Integrand size = 19, antiderivative size = 80 \[ \int e^{c+b^2 x^2} x^3 \text {erfc}(b x) \, dx=-\frac {e^c x}{b^3 \sqrt {\pi }}+\frac {e^c x^3}{3 b \sqrt {\pi }}-\frac {e^{c+b^2 x^2} \text {erfc}(b x)}{2 b^4}+\frac {e^{c+b^2 x^2} x^2 \text {erfc}(b x)}{2 b^2} \]
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Time = 0.07 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6521, 6518, 8, 12, 30} \[ \int e^{c+b^2 x^2} x^3 \text {erfc}(b x) \, dx=-\frac {e^c x}{\sqrt {\pi } b^3}+\frac {x^2 e^{b^2 x^2+c} \text {erfc}(b x)}{2 b^2}-\frac {e^{b^2 x^2+c} \text {erfc}(b x)}{2 b^4}+\frac {e^c x^3}{3 \sqrt {\pi } b} \]
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Rule 8
Rule 12
Rule 30
Rule 6518
Rule 6521
Rubi steps \begin{align*} \text {integral}& = \frac {e^{c+b^2 x^2} x^2 \text {erfc}(b x)}{2 b^2}-\frac {\int e^{c+b^2 x^2} x \text {erfc}(b x) \, dx}{b^2}+\frac {\int e^c x^2 \, dx}{b \sqrt {\pi }} \\ & = -\frac {e^{c+b^2 x^2} \text {erfc}(b x)}{2 b^4}+\frac {e^{c+b^2 x^2} x^2 \text {erfc}(b x)}{2 b^2}-\frac {\int e^c \, dx}{b^3 \sqrt {\pi }}+\frac {e^c \int x^2 \, dx}{b \sqrt {\pi }} \\ & = -\frac {e^c x}{b^3 \sqrt {\pi }}+\frac {e^c x^3}{3 b \sqrt {\pi }}-\frac {e^{c+b^2 x^2} \text {erfc}(b x)}{2 b^4}+\frac {e^{c+b^2 x^2} x^2 \text {erfc}(b x)}{2 b^2} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.72 \[ \int e^{c+b^2 x^2} x^3 \text {erfc}(b x) \, dx=\frac {e^c \left (2 b x \left (-3+b^2 x^2\right )+3 e^{b^2 x^2} \sqrt {\pi } \left (-1+b^2 x^2\right ) \text {erfc}(b x)\right )}{6 b^4 \sqrt {\pi }} \]
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Time = 1.01 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.24
method | result | size |
default | \(\frac {\frac {{\mathrm e}^{c} \left (\frac {b^{2} x^{2} {\mathrm e}^{b^{2} x^{2}}}{2}-\frac {{\mathrm e}^{b^{2} x^{2}}}{2}\right )}{b^{3}}-\frac {\operatorname {erf}\left (b x \right ) {\mathrm e}^{c} \left (\frac {b^{2} x^{2} {\mathrm e}^{b^{2} x^{2}}}{2}-\frac {{\mathrm e}^{b^{2} x^{2}}}{2}\right )}{b^{3}}+\frac {{\mathrm e}^{c} \left (\frac {1}{3} b^{3} x^{3}-b x \right )}{\sqrt {\pi }\, b^{3}}}{b}\) | \(99\) |
parallelrisch | \(\frac {2 \,{\mathrm e}^{b^{2} x^{2}+c} x^{3} {\mathrm e}^{-b^{2} x^{2}} b^{3}+3 \,{\mathrm e}^{b^{2} x^{2}+c} x^{2} \operatorname {erfc}\left (b x \right ) b^{2} \sqrt {\pi }-6 \,{\mathrm e}^{b^{2} x^{2}+c} x \,{\mathrm e}^{-b^{2} x^{2}} b -3 \,{\mathrm e}^{b^{2} x^{2}+c} \operatorname {erfc}\left (b x \right ) \sqrt {\pi }}{6 b^{4} \sqrt {\pi }}\) | \(104\) |
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Time = 0.25 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.85 \[ \int e^{c+b^2 x^2} x^3 \text {erfc}(b x) \, dx=\frac {2 \, \sqrt {\pi } {\left (b^{3} x^{3} - 3 \, b x\right )} e^{c} - 3 \, {\left (\pi - \pi b^{2} x^{2} - {\left (\pi - \pi b^{2} x^{2}\right )} \operatorname {erf}\left (b x\right )\right )} e^{\left (b^{2} x^{2} + c\right )}}{6 \, \pi b^{4}} \]
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Time = 6.42 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.04 \[ \int e^{c+b^2 x^2} x^3 \text {erfc}(b x) \, dx=\begin {cases} \frac {x^{3} e^{c}}{3 \sqrt {\pi } b} + \frac {x^{2} e^{c} e^{b^{2} x^{2}} \operatorname {erfc}{\left (b x \right )}}{2 b^{2}} - \frac {x e^{c}}{\sqrt {\pi } b^{3}} - \frac {e^{c} e^{b^{2} x^{2}} \operatorname {erfc}{\left (b x \right )}}{2 b^{4}} & \text {for}\: b \neq 0 \\\frac {x^{4} e^{c}}{4} & \text {otherwise} \end {cases} \]
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\[ \int e^{c+b^2 x^2} x^3 \text {erfc}(b x) \, dx=\int { x^{3} \operatorname {erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )} \,d x } \]
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\[ \int e^{c+b^2 x^2} x^3 \text {erfc}(b x) \, dx=\int { x^{3} \operatorname {erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )} \,d x } \]
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Time = 0.15 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.79 \[ \int e^{c+b^2 x^2} x^3 \text {erfc}(b x) \, dx=-\frac {{\mathrm {e}}^c\,\left (6\,b\,x-2\,b^3\,x^3+3\,\sqrt {\pi }\,{\mathrm {e}}^{b^2\,x^2}\,\mathrm {erfc}\left (b\,x\right )-3\,b^2\,x^2\,\sqrt {\pi }\,{\mathrm {e}}^{b^2\,x^2}\,\mathrm {erfc}\left (b\,x\right )\right )}{6\,b^4\,\sqrt {\pi }} \]
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