Integrand size = 8, antiderivative size = 57 \[ \int x^2 \text {erfi}(b x) \, dx=\frac {e^{b^2 x^2}}{3 b^3 \sqrt {\pi }}-\frac {e^{b^2 x^2} x^2}{3 b \sqrt {\pi }}+\frac {1}{3} x^3 \text {erfi}(b x) \]
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Time = 0.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6498, 2243, 2240} \[ \int x^2 \text {erfi}(b x) \, dx=-\frac {x^2 e^{b^2 x^2}}{3 \sqrt {\pi } b}+\frac {e^{b^2 x^2}}{3 \sqrt {\pi } b^3}+\frac {1}{3} x^3 \text {erfi}(b x) \]
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Rule 2240
Rule 2243
Rule 6498
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x^3 \text {erfi}(b x)-\frac {(2 b) \int e^{b^2 x^2} x^3 \, dx}{3 \sqrt {\pi }} \\ & = -\frac {e^{b^2 x^2} x^2}{3 b \sqrt {\pi }}+\frac {1}{3} x^3 \text {erfi}(b x)+\frac {2 \int e^{b^2 x^2} x \, dx}{3 b \sqrt {\pi }} \\ & = \frac {e^{b^2 x^2}}{3 b^3 \sqrt {\pi }}-\frac {e^{b^2 x^2} x^2}{3 b \sqrt {\pi }}+\frac {1}{3} x^3 \text {erfi}(b x) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.72 \[ \int x^2 \text {erfi}(b x) \, dx=\frac {1}{3} \left (\frac {e^{b^2 x^2} \left (1-b^2 x^2\right )}{b^3 \sqrt {\pi }}+x^3 \text {erfi}(b x)\right ) \]
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Time = 0.48 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.81
method | result | size |
meijerg | \(\frac {-\frac {2}{3}+\frac {\left (-2 b^{2} x^{2}+2\right ) {\mathrm e}^{b^{2} x^{2}}}{3}+\frac {2 b^{3} x^{3} \sqrt {\pi }\, \operatorname {erfi}\left (b x \right )}{3}}{2 b^{3} \sqrt {\pi }}\) | \(46\) |
parts | \(\frac {x^{3} \operatorname {erfi}\left (b x \right )}{3}-\frac {2 b \left (\frac {x^{2} {\mathrm e}^{b^{2} x^{2}}}{2 b^{2}}-\frac {{\mathrm e}^{b^{2} x^{2}}}{2 b^{4}}\right )}{3 \sqrt {\pi }}\) | \(47\) |
parallelrisch | \(\frac {b^{3} x^{3} \sqrt {\pi }\, \operatorname {erfi}\left (b x \right )-b^{2} x^{2} {\mathrm e}^{b^{2} x^{2}}+{\mathrm e}^{b^{2} x^{2}}}{3 b^{3} \sqrt {\pi }}\) | \(48\) |
derivativedivides | \(\frac {\frac {b^{3} x^{3} \operatorname {erfi}\left (b x \right )}{3}-\frac {2 \left (\frac {b^{2} x^{2} {\mathrm e}^{b^{2} x^{2}}}{2}-\frac {{\mathrm e}^{b^{2} x^{2}}}{2}\right )}{3 \sqrt {\pi }}}{b^{3}}\) | \(50\) |
default | \(\frac {\frac {b^{3} x^{3} \operatorname {erfi}\left (b x \right )}{3}-\frac {2 \left (\frac {b^{2} x^{2} {\mathrm e}^{b^{2} x^{2}}}{2}-\frac {{\mathrm e}^{b^{2} x^{2}}}{2}\right )}{3 \sqrt {\pi }}}{b^{3}}\) | \(50\) |
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Time = 0.24 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.75 \[ \int x^2 \text {erfi}(b x) \, dx=\frac {\pi b^{3} x^{3} \operatorname {erfi}\left (b x\right ) - \sqrt {\pi } {\left (b^{2} x^{2} - 1\right )} e^{\left (b^{2} x^{2}\right )}}{3 \, \pi b^{3}} \]
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Time = 0.19 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.86 \[ \int x^2 \text {erfi}(b x) \, dx=\begin {cases} \frac {x^{3} \operatorname {erfi}{\left (b x \right )}}{3} - \frac {x^{2} e^{b^{2} x^{2}}}{3 \sqrt {\pi } b} + \frac {e^{b^{2} x^{2}}}{3 \sqrt {\pi } b^{3}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.61 \[ \int x^2 \text {erfi}(b x) \, dx=\frac {1}{3} \, x^{3} \operatorname {erfi}\left (b x\right ) - \frac {{\left (b^{2} x^{2} - 1\right )} e^{\left (b^{2} x^{2}\right )}}{3 \, \sqrt {\pi } b^{3}} \]
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\[ \int x^2 \text {erfi}(b x) \, dx=\int { x^{2} \operatorname {erfi}\left (b x\right ) \,d x } \]
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Time = 0.08 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.61 \[ \int x^2 \text {erfi}(b x) \, dx=\frac {x^3\,\mathrm {erfi}\left (b\,x\right )}{3}-\frac {{\mathrm {e}}^{b^2\,x^2}\,\left (b^2\,x^2-1\right )}{3\,b^3\,\sqrt {\pi }} \]
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