Integrand size = 8, antiderivative size = 84 \[ \int x^4 \text {erfc}(b x) \, dx=-\frac {2 e^{-b^2 x^2}}{5 b^5 \sqrt {\pi }}-\frac {2 e^{-b^2 x^2} x^2}{5 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} x^4}{5 b \sqrt {\pi }}+\frac {1}{5} x^5 \text {erfc}(b x) \]
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Time = 0.06 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6497, 2243, 2240} \[ \int x^4 \text {erfc}(b x) \, dx=-\frac {x^4 e^{-b^2 x^2}}{5 \sqrt {\pi } b}-\frac {2 e^{-b^2 x^2}}{5 \sqrt {\pi } b^5}-\frac {2 x^2 e^{-b^2 x^2}}{5 \sqrt {\pi } b^3}+\frac {1}{5} x^5 \text {erfc}(b x) \]
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Rule 2240
Rule 2243
Rule 6497
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} x^5 \text {erfc}(b x)+\frac {(2 b) \int e^{-b^2 x^2} x^5 \, dx}{5 \sqrt {\pi }} \\ & = -\frac {e^{-b^2 x^2} x^4}{5 b \sqrt {\pi }}+\frac {1}{5} x^5 \text {erfc}(b x)+\frac {4 \int e^{-b^2 x^2} x^3 \, dx}{5 b \sqrt {\pi }} \\ & = -\frac {2 e^{-b^2 x^2} x^2}{5 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} x^4}{5 b \sqrt {\pi }}+\frac {1}{5} x^5 \text {erfc}(b x)+\frac {4 \int e^{-b^2 x^2} x \, dx}{5 b^3 \sqrt {\pi }} \\ & = -\frac {2 e^{-b^2 x^2}}{5 b^5 \sqrt {\pi }}-\frac {2 e^{-b^2 x^2} x^2}{5 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} x^4}{5 b \sqrt {\pi }}+\frac {1}{5} x^5 \text {erfc}(b x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.79 \[ \int x^4 \text {erfc}(b x) \, dx=e^{-b^2 x^2} \left (-\frac {2}{5 b^5 \sqrt {\pi }}-\frac {2 x^2}{5 b^3 \sqrt {\pi }}-\frac {x^4}{5 b \sqrt {\pi }}\right )+\frac {1}{5} x^5 \text {erfc}(b x) \]
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Time = 0.18 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.82
method | result | size |
parallelrisch | \(\frac {x^{5} \operatorname {erfc}\left (b x \right ) b^{5} \sqrt {\pi }-{\mathrm e}^{-b^{2} x^{2}} x^{4} b^{4}-2 x^{2} {\mathrm e}^{-b^{2} x^{2}} b^{2}-2 \,{\mathrm e}^{-b^{2} x^{2}}}{5 b^{5} \sqrt {\pi }}\) | \(69\) |
derivativedivides | \(\frac {\frac {b^{5} x^{5} \operatorname {erfc}\left (b x \right )}{5}+\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}} x^{4} b^{4}}{5}-\frac {2 x^{2} {\mathrm e}^{-b^{2} x^{2}} b^{2}}{5}-\frac {2 \,{\mathrm e}^{-b^{2} x^{2}}}{5}}{\sqrt {\pi }}}{b^{5}}\) | \(72\) |
default | \(\frac {\frac {b^{5} x^{5} \operatorname {erfc}\left (b x \right )}{5}+\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}} x^{4} b^{4}}{5}-\frac {2 x^{2} {\mathrm e}^{-b^{2} x^{2}} b^{2}}{5}-\frac {2 \,{\mathrm e}^{-b^{2} x^{2}}}{5}}{\sqrt {\pi }}}{b^{5}}\) | \(72\) |
parts | \(\frac {x^{5} \operatorname {erfc}\left (b x \right )}{5}+\frac {2 b \left (-\frac {x^{4} {\mathrm e}^{-b^{2} x^{2}}}{2 b^{2}}+\frac {-\frac {x^{2} {\mathrm e}^{-b^{2} x^{2}}}{b^{2}}-\frac {{\mathrm e}^{-b^{2} x^{2}}}{b^{4}}}{b^{2}}\right )}{5 \sqrt {\pi }}\) | \(72\) |
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Time = 0.25 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.71 \[ \int x^4 \text {erfc}(b x) \, dx=-\frac {\pi b^{5} x^{5} \operatorname {erf}\left (b x\right ) - \pi b^{5} x^{5} + \sqrt {\pi } {\left (b^{4} x^{4} + 2 \, b^{2} x^{2} + 2\right )} e^{\left (-b^{2} x^{2}\right )}}{5 \, \pi b^{5}} \]
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Time = 0.29 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.93 \[ \int x^4 \text {erfc}(b x) \, dx=\begin {cases} \frac {x^{5} \operatorname {erfc}{\left (b x \right )}}{5} - \frac {x^{4} e^{- b^{2} x^{2}}}{5 \sqrt {\pi } b} - \frac {2 x^{2} e^{- b^{2} x^{2}}}{5 \sqrt {\pi } b^{3}} - \frac {2 e^{- b^{2} x^{2}}}{5 \sqrt {\pi } b^{5}} & \text {for}\: b \neq 0 \\\frac {x^{5}}{5} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.52 \[ \int x^4 \text {erfc}(b x) \, dx=\frac {1}{5} \, x^{5} \operatorname {erfc}\left (b x\right ) - \frac {{\left (b^{4} x^{4} + 2 \, b^{2} x^{2} + 2\right )} e^{\left (-b^{2} x^{2}\right )}}{5 \, \sqrt {\pi } b^{5}} \]
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Time = 0.27 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.58 \[ \int x^4 \text {erfc}(b x) \, dx=-\frac {1}{5} \, x^{5} \operatorname {erf}\left (b x\right ) + \frac {1}{5} \, x^{5} - \frac {{\left (b^{4} x^{4} + 2 \, b^{2} x^{2} + 2\right )} e^{\left (-b^{2} x^{2}\right )}}{5 \, \sqrt {\pi } b^{5}} \]
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Time = 5.10 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.83 \[ \int x^4 \text {erfc}(b x) \, dx=\frac {x^5\,\mathrm {erfc}\left (b\,x\right )}{5}-\frac {\frac {2\,{\mathrm {e}}^{-b^2\,x^2}}{5\,\sqrt {\pi }}+\frac {2\,b^2\,x^2\,{\mathrm {e}}^{-b^2\,x^2}}{5\,\sqrt {\pi }}+\frac {b^4\,x^4\,{\mathrm {e}}^{-b^2\,x^2}}{5\,\sqrt {\pi }}}{b^5} \]
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