} x}{\sqrt {\pi }}-8 e^{-b^2 x^2} \left (1+b^2 x^2\right ) \text {erf}(b x)+5 \sqrt {2} \text {erf}\left (\sqrt {2} b x\right )}{16 b^4} \]
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Time = 0.91 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.92
method | result | size |
default | \(\frac {\frac {\operatorname {erf}\left (b x \right ) \left (-\frac {x^{2} {\mathrm e}^{-b^{2} x^{2}} b^{2}}{2}-\frac {{\mathrm e}^{-b^{2} x^{2}}}{2}\right )}{b^{3}}-\frac {-\frac {5 \sqrt {2}\, \sqrt {\pi }\, \operatorname {erf}\left (b x \sqrt {2}\right )}{16}+\frac {{\mathrm e}^{-2 b^{2} x^{2}} b x}{4}}{\sqrt {\pi }\, b^{3}}}{b}\) | \(83\) |
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Time = 0.26 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.84 \[ \int e^{-b^2 x^2} x^3 \text {erf}(b x) \, dx=-\frac {4 \, \sqrt {\pi } b^{2} x e^{\left (-2 \, b^{2} x^{2}\right )} - 5 \, \sqrt {2} \pi \sqrt {b^{2}} \operatorname {erf}\left (\sqrt {2} \sqrt {b^{2}} x\right ) + 8 \, {\left (\pi b^{3} x^{2} + \pi b\right )} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{16 \, \pi b^{5}} \]
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\[ \int e^{-b^2 x^2} x^3 \text {erf}(b x) \, dx=\int x^{3} e^{- b^{2} x^{2}} \operatorname {erf}{\left (b x \right )}\, dx \]
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\[ \int e^{-b^2 x^2} x^3 \text {erf}(b x) \, dx=\int { x^{3} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.04 \[ \int e^{-b^2 x^2} x^3 \text {erf}(b x) \, dx=-\frac {{\left (b^{2} x^{2} + 1\right )} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{2 \, b^{4}} - \frac {b^{2} {\left (\frac {4 \, x e^{\left (-2 \, b^{2} x^{2}\right )}}{b^{2}} + \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {2} b x\right )}{b^{3}}\right )} + \frac {4 \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {2} b x\right )}{b}}{16 \, \sqrt {\pi } b^{3}} \]
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Time = 5.29 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.18 \[ \int e^{-b^2 x^2} x^3 \text {erf}(b x) \, dx=\frac {\sqrt {2}\,\mathrm {erf}\left (\sqrt {2}\,x\,\sqrt {b^2}\right )}{4\,b\,{\left (b^2\right )}^{3/2}}-\frac {\mathrm {erfi}\left (\sqrt {2}\,x\,\sqrt {-b^2}\right )}{4\,b\,{\left (-2\,b^2\right )}^{3/2}}-\mathrm {erf}\left (b\,x\right )\,\left (\frac {{\mathrm {e}}^{-b^2\,x^2}}{2\,b^4}+\frac {x^2\,{\mathrm {e}}^{-b^2\,x^2}}{2\,b^2}\right )-\frac {x\,{\mathrm {e}}^{-2\,b^2\,x^2}}{4\,b^3\,\sqrt {\pi }} \]
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