Integrand size = 19, antiderivative size = 121 \[ \int e^{c+b^2 x^2} x^4 \text {erfi}(b x) \, dx=\frac {e^{c+2 b^2 x^2}}{2 b^5 \sqrt {\pi }}-\frac {e^{c+2 b^2 x^2} x^2}{4 b^3 \sqrt {\pi }}-\frac {3 e^{c+b^2 x^2} x \text {erfi}(b x)}{4 b^4}+\frac {e^{c+b^2 x^2} x^3 \text {erfi}(b x)}{2 b^2}+\frac {3 e^c \sqrt {\pi } \text {erfi}(b x)^2}{16 b^5} \]
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Time = 0.12 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6522, 6510, 30, 2240, 2243} \[ \int e^{c+b^2 x^2} x^4 \text {erfi}(b x) \, dx=\frac {3 \sqrt {\pi } e^c \text {erfi}(b x)^2}{16 b^5}+\frac {x^3 e^{b^2 x^2+c} \text {erfi}(b x)}{2 b^2}+\frac {e^{2 b^2 x^2+c}}{2 \sqrt {\pi } b^5}-\frac {3 x e^{b^2 x^2+c} \text {erfi}(b x)}{4 b^4}-\frac {x^2 e^{2 b^2 x^2+c}}{4 \sqrt {\pi } b^3} \]
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Rule 30
Rule 2240
Rule 2243
Rule 6510
Rule 6522
Rubi steps \begin{align*} \text {integral}& = \frac {e^{c+b^2 x^2} x^3 \text {erfi}(b x)}{2 b^2}-\frac {3 \int e^{c+b^2 x^2} x^2 \text {erfi}(b x) \, dx}{2 b^2}-\frac {\int e^{c+2 b^2 x^2} x^3 \, dx}{b \sqrt {\pi }} \\ & = -\frac {e^{c+2 b^2 x^2} x^2}{4 b^3 \sqrt {\pi }}-\frac {3 e^{c+b^2 x^2} x \text {erfi}(b x)}{4 b^4}+\frac {e^{c+b^2 x^2} x^3 \text {erfi}(b x)}{2 b^2}+\frac {3 \int e^{c+b^2 x^2} \text {erfi}(b x) \, dx}{4 b^4}+\frac {\int e^{c+2 b^2 x^2} x \, dx}{2 b^3 \sqrt {\pi }}+\frac {3 \int e^{c+2 b^2 x^2} x \, dx}{2 b^3 \sqrt {\pi }} \\ & = \frac {e^{c+2 b^2 x^2}}{2 b^5 \sqrt {\pi }}-\frac {e^{c+2 b^2 x^2} x^2}{4 b^3 \sqrt {\pi }}-\frac {3 e^{c+b^2 x^2} x \text {erfi}(b x)}{4 b^4}+\frac {e^{c+b^2 x^2} x^3 \text {erfi}(b x)}{2 b^2}+\frac {\left (3 e^c \sqrt {\pi }\right ) \text {Subst}(\int x \, dx,x,\text {erfi}(b x))}{8 b^5} \\ & = \frac {e^{c+2 b^2 x^2}}{2 b^5 \sqrt {\pi }}-\frac {e^{c+2 b^2 x^2} x^2}{4 b^3 \sqrt {\pi }}-\frac {3 e^{c+b^2 x^2} x \text {erfi}(b x)}{4 b^4}+\frac {e^{c+b^2 x^2} x^3 \text {erfi}(b x)}{2 b^2}+\frac {3 e^c \sqrt {\pi } \text {erfi}(b x)^2}{16 b^5} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.64 \[ \int e^{c+b^2 x^2} x^4 \text {erfi}(b x) \, dx=\frac {e^c \left (-4 e^{2 b^2 x^2} \left (-2+b^2 x^2\right )+4 b e^{b^2 x^2} \sqrt {\pi } x \left (-3+2 b^2 x^2\right ) \text {erfi}(b x)+3 \pi \text {erfi}(b x)^2\right )}{16 b^5 \sqrt {\pi }} \]
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\[\int {\mathrm e}^{b^{2} x^{2}+c} x^{4} \operatorname {erfi}\left (b x \right )d x\]
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none
Time = 0.25 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.61 \[ \int e^{c+b^2 x^2} x^4 \text {erfi}(b x) \, dx=\frac {{\left (4 \, {\left (2 \, \pi b^{3} x^{3} - 3 \, \pi b x\right )} \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2}\right )} + \sqrt {\pi } {\left (3 \, \pi \operatorname {erfi}\left (b x\right )^{2} - 4 \, {\left (b^{2} x^{2} - 2\right )} e^{\left (2 \, b^{2} x^{2}\right )}\right )}\right )} e^{c}}{16 \, \pi b^{5}} \]
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Time = 1.42 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.02 \[ \int e^{c+b^2 x^2} x^4 \text {erfi}(b x) \, dx=\begin {cases} \frac {x^{3} e^{c} e^{b^{2} x^{2}} \operatorname {erfi}{\left (b x \right )}}{2 b^{2}} - \frac {x^{2} e^{c} e^{2 b^{2} x^{2}}}{4 \sqrt {\pi } b^{3}} - \frac {3 x e^{c} e^{b^{2} x^{2}} \operatorname {erfi}{\left (b x \right )}}{4 b^{4}} + \frac {e^{c} e^{2 b^{2} x^{2}}}{2 \sqrt {\pi } b^{5}} + \frac {3 \sqrt {\pi } e^{c} \operatorname {erfi}^{2}{\left (b x \right )}}{16 b^{5}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]
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\[ \int e^{c+b^2 x^2} x^4 \text {erfi}(b x) \, dx=\int { x^{4} \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )} \,d x } \]
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\[ \int e^{c+b^2 x^2} x^4 \text {erfi}(b x) \, dx=\int { x^{4} \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )} \,d x } \]
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Time = 5.15 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.04 \[ \int e^{c+b^2 x^2} x^4 \text {erfi}(b x) \, dx=\mathrm {erfi}\left (b\,x\right )\,\left (\frac {x^3\,{\mathrm {e}}^{b^2\,x^2+c}}{2\,b^2}-\frac {3\,x\,{\mathrm {e}}^{b^2\,x^2+c}}{4\,b^4}+\frac {3\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b^2\,x}{\sqrt {b^2}}\right )\,{\mathrm {e}}^c}{8\,{\left (b^2\right )}^{5/2}}\right )+\frac {8\,{\mathrm {e}}^{2\,b^2\,x^2+c}-3\,\pi \,{\mathrm {erfi}\left (\frac {b^2\,x}{\sqrt {b^2}}\right )}^2\,{\mathrm {e}}^c}{16\,b^5\,\sqrt {\pi }}-\frac {x^2\,{\mathrm {e}}^{2\,b^2\,x^2+c}}{4\,b^3\,\sqrt {\pi }} \]
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