Integrand size = 10, antiderivative size = 165 \[ \int x^4 \text {erfc}(b x)^2 \, dx=\frac {11 e^{-2 b^2 x^2} x}{20 b^4 \pi }+\frac {e^{-2 b^2 x^2} x^3}{5 b^2 \pi }-\frac {43 \text {erf}\left (\sqrt {2} b x\right )}{40 b^5 \sqrt {2 \pi }}-\frac {4 e^{-b^2 x^2} \text {erfc}(b x)}{5 b^5 \sqrt {\pi }}-\frac {4 e^{-b^2 x^2} x^2 \text {erfc}(b x)}{5 b^3 \sqrt {\pi }}-\frac {2 e^{-b^2 x^2} x^4 \text {erfc}(b x)}{5 b \sqrt {\pi }}+\frac {1}{5} x^5 \text {erfc}(b x)^2 \]
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Time = 0.17 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6500, 6521, 6518, 2236, 2243} \[ \int x^4 \text {erfc}(b x)^2 \, dx=-\frac {43 \text {erf}\left (\sqrt {2} b x\right )}{40 \sqrt {2 \pi } b^5}-\frac {2 x^4 e^{-b^2 x^2} \text {erfc}(b x)}{5 \sqrt {\pi } b}+\frac {x^3 e^{-2 b^2 x^2}}{5 \pi b^2}-\frac {4 e^{-b^2 x^2} \text {erfc}(b x)}{5 \sqrt {\pi } b^5}+\frac {11 x e^{-2 b^2 x^2}}{20 \pi b^4}-\frac {4 x^2 e^{-b^2 x^2} \text {erfc}(b x)}{5 \sqrt {\pi } b^3}+\frac {1}{5} x^5 \text {erfc}(b x)^2 \]
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Rule 2236
Rule 2243
Rule 6500
Rule 6518
Rule 6521
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} x^5 \text {erfc}(b x)^2+\frac {(4 b) \int e^{-b^2 x^2} x^5 \text {erfc}(b x) \, dx}{5 \sqrt {\pi }} \\ & = -\frac {2 e^{-b^2 x^2} x^4 \text {erfc}(b x)}{5 b \sqrt {\pi }}+\frac {1}{5} x^5 \text {erfc}(b x)^2-\frac {4 \int e^{-2 b^2 x^2} x^4 \, dx}{5 \pi }+\frac {8 \int e^{-b^2 x^2} x^3 \text {erfc}(b x) \, dx}{5 b \sqrt {\pi }} \\ & = \frac {e^{-2 b^2 x^2} x^3}{5 b^2 \pi }-\frac {4 e^{-b^2 x^2} x^2 \text {erfc}(b x)}{5 b^3 \sqrt {\pi }}-\frac {2 e^{-b^2 x^2} x^4 \text {erfc}(b x)}{5 b \sqrt {\pi }}+\frac {1}{5} x^5 \text {erfc}(b x)^2-\frac {3 \int e^{-2 b^2 x^2} x^2 \, dx}{5 b^2 \pi }-\frac {8 \int e^{-2 b^2 x^2} x^2 \, dx}{5 b^2 \pi }+\frac {8 \int e^{-b^2 x^2} x \text {erfc}(b x) \, dx}{5 b^3 \sqrt {\pi }} \\ & = \frac {11 e^{-2 b^2 x^2} x}{20 b^4 \pi }+\frac {e^{-2 b^2 x^2} x^3}{5 b^2 \pi }-\frac {4 e^{-b^2 x^2} \text {erfc}(b x)}{5 b^5 \sqrt {\pi }}-\frac {4 e^{-b^2 x^2} x^2 \text {erfc}(b x)}{5 b^3 \sqrt {\pi }}-\frac {2 e^{-b^2 x^2} x^4 \text {erfc}(b x)}{5 b \sqrt {\pi }}+\frac {1}{5} x^5 \text {erfc}(b x)^2-\frac {3 \int e^{-2 b^2 x^2} \, dx}{20 b^4 \pi }-\frac {2 \int e^{-2 b^2 x^2} \, dx}{5 b^4 \pi }-\frac {8 \int e^{-2 b^2 x^2} \, dx}{5 b^4 \pi } \\ & = \frac {11 e^{-2 b^2 x^2} x}{20 b^4 \pi }+\frac {e^{-2 b^2 x^2} x^3}{5 b^2 \pi }-\frac {2 \sqrt {\frac {2}{\pi }} \text {erf}\left (\sqrt {2} b x\right )}{5 b^5}-\frac {11 \text {erf}\left (\sqrt {2} b x\right )}{40 b^5 \sqrt {2 \pi }}-\frac {4 e^{-b^2 x^2} \text {erfc}(b x)}{5 b^5 \sqrt {\pi }}-\frac {4 e^{-b^2 x^2} x^2 \text {erfc}(b x)}{5 b^3 \sqrt {\pi }}-\frac {2 e^{-b^2 x^2} x^4 \text {erfc}(b x)}{5 b \sqrt {\pi }}+\frac {1}{5} x^5 \text {erfc}(b x)^2 \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.65 \[ \int x^4 \text {erfc}(b x)^2 \, dx=\frac {-43 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} b x\right )+4 \left (b e^{-2 b^2 x^2} x \left (11+4 b^2 x^2\right )-8 e^{-b^2 x^2} \sqrt {\pi } \left (2+2 b^2 x^2+b^4 x^4\right ) \text {erfc}(b x)+4 b^5 \pi x^5 \text {erfc}(b x)^2\right )}{80 b^5 \pi } \]
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Time = 0.54 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.24
method | result | size |
derivativedivides | \(\frac {\frac {b^{5} x^{5}}{5}-\frac {2 \,\operatorname {erf}\left (b x \right ) b^{5} x^{5}}{5}+\frac {-\frac {2 \,{\mathrm e}^{-b^{2} x^{2}} x^{4} b^{4}}{5}-\frac {4 x^{2} {\mathrm e}^{-b^{2} x^{2}} b^{2}}{5}-\frac {4 \,{\mathrm e}^{-b^{2} x^{2}}}{5}}{\sqrt {\pi }}+\frac {\operatorname {erf}\left (b x \right )^{2} b^{5} x^{5}}{5}-\frac {4 \,\operatorname {erf}\left (b x \right ) \left (-\frac {{\mathrm e}^{-b^{2} x^{2}} x^{4} b^{4}}{2}-x^{2} {\mathrm e}^{-b^{2} x^{2}} b^{2}-{\mathrm e}^{-b^{2} x^{2}}\right )}{5 \sqrt {\pi }}+\frac {-\frac {43 \sqrt {2}\, \sqrt {\pi }\, \operatorname {erf}\left (b x \sqrt {2}\right )}{80}+\frac {11 \,{\mathrm e}^{-2 b^{2} x^{2}} b x}{20}+\frac {{\mathrm e}^{-2 b^{2} x^{2}} b^{3} x^{3}}{5}}{\pi }}{b^{5}}\) | \(205\) |
default | \(\frac {\frac {b^{5} x^{5}}{5}-\frac {2 \,\operatorname {erf}\left (b x \right ) b^{5} x^{5}}{5}+\frac {-\frac {2 \,{\mathrm e}^{-b^{2} x^{2}} x^{4} b^{4}}{5}-\frac {4 x^{2} {\mathrm e}^{-b^{2} x^{2}} b^{2}}{5}-\frac {4 \,{\mathrm e}^{-b^{2} x^{2}}}{5}}{\sqrt {\pi }}+\frac {\operatorname {erf}\left (b x \right )^{2} b^{5} x^{5}}{5}-\frac {4 \,\operatorname {erf}\left (b x \right ) \left (-\frac {{\mathrm e}^{-b^{2} x^{2}} x^{4} b^{4}}{2}-x^{2} {\mathrm e}^{-b^{2} x^{2}} b^{2}-{\mathrm e}^{-b^{2} x^{2}}\right )}{5 \sqrt {\pi }}+\frac {-\frac {43 \sqrt {2}\, \sqrt {\pi }\, \operatorname {erf}\left (b x \sqrt {2}\right )}{80}+\frac {11 \,{\mathrm e}^{-2 b^{2} x^{2}} b x}{20}+\frac {{\mathrm e}^{-2 b^{2} x^{2}} b^{3} x^{3}}{5}}{\pi }}{b^{5}}\) | \(205\) |
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Time = 0.27 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.93 \[ \int x^4 \text {erfc}(b x)^2 \, dx=\frac {16 \, \pi b^{6} x^{5} \operatorname {erf}\left (b x\right )^{2} - 32 \, \pi b^{6} x^{5} \operatorname {erf}\left (b x\right ) + 16 \, \pi b^{6} x^{5} - 43 \, \sqrt {2} \sqrt {\pi } \sqrt {b^{2}} \operatorname {erf}\left (\sqrt {2} \sqrt {b^{2}} x\right ) - 32 \, \sqrt {\pi } {\left (b^{5} x^{4} + 2 \, b^{3} x^{2} - {\left (b^{5} x^{4} + 2 \, b^{3} x^{2} + 2 \, b\right )} \operatorname {erf}\left (b x\right ) + 2 \, b\right )} e^{\left (-b^{2} x^{2}\right )} + 4 \, {\left (4 \, b^{4} x^{3} + 11 \, b^{2} x\right )} e^{\left (-2 \, b^{2} x^{2}\right )}}{80 \, \pi b^{6}} \]
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\[ \int x^4 \text {erfc}(b x)^2 \, dx=\int x^{4} \operatorname {erfc}^{2}{\left (b x \right )}\, dx \]
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\[ \int x^4 \text {erfc}(b x)^2 \, dx=\int { x^{4} \operatorname {erfc}\left (b x\right )^{2} \,d x } \]
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Time = 0.34 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.32 \[ \int x^4 \text {erfc}(b x)^2 \, dx=\frac {1}{5} \, x^{5} \operatorname {erf}\left (b x\right )^{2} - \frac {2}{5} \, x^{5} \operatorname {erf}\left (b x\right ) + \frac {1}{5} \, x^{5} + \frac {b {\left (\frac {32 \, {\left (b^{4} x^{4} + 2 \, b^{2} x^{2} + 2\right )} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{b^{6}} + \frac {b^{4} {\left (\frac {4 \, {\left (4 \, b^{2} x^{3} + 3 \, x\right )} e^{\left (-2 \, b^{2} x^{2}\right )}}{b^{4}} + \frac {3 \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {2} b x\right )}{b^{5}}\right )} + 8 \, b^{2} {\left (\frac {4 \, x e^{\left (-2 \, b^{2} x^{2}\right )}}{b^{2}} + \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {2} b x\right )}{b^{3}}\right )} + \frac {32 \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {2} b x\right )}{b}}{\sqrt {\pi } b^{5}}\right )}}{80 \, \sqrt {\pi }} - \frac {2 \, {\left (b^{4} x^{4} + 2 \, b^{2} x^{2} + 2\right )} e^{\left (-b^{2} x^{2}\right )}}{5 \, \sqrt {\pi } b^{5}} \]
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Timed out. \[ \int x^4 \text {erfc}(b x)^2 \, dx=\int x^4\,{\mathrm {erfc}\left (b\,x\right )}^2 \,d x \]
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