Integrand size = 10, antiderivative size = 124 \[ \int x^3 \text {erfi}(b x)^2 \, dx=-\frac {e^{2 b^2 x^2}}{2 b^4 \pi }+\frac {e^{2 b^2 x^2} x^2}{4 b^2 \pi }+\frac {3 e^{b^2 x^2} x \text {erfi}(b x)}{4 b^3 \sqrt {\pi }}-\frac {e^{b^2 x^2} x^3 \text {erfi}(b x)}{2 b \sqrt {\pi }}-\frac {3 \text {erfi}(b x)^2}{16 b^4}+\frac {1}{4} x^4 \text {erfi}(b x)^2 \]
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Time = 0.11 (sec) , antiderivative size = 124, 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 = {6501, 6522, 6510, 30, 2240, 2243} \[ \int x^3 \text {erfi}(b x)^2 \, dx=-\frac {3 \text {erfi}(b x)^2}{16 b^4}-\frac {x^3 e^{b^2 x^2} \text {erfi}(b x)}{2 \sqrt {\pi } b}+\frac {x^2 e^{2 b^2 x^2}}{4 \pi b^2}-\frac {e^{2 b^2 x^2}}{2 \pi b^4}+\frac {3 x e^{b^2 x^2} \text {erfi}(b x)}{4 \sqrt {\pi } b^3}+\frac {1}{4} x^4 \text {erfi}(b x)^2 \]
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Rule 30
Rule 2240
Rule 2243
Rule 6501
Rule 6510
Rule 6522
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x^4 \text {erfi}(b x)^2-\frac {b \int e^{b^2 x^2} x^4 \text {erfi}(b x) \, dx}{\sqrt {\pi }} \\ & = -\frac {e^{b^2 x^2} x^3 \text {erfi}(b x)}{2 b \sqrt {\pi }}+\frac {1}{4} x^4 \text {erfi}(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 {erfi}(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 {erfi}(b x)}{4 b^3 \sqrt {\pi }}-\frac {e^{b^2 x^2} x^3 \text {erfi}(b x)}{2 b \sqrt {\pi }}+\frac {1}{4} x^4 \text {erfi}(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 {erfi}(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 {erfi}(b x)}{4 b^3 \sqrt {\pi }}-\frac {e^{b^2 x^2} x^3 \text {erfi}(b x)}{2 b \sqrt {\pi }}+\frac {1}{4} x^4 \text {erfi}(b x)^2-\frac {3 \text {Subst}(\int x \, dx,x,\text {erfi}(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 {erfi}(b x)}{4 b^3 \sqrt {\pi }}-\frac {e^{b^2 x^2} x^3 \text {erfi}(b x)}{2 b \sqrt {\pi }}-\frac {3 \text {erfi}(b x)^2}{16 b^4}+\frac {1}{4} x^4 \text {erfi}(b x)^2 \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.66 \[ \int x^3 \text {erfi}(b x)^2 \, dx=\frac {4 e^{2 b^2 x^2} \left (-2+b^2 x^2\right )-4 b e^{b^2 x^2} \sqrt {\pi } x \left (-3+2 b^2 x^2\right ) \text {erfi}(b x)+\pi \left (-3+4 b^4 x^4\right ) \text {erfi}(b x)^2}{16 b^4 \pi } \]
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Time = 0.16 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.90
method | result | size |
parallelrisch | \(\frac {4 \operatorname {erfi}\left (b x \right )^{2} x^{4} \pi ^{\frac {3}{2}} b^{4}-8 \,\operatorname {erfi}\left (b x \right ) {\mathrm e}^{b^{2} x^{2}} x^{3} b^{3} \pi +4 \,{\mathrm e}^{2 b^{2} x^{2}} x^{2} b^{2} \sqrt {\pi }+12 \,\operatorname {erfi}\left (b x \right ) x \,{\mathrm e}^{b^{2} x^{2}} b \pi -3 \operatorname {erfi}\left (b x \right )^{2} \pi ^{\frac {3}{2}}-8 \,{\mathrm e}^{2 b^{2} x^{2}} \sqrt {\pi }}{16 \pi ^{\frac {3}{2}} b^{4}}\) | \(112\) |
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Time = 0.26 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.64 \[ \int x^3 \text {erfi}(b x)^2 \, dx=-\frac {4 \, \sqrt {\pi } {\left (2 \, b^{3} x^{3} - 3 \, b x\right )} \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2}\right )} + {\left (3 \, \pi - 4 \, \pi b^{4} x^{4}\right )} \operatorname {erfi}\left (b x\right )^{2} - 4 \, {\left (b^{2} x^{2} - 2\right )} e^{\left (2 \, b^{2} x^{2}\right )}}{16 \, \pi b^{4}} \]
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Time = 0.31 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.94 \[ \int x^3 \text {erfi}(b x)^2 \, dx=\begin {cases} \frac {x^{4} \operatorname {erfi}^{2}{\left (b x \right )}}{4} - \frac {x^{3} e^{b^{2} x^{2}} \operatorname {erfi}{\left (b x \right )}}{2 \sqrt {\pi } b} + \frac {x^{2} e^{2 b^{2} x^{2}}}{4 \pi b^{2}} + \frac {3 x e^{b^{2} x^{2}} \operatorname {erfi}{\left (b x \right )}}{4 \sqrt {\pi } b^{3}} - \frac {e^{2 b^{2} x^{2}}}{2 \pi b^{4}} - \frac {3 \operatorname {erfi}^{2}{\left (b x \right )}}{16 b^{4}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]
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\[ \int x^3 \text {erfi}(b x)^2 \, dx=\int { x^{3} \operatorname {erfi}\left (b x\right )^{2} \,d x } \]
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\[ \int x^3 \text {erfi}(b x)^2 \, dx=\int { x^{3} \operatorname {erfi}\left (b x\right )^{2} \,d x } \]
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Time = 4.89 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.81 \[ \int x^3 \text {erfi}(b x)^2 \, dx=\frac {x^4\,{\mathrm {erfi}\left (b\,x\right )}^2}{4}-\frac {\frac {{\mathrm {e}}^{2\,b^2\,x^2}}{2}+\frac {3\,\pi \,{\mathrm {erfi}\left (b\,x\right )}^2}{16}-\frac {b^2\,x^2\,{\mathrm {e}}^{2\,b^2\,x^2}}{4}+\frac {b^3\,x^3\,\sqrt {\pi }\,{\mathrm {e}}^{b^2\,x^2}\,\mathrm {erfi}\left (b\,x\right )}{2}-\frac {3\,b\,x\,\sqrt {\pi }\,{\mathrm {e}}^{b^2\,x^2}\,\mathrm {erfi}\left (b\,x\right )}{4}}{b^4\,\pi } \]
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