Integrand size = 18, antiderivative size = 90 \[ \int e^{-b^2 x^2} x^3 \text {erfc}(b x) \, dx=\frac {e^{-2 b^2 x^2} x}{4 b^3 \sqrt {\pi }}-\frac {5 \text {erf}\left (\sqrt {2} b x\right )}{8 \sqrt {2} b^4}-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{2 b^4}-\frac {e^{-b^2 x^2} x^2 \text {erfc}(b x)}{2 b^2} \]
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Time = 0.08 (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 = {6521, 6518, 2236, 2243} \[ \int e^{-b^2 x^2} x^3 \text {erfc}(b x) \, dx=-\frac {5 \text {erf}\left (\sqrt {2} b x\right )}{8 \sqrt {2} b^4}-\frac {x^2 e^{-b^2 x^2} \text {erfc}(b x)}{2 b^2}-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{2 b^4}+\frac {x e^{-2 b^2 x^2}}{4 \sqrt {\pi } b^3} \]
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Rule 2236
Rule 2243
Rule 6518
Rule 6521
Rubi steps \begin{align*} \text {integral}& = -\frac {e^{-b^2 x^2} x^2 \text {erfc}(b x)}{2 b^2}+\frac {\int e^{-b^2 x^2} x \text {erfc}(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 {erfc}(b x)}{2 b^4}-\frac {e^{-b^2 x^2} x^2 \text {erfc}(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 {5 \text {erf}\left (\sqrt {2} b x\right )}{8 \sqrt {2} b^4}-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{2 b^4}-\frac {e^{-b^2 x^2} x^2 \text {erfc}(b x)}{2 b^2} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.77 \[ \int e^{-b^2 x^2} x^3 \text {erfc}(b x) \, dx=\frac {-5 \sqrt {2} \text {erf}\left (\sqrt {2} b x\right )+4 e^{-2 b^2 x^2} \left (\frac {b x}{\sqrt {\pi }}-2 e^{b^2 x^2} \left (1+b^2 x^2\right ) \text {erfc}(b x)\right )}{16 b^4} \]
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Time = 0.97 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.31
method | result | size |
default | \(\frac {\frac {-\frac {x^{2} {\mathrm e}^{-b^{2} x^{2}} b^{2}}{2}-\frac {{\mathrm e}^{-b^{2} x^{2}}}{2}}{b^{3}}-\frac {\operatorname {erf}\left (b x \right ) \left (-\frac {x^{2} {\mathrm e}^{-b^{2} x^{2}} b^{2}}{2}-\frac {{\mathrm e}^{-b^{2} x^{2}}}{2}\right )}{b^{3}}+\frac {-\frac {5 \sqrt {2}\, \sqrt {\pi }\, \operatorname {erf}\left (b x \sqrt {2}\right )}{16}+\frac {{\mathrm e}^{-2 b^{2} x^{2}} b x}{4}}{\sqrt {\pi }\, b^{3}}}{b}\) | \(118\) |
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Time = 0.26 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00 \[ \int e^{-b^2 x^2} x^3 \text {erfc}(b x) \, dx=\frac {4 \, \sqrt {\pi } b^{2} x e^{\left (-2 \, b^{2} x^{2}\right )} - 5 \, \sqrt {2} \pi \sqrt {b^{2}} \operatorname {erf}\left (\sqrt {2} \sqrt {b^{2}} x\right ) - 8 \, {\left (\pi b^{3} x^{2} + \pi b - {\left (\pi b^{3} x^{2} + \pi b\right )} \operatorname {erf}\left (b x\right )\right )} e^{\left (-b^{2} x^{2}\right )}}{16 \, \pi b^{5}} \]
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\[ \int e^{-b^2 x^2} x^3 \text {erfc}(b x) \, dx=\int x^{3} e^{- b^{2} x^{2}} \operatorname {erfc}{\left (b x \right )}\, dx \]
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\[ \int e^{-b^2 x^2} x^3 \text {erfc}(b x) \, dx=\int { x^{3} \operatorname {erfc}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} \,d x } \]
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\[ \int e^{-b^2 x^2} x^3 \text {erfc}(b x) \, dx=\int { x^{3} \operatorname {erfc}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} \,d x } \]
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Timed out. \[ \int e^{-b^2 x^2} x^3 \text {erfc}(b x) \, dx=\int x^3\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erfc}\left (b\,x\right ) \,d x \]
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