Integrand size = 19, antiderivative size = 79 \[ \int e^{c+b^2 x^2} x^3 \text {erf}(b x) \, dx=\frac {e^c x}{b^3 \sqrt {\pi }}-\frac {e^c x^3}{3 b \sqrt {\pi }}-\frac {e^{c+b^2 x^2} \text {erf}(b x)}{2 b^4}+\frac {e^{c+b^2 x^2} x^2 \text {erf}(b x)}{2 b^2} \]
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Time = 0.06 (sec) , antiderivative size = 79, 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 = {6520, 6517, 8, 12, 30} \[ \int e^{c+b^2 x^2} x^3 \text {erf}(b x) \, dx=\frac {e^c x}{\sqrt {\pi } b^3}+\frac {x^2 e^{b^2 x^2+c} \text {erf}(b x)}{2 b^2}-\frac {e^{b^2 x^2+c} \text {erf}(b x)}{2 b^4}-\frac {e^c x^3}{3 \sqrt {\pi } b} \]
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Rule 8
Rule 12
Rule 30
Rule 6517
Rule 6520
Rubi steps \begin{align*} \text {integral}& = \frac {e^{c+b^2 x^2} x^2 \text {erf}(b x)}{2 b^2}-\frac {\int e^{c+b^2 x^2} x \text {erf}(b x) \, dx}{b^2}-\frac {\int e^c x^2 \, dx}{b \sqrt {\pi }} \\ & = -\frac {e^{c+b^2 x^2} \text {erf}(b x)}{2 b^4}+\frac {e^{c+b^2 x^2} x^2 \text {erf}(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 {erf}(b x)}{2 b^4}+\frac {e^{c+b^2 x^2} x^2 \text {erf}(b x)}{2 b^2} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.72 \[ \int e^{c+b^2 x^2} x^3 \text {erf}(b x) \, dx=\frac {e^c \left (6 b x-2 b^3 x^3+3 e^{b^2 x^2} \sqrt {\pi } \left (-1+b^2 x^2\right ) \text {erf}(b x)\right )}{6 b^4 \sqrt {\pi }} \]
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Time = 0.96 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.84
method | result | size |
default | \(\frac {\frac {\operatorname {erf}\left (b x \right ) {\mathrm e}^{c} \left (\frac {b^{2} x^{2} {\mathrm e}^{b^{2} x^{2}}}{2}-\frac {{\mathrm e}^{b^{2} x^{2}}}{2}\right )}{b^{3}}-\frac {{\mathrm e}^{c} \left (\frac {1}{3} b^{3} x^{3}-b x \right )}{\sqrt {\pi }\, b^{3}}}{b}\) | \(66\) |
parallelrisch | \(\frac {-2 \,{\mathrm e}^{b^{2} x^{2}+c} x^{3} {\mathrm e}^{-b^{2} x^{2}} b^{3}+3 \,{\mathrm e}^{b^{2} x^{2}+c} x^{2} \operatorname {erf}\left (b x \right ) b^{2} \sqrt {\pi }+6 \,{\mathrm e}^{b^{2} x^{2}+c} x \,{\mathrm e}^{-b^{2} x^{2}} b -3 \,{\mathrm e}^{b^{2} x^{2}+c} \operatorname {erf}\left (b x \right ) \sqrt {\pi }}{6 b^{4} \sqrt {\pi }}\) | \(104\) |
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Time = 0.24 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.70 \[ \int e^{c+b^2 x^2} x^3 \text {erf}(b x) \, dx=-\frac {3 \, {\left (\pi - \pi b^{2} x^{2}\right )} \operatorname {erf}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )} + 2 \, \sqrt {\pi } {\left (b^{3} x^{3} - 3 \, b x\right )} e^{c}}{6 \, \pi b^{4}} \]
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Time = 6.51 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.96 \[ \int e^{c+b^2 x^2} x^3 \text {erf}(b x) \, dx=\begin {cases} - \frac {x^{3} e^{c}}{3 \sqrt {\pi } b} + \frac {x^{2} e^{c} e^{b^{2} x^{2}} \operatorname {erf}{\left (b x \right )}}{2 b^{2}} + \frac {x e^{c}}{\sqrt {\pi } b^{3}} - \frac {e^{c} e^{b^{2} x^{2}} \operatorname {erf}{\left (b x \right )}}{2 b^{4}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.75 \[ \int e^{c+b^2 x^2} x^3 \text {erf}(b x) \, dx=-\frac {2 \, b^{3} x^{3} e^{c} - 3 \, {\left (\sqrt {\pi } b^{2} x^{2} e^{c} - \sqrt {\pi } e^{c}\right )} \operatorname {erf}\left (b x\right ) e^{\left (b^{2} x^{2}\right )} - 6 \, b x e^{c}}{6 \, \sqrt {\pi } b^{4}} \]
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Time = 0.28 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.90 \[ \int e^{c+b^2 x^2} x^3 \text {erf}(b x) \, dx=\frac {1}{2} \, {\left (\frac {{\left (b^{2} x^{2} + c - 1\right )} e^{\left (b^{2} x^{2} + c\right )}}{b^{4}} - \frac {c e^{\left (b^{2} x^{2} + c\right )}}{b^{4}}\right )} \operatorname {erf}\left (b x\right ) - \frac {b^{2} x^{3} e^{c} - 3 \, x e^{c}}{3 \, \sqrt {\pi } b^{3}} \]
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Time = 0.23 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.82 \[ \int e^{c+b^2 x^2} x^3 \text {erf}(b x) \, dx=\frac {3\,x\,{\mathrm {e}}^c-b^2\,x^3\,{\mathrm {e}}^c}{3\,b^3\,\sqrt {\pi }}-\mathrm {erf}\left (b\,x\right )\,\left (\frac {{\mathrm {e}}^{b^2\,x^2+c}}{2\,b^4}-\frac {x^2\,{\mathrm {e}}^{b^2\,x^2+c}}{2\,b^2}\right ) \]
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