Integrand size = 18, antiderivative size = 135 \[ \int e^{-b^2 x^2} x^5 \text {erf}(b x) \, dx=-\frac {11 e^{-2 b^2 x^2} x}{16 b^5 \sqrt {\pi }}-\frac {e^{-2 b^2 x^2} x^3}{4 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erf}(b x)}{b^6}-\frac {e^{-b^2 x^2} x^2 \text {erf}(b x)}{b^4}-\frac {e^{-b^2 x^2} x^4 \text {erf}(b x)}{2 b^2}+\frac {43 \text {erf}\left (\sqrt {2} b x\right )}{32 \sqrt {2} b^6} \]
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Time = 0.15 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6520, 6517, 2236, 2243} \[ \int e^{-b^2 x^2} x^5 \text {erf}(b x) \, dx=\frac {43 \text {erf}\left (\sqrt {2} b x\right )}{32 \sqrt {2} b^6}-\frac {x^4 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}-\frac {e^{-b^2 x^2} \text {erf}(b x)}{b^6}-\frac {11 x e^{-2 b^2 x^2}}{16 \sqrt {\pi } b^5}-\frac {x^2 e^{-b^2 x^2} \text {erf}(b x)}{b^4}-\frac {x^3 e^{-2 b^2 x^2}}{4 \sqrt {\pi } b^3} \]
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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^4 \text {erf}(b x)}{2 b^2}+\frac {2 \int e^{-b^2 x^2} x^3 \text {erf}(b x) \, dx}{b^2}+\frac {\int e^{-2 b^2 x^2} x^4 \, dx}{b \sqrt {\pi }} \\ & = -\frac {e^{-2 b^2 x^2} x^3}{4 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} x^2 \text {erf}(b x)}{b^4}-\frac {e^{-b^2 x^2} x^4 \text {erf}(b x)}{2 b^2}+\frac {2 \int e^{-b^2 x^2} x \text {erf}(b x) \, dx}{b^4}+\frac {3 \int e^{-2 b^2 x^2} x^2 \, dx}{4 b^3 \sqrt {\pi }}+\frac {2 \int e^{-2 b^2 x^2} x^2 \, dx}{b^3 \sqrt {\pi }} \\ & = -\frac {11 e^{-2 b^2 x^2} x}{16 b^5 \sqrt {\pi }}-\frac {e^{-2 b^2 x^2} x^3}{4 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erf}(b x)}{b^6}-\frac {e^{-b^2 x^2} x^2 \text {erf}(b x)}{b^4}-\frac {e^{-b^2 x^2} x^4 \text {erf}(b x)}{2 b^2}+\frac {3 \int e^{-2 b^2 x^2} \, dx}{16 b^5 \sqrt {\pi }}+\frac {\int e^{-2 b^2 x^2} \, dx}{2 b^5 \sqrt {\pi }}+\frac {2 \int e^{-2 b^2 x^2} \, dx}{b^5 \sqrt {\pi }} \\ & = -\frac {11 e^{-2 b^2 x^2} x}{16 b^5 \sqrt {\pi }}-\frac {e^{-2 b^2 x^2} x^3}{4 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erf}(b x)}{b^6}-\frac {e^{-b^2 x^2} x^2 \text {erf}(b x)}{b^4}-\frac {e^{-b^2 x^2} x^4 \text {erf}(b x)}{2 b^2}+\frac {43 \text {erf}\left (\sqrt {2} b x\right )}{32 \sqrt {2} b^6} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.64 \[ \int e^{-b^2 x^2} x^5 \text {erf}(b x) \, dx=\frac {-\frac {4 b e^{-2 b^2 x^2} x \left (11+4 b^2 x^2\right )}{\sqrt {\pi }}-32 e^{-b^2 x^2} \left (2+2 b^2 x^2+b^4 x^4\right ) \text {erf}(b x)+43 \sqrt {2} \text {erf}\left (\sqrt {2} b x\right )}{64 b^6} \]
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Time = 2.31 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.88
method | result | size |
default | \(\frac {\frac {\operatorname {erf}\left (b x \right ) \left (-\frac {{\mathrm e}^{-b^{2} x^{2}} x^{4} b^{4}}{2}-x^{2} {\mathrm e}^{-b^{2} x^{2}} b^{2}-{\mathrm e}^{-b^{2} x^{2}}\right )}{b^{5}}-\frac {-\frac {43 \sqrt {2}\, \sqrt {\pi }\, \operatorname {erf}\left (b x \sqrt {2}\right )}{64}+\frac {{\mathrm e}^{-2 b^{2} x^{2}} b^{3} x^{3}}{4}+\frac {11 \,{\mathrm e}^{-2 b^{2} x^{2}} b x}{16}}{\sqrt {\pi }\, b^{5}}}{b}\) | \(119\) |
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Time = 0.25 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.72 \[ \int e^{-b^2 x^2} x^5 \text {erf}(b x) \, dx=\frac {43 \, \sqrt {2} \pi \sqrt {b^{2}} \operatorname {erf}\left (\sqrt {2} \sqrt {b^{2}} x\right ) - 32 \, {\left (\pi b^{5} x^{4} + 2 \, \pi b^{3} x^{2} + 2 \, \pi b\right )} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} - 4 \, \sqrt {\pi } {\left (4 \, b^{4} x^{3} + 11 \, b^{2} x\right )} e^{\left (-2 \, b^{2} x^{2}\right )}}{64 \, \pi b^{7}} \]
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\[ \int e^{-b^2 x^2} x^5 \text {erf}(b x) \, dx=\int x^{5} e^{- b^{2} x^{2}} \operatorname {erf}{\left (b x \right )}\, dx \]
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\[ \int e^{-b^2 x^2} x^5 \text {erf}(b x) \, dx=\int { x^{5} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.13 \[ \int e^{-b^2 x^2} x^5 \text {erf}(b x) \, dx=-\frac {{\left (b^{4} x^{4} + 2 \, b^{2} x^{2} + 2\right )} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{2 \, b^{6}} - \frac {b^{4} {\left (\frac {4 \, {\left (4 \, b^{2} x^{3} + 3 \, x\right )} e^{\left (-2 \, b^{2} x^{2}\right )}}{b^{4}} + \frac {3 \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {2} b x\right )}{b^{5}}\right )} + 8 \, b^{2} {\left (\frac {4 \, x e^{\left (-2 \, b^{2} x^{2}\right )}}{b^{2}} + \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {2} b x\right )}{b^{3}}\right )} + \frac {32 \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {2} b x\right )}{b}}{64 \, \sqrt {\pi } b^{5}} \]
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Time = 0.48 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.42 \[ \int e^{-b^2 x^2} x^5 \text {erf}(b x) \, dx=\frac {\sqrt {2}\,\mathrm {erf}\left (\sqrt {2}\,x\,\sqrt {b^2}\right )}{2\,b\,{\left (b^2\right )}^{5/2}}-\frac {\mathrm {erfi}\left (x\,\sqrt {-2\,b^2}\right )}{2\,b^3\,{\left (-2\,b^2\right )}^{3/2}}-\frac {x^3\,{\mathrm {e}}^{-2\,b^2\,x^2}}{4\,b^3\,\sqrt {\pi }}-\mathrm {erf}\left (b\,x\right )\,\left (\frac {{\mathrm {e}}^{-b^2\,x^2}}{b^6}+\frac {x^4\,{\mathrm {e}}^{-b^2\,x^2}}{2\,b^2}+\frac {x^2\,{\mathrm {e}}^{-b^2\,x^2}}{b^4}\right )-\frac {11\,x\,{\mathrm {e}}^{-2\,b^2\,x^2}}{16\,b^5\,\sqrt {\pi }}+\frac {3\,\sqrt {2}\,x^5}{64\,b\,{\left (b^2\,x^2\right )}^{5/2}}-\frac {3\,\sqrt {2}\,x^5\,\mathrm {erfc}\left (\sqrt {2\,b^2\,x^2}\right )}{64\,b\,{\left (b^2\,x^2\right )}^{5/2}} \]
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