Integrand size = 8, antiderivative size = 96 \[ \int x^5 \text {erfc}(b x) \, dx=-\frac {5 e^{-b^2 x^2} x}{8 b^5 \sqrt {\pi }}-\frac {5 e^{-b^2 x^2} x^3}{12 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} x^5}{6 b \sqrt {\pi }}+\frac {5 \text {erf}(b x)}{16 b^6}+\frac {1}{6} x^6 \text {erfc}(b x) \]
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Time = 0.06 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6497, 2243, 2236} \[ \int x^5 \text {erfc}(b x) \, dx=\frac {5 \text {erf}(b x)}{16 b^6}-\frac {x^5 e^{-b^2 x^2}}{6 \sqrt {\pi } b}-\frac {5 x e^{-b^2 x^2}}{8 \sqrt {\pi } b^5}-\frac {5 x^3 e^{-b^2 x^2}}{12 \sqrt {\pi } b^3}+\frac {1}{6} x^6 \text {erfc}(b x) \]
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Rule 2236
Rule 2243
Rule 6497
Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} x^6 \text {erfc}(b x)+\frac {b \int e^{-b^2 x^2} x^6 \, dx}{3 \sqrt {\pi }} \\ & = -\frac {e^{-b^2 x^2} x^5}{6 b \sqrt {\pi }}+\frac {1}{6} x^6 \text {erfc}(b x)+\frac {5 \int e^{-b^2 x^2} x^4 \, dx}{6 b \sqrt {\pi }} \\ & = -\frac {5 e^{-b^2 x^2} x^3}{12 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} x^5}{6 b \sqrt {\pi }}+\frac {1}{6} x^6 \text {erfc}(b x)+\frac {5 \int e^{-b^2 x^2} x^2 \, dx}{4 b^3 \sqrt {\pi }} \\ & = -\frac {5 e^{-b^2 x^2} x}{8 b^5 \sqrt {\pi }}-\frac {5 e^{-b^2 x^2} x^3}{12 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} x^5}{6 b \sqrt {\pi }}+\frac {1}{6} x^6 \text {erfc}(b x)+\frac {5 \int e^{-b^2 x^2} \, dx}{8 b^5 \sqrt {\pi }} \\ & = -\frac {5 e^{-b^2 x^2} x}{8 b^5 \sqrt {\pi }}-\frac {5 e^{-b^2 x^2} x^3}{12 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} x^5}{6 b \sqrt {\pi }}+\frac {5 \text {erf}(b x)}{16 b^6}+\frac {1}{6} x^6 \text {erfc}(b x) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.65 \[ \int x^5 \text {erfc}(b x) \, dx=\frac {1}{48} \left (-\frac {2 e^{-b^2 x^2} x \left (15+10 b^2 x^2+4 b^4 x^4\right )}{b^5 \sqrt {\pi }}+\frac {15 \text {erf}(b x)}{b^6}+8 x^6 \text {erfc}(b x)\right ) \]
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Time = 0.25 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.84
method | result | size |
parallelrisch | \(\frac {8 \,\operatorname {erfc}\left (b x \right ) x^{6} b^{6} \sqrt {\pi }-8 \,{\mathrm e}^{-b^{2} x^{2}} x^{5} b^{5}-20 x^{3} {\mathrm e}^{-b^{2} x^{2}} b^{3}-30 \,{\mathrm e}^{-b^{2} x^{2}} b x -15 \,\operatorname {erfc}\left (b x \right ) \sqrt {\pi }}{48 b^{6} \sqrt {\pi }}\) | \(81\) |
derivativedivides | \(\frac {\frac {b^{6} x^{6} \operatorname {erfc}\left (b x \right )}{6}+\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}} x^{5} b^{5}}{2}-\frac {5 x^{3} {\mathrm e}^{-b^{2} x^{2}} b^{3}}{4}-\frac {15 \,{\mathrm e}^{-b^{2} x^{2}} b x}{8}+\frac {15 \,\operatorname {erf}\left (b x \right ) \sqrt {\pi }}{16}}{3 \sqrt {\pi }}}{b^{6}}\) | \(83\) |
default | \(\frac {\frac {b^{6} x^{6} \operatorname {erfc}\left (b x \right )}{6}+\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}} x^{5} b^{5}}{2}-\frac {5 x^{3} {\mathrm e}^{-b^{2} x^{2}} b^{3}}{4}-\frac {15 \,{\mathrm e}^{-b^{2} x^{2}} b x}{8}+\frac {15 \,\operatorname {erf}\left (b x \right ) \sqrt {\pi }}{16}}{3 \sqrt {\pi }}}{b^{6}}\) | \(83\) |
parts | \(\frac {x^{6} \operatorname {erfc}\left (b x \right )}{6}+\frac {b \left (-\frac {x^{5} {\mathrm e}^{-b^{2} x^{2}}}{2 b^{2}}+\frac {-\frac {5 x^{3} {\mathrm e}^{-b^{2} x^{2}}}{4 b^{2}}+\frac {5 \left (-\frac {3 x \,{\mathrm e}^{-b^{2} x^{2}}}{4 b^{2}}+\frac {3 \sqrt {\pi }\, \operatorname {erf}\left (b x \right )}{8 b^{3}}\right )}{2 b^{2}}}{b^{2}}\right )}{3 \sqrt {\pi }}\) | \(91\) |
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Time = 0.26 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.74 \[ \int x^5 \text {erfc}(b x) \, dx=\frac {8 \, \pi b^{6} x^{6} - 2 \, \sqrt {\pi } {\left (4 \, b^{5} x^{5} + 10 \, b^{3} x^{3} + 15 \, b x\right )} e^{\left (-b^{2} x^{2}\right )} + {\left (15 \, \pi - 8 \, \pi b^{6} x^{6}\right )} \operatorname {erf}\left (b x\right )}{48 \, \pi b^{6}} \]
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Time = 0.40 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.96 \[ \int x^5 \text {erfc}(b x) \, dx=\begin {cases} \frac {x^{6} \operatorname {erfc}{\left (b x \right )}}{6} - \frac {x^{5} e^{- b^{2} x^{2}}}{6 \sqrt {\pi } b} - \frac {5 x^{3} e^{- b^{2} x^{2}}}{12 \sqrt {\pi } b^{3}} - \frac {5 x e^{- b^{2} x^{2}}}{8 \sqrt {\pi } b^{5}} - \frac {5 \operatorname {erfc}{\left (b x \right )}}{16 b^{6}} & \text {for}\: b \neq 0 \\\frac {x^{6}}{6} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.66 \[ \int x^5 \text {erfc}(b x) \, dx=\frac {1}{6} \, x^{6} \operatorname {erfc}\left (b x\right ) - \frac {b {\left (\frac {2 \, {\left (4 \, b^{4} x^{5} + 10 \, b^{2} x^{3} + 15 \, x\right )} e^{\left (-b^{2} x^{2}\right )}}{b^{6}} - \frac {15 \, \sqrt {\pi } \operatorname {erf}\left (b x\right )}{b^{7}}\right )}}{48 \, \sqrt {\pi }} \]
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Time = 0.27 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.72 \[ \int x^5 \text {erfc}(b x) \, dx=-\frac {1}{6} \, x^{6} \operatorname {erf}\left (b x\right ) + \frac {1}{6} \, x^{6} - \frac {b {\left (\frac {2 \, {\left (4 \, b^{4} x^{5} + 10 \, b^{2} x^{3} + 15 \, x\right )} e^{\left (-b^{2} x^{2}\right )}}{b^{6}} + \frac {15 \, \sqrt {\pi } \operatorname {erf}\left (-b x\right )}{b^{7}}\right )}}{48 \, \sqrt {\pi }} \]
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Time = 4.95 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.81 \[ \int x^5 \text {erfc}(b x) \, dx=\frac {x^6\,\mathrm {erfc}\left (b\,x\right )}{6}-\frac {\frac {5\,\mathrm {erfc}\left (b\,x\right )}{16}+\frac {5\,b^3\,x^3\,{\mathrm {e}}^{-b^2\,x^2}}{12\,\sqrt {\pi }}+\frac {b^5\,x^5\,{\mathrm {e}}^{-b^2\,x^2}}{6\,\sqrt {\pi }}+\frac {5\,b\,x\,{\mathrm {e}}^{-b^2\,x^2}}{8\,\sqrt {\pi }}}{b^6} \]
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