Integrand size = 10, antiderivative size = 126 \[ \int x^3 \text {erfc}(b x)^2 \, dx=\frac {e^{-2 b^2 x^2}}{2 b^4 \pi }+\frac {e^{-2 b^2 x^2} x^2}{4 b^2 \pi }-\frac {3 e^{-b^2 x^2} x \text {erfc}(b x)}{4 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} x^3 \text {erfc}(b x)}{2 b \sqrt {\pi }}-\frac {3 \text {erfc}(b x)^2}{16 b^4}+\frac {1}{4} x^4 \text {erfc}(b x)^2 \]
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Time = 0.13 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6500, 6521, 6509, 30, 2240, 2243} \[ \int x^3 \text {erfc}(b x)^2 \, dx=-\frac {3 \text {erfc}(b x)^2}{16 b^4}-\frac {x^3 e^{-b^2 x^2} \text {erfc}(b x)}{2 \sqrt {\pi } b}+\frac {x^2 e^{-2 b^2 x^2}}{4 \pi b^2}+\frac {e^{-2 b^2 x^2}}{2 \pi b^4}-\frac {3 x e^{-b^2 x^2} \text {erfc}(b x)}{4 \sqrt {\pi } b^3}+\frac {1}{4} x^4 \text {erfc}(b x)^2 \]
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Rule 30
Rule 2240
Rule 2243
Rule 6500
Rule 6509
Rule 6521
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x^4 \text {erfc}(b x)^2+\frac {b \int e^{-b^2 x^2} x^4 \text {erfc}(b x) \, dx}{\sqrt {\pi }} \\ & = -\frac {e^{-b^2 x^2} x^3 \text {erfc}(b x)}{2 b \sqrt {\pi }}+\frac {1}{4} x^4 \text {erfc}(b x)^2-\frac {\int e^{-2 b^2 x^2} x^3 \, dx}{\pi }+\frac {3 \int e^{-b^2 x^2} x^2 \text {erfc}(b x) \, dx}{2 b \sqrt {\pi }} \\ & = \frac {e^{-2 b^2 x^2} x^2}{4 b^2 \pi }-\frac {3 e^{-b^2 x^2} x \text {erfc}(b x)}{4 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} x^3 \text {erfc}(b x)}{2 b \sqrt {\pi }}+\frac {1}{4} x^4 \text {erfc}(b x)^2-\frac {\int e^{-2 b^2 x^2} x \, dx}{2 b^2 \pi }-\frac {3 \int e^{-2 b^2 x^2} x \, dx}{2 b^2 \pi }+\frac {3 \int e^{-b^2 x^2} \text {erfc}(b x) \, dx}{4 b^3 \sqrt {\pi }} \\ & = \frac {e^{-2 b^2 x^2}}{2 b^4 \pi }+\frac {e^{-2 b^2 x^2} x^2}{4 b^2 \pi }-\frac {3 e^{-b^2 x^2} x \text {erfc}(b x)}{4 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} x^3 \text {erfc}(b x)}{2 b \sqrt {\pi }}+\frac {1}{4} x^4 \text {erfc}(b x)^2-\frac {3 \text {Subst}(\int x \, dx,x,\text {erfc}(b x))}{8 b^4} \\ & = \frac {e^{-2 b^2 x^2}}{2 b^4 \pi }+\frac {e^{-2 b^2 x^2} x^2}{4 b^2 \pi }-\frac {3 e^{-b^2 x^2} x \text {erfc}(b x)}{4 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} x^3 \text {erfc}(b x)}{2 b \sqrt {\pi }}-\frac {3 \text {erfc}(b x)^2}{16 b^4}+\frac {1}{4} x^4 \text {erfc}(b x)^2 \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.14 \[ \int x^3 \text {erfc}(b x)^2 \, dx=\frac {1}{8} \left (2 x^4-4 x^4 \text {erf}(b x)+2 x^4 \text {erf}(b x)^2+\frac {e^{-2 b^2 x^2} \left (8+4 b^2 x^2+4 b e^{b^2 x^2} \sqrt {\pi } x \left (3+2 b^2 x^2\right ) \text {erf}(b x)-3 e^{2 b^2 x^2} \pi \text {erf}(b x)^2\right )}{2 b^4 \pi }-\frac {4 x \Gamma \left (\frac {5}{2},b^2 x^2\right )}{b^3 \sqrt {\pi } \sqrt {b^2 x^2}}\right ) \]
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Time = 0.37 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.92
method | result | size |
parallelrisch | \(\frac {4 \operatorname {erfc}\left (b x \right )^{2} x^{4} \pi ^{\frac {3}{2}} b^{4}-8 \,{\mathrm e}^{-b^{2} x^{2}} \operatorname {erfc}\left (b x \right ) x^{3} b^{3} \pi +4 x^{2} {\mathrm e}^{-2 b^{2} x^{2}} b^{2} \sqrt {\pi }-12 \,{\mathrm e}^{-b^{2} x^{2}} x \,\operatorname {erfc}\left (b x \right ) b \pi -3 \operatorname {erfc}\left (b x \right )^{2} \pi ^{\frac {3}{2}}+8 \,{\mathrm e}^{-2 b^{2} x^{2}} \sqrt {\pi }}{16 \pi ^{\frac {3}{2}} b^{4}}\) | \(116\) |
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Time = 0.25 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.98 \[ \int x^3 \text {erfc}(b x)^2 \, dx=\frac {4 \, \pi b^{4} x^{4} - {\left (3 \, \pi - 4 \, \pi b^{4} x^{4}\right )} \operatorname {erf}\left (b x\right )^{2} - 4 \, \sqrt {\pi } {\left (2 \, b^{3} x^{3} + 3 \, b x - {\left (2 \, b^{3} x^{3} + 3 \, b x\right )} \operatorname {erf}\left (b x\right )\right )} e^{\left (-b^{2} x^{2}\right )} + 2 \, {\left (3 \, \pi - 4 \, \pi b^{4} x^{4}\right )} \operatorname {erf}\left (b x\right ) + 4 \, {\left (b^{2} x^{2} + 2\right )} e^{\left (-2 \, b^{2} x^{2}\right )}}{16 \, \pi b^{4}} \]
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Time = 0.33 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.96 \[ \int x^3 \text {erfc}(b x)^2 \, dx=\begin {cases} \frac {x^{4} \operatorname {erfc}^{2}{\left (b x \right )}}{4} - \frac {x^{3} e^{- b^{2} x^{2}} \operatorname {erfc}{\left (b x \right )}}{2 \sqrt {\pi } b} + \frac {x^{2} e^{- 2 b^{2} x^{2}}}{4 \pi b^{2}} - \frac {3 x e^{- b^{2} x^{2}} \operatorname {erfc}{\left (b x \right )}}{4 \sqrt {\pi } b^{3}} - \frac {3 \operatorname {erfc}^{2}{\left (b x \right )}}{16 b^{4}} + \frac {e^{- 2 b^{2} x^{2}}}{2 \pi b^{4}} & \text {for}\: b \neq 0 \\\frac {x^{4}}{4} & \text {otherwise} \end {cases} \]
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\[ \int x^3 \text {erfc}(b x)^2 \, dx=\int { x^{3} \operatorname {erfc}\left (b x\right )^{2} \,d x } \]
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\[ \int x^3 \text {erfc}(b x)^2 \, dx=\int { x^{3} \operatorname {erfc}\left (b x\right )^{2} \,d x } \]
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Time = 4.91 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.81 \[ \int x^3 \text {erfc}(b x)^2 \, dx=\frac {x^4\,{\mathrm {erfc}\left (b\,x\right )}^2}{4}-\frac {\frac {3\,\pi \,{\mathrm {erfc}\left (b\,x\right )}^2}{16}-\frac {{\mathrm {e}}^{-2\,b^2\,x^2}}{2}-\frac {b^2\,x^2\,{\mathrm {e}}^{-2\,b^2\,x^2}}{4}+\frac {b^3\,x^3\,\sqrt {\pi }\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erfc}\left (b\,x\right )}{2}+\frac {3\,b\,x\,\sqrt {\pi }\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erfc}\left (b\,x\right )}{4}}{b^4\,\pi } \]
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