Integrand size = 6, antiderivative size = 45 \[ \int x \text {erfi}(b x) \, dx=-\frac {e^{b^2 x^2} x}{2 b \sqrt {\pi }}+\frac {\text {erfi}(b x)}{4 b^2}+\frac {1}{2} x^2 \text {erfi}(b x) \]
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Time = 0.02 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6498, 2243, 2235} \[ \int x \text {erfi}(b x) \, dx=\frac {\text {erfi}(b x)}{4 b^2}-\frac {x e^{b^2 x^2}}{2 \sqrt {\pi } b}+\frac {1}{2} x^2 \text {erfi}(b x) \]
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Rule 2235
Rule 2243
Rule 6498
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x^2 \text {erfi}(b x)-\frac {b \int e^{b^2 x^2} x^2 \, dx}{\sqrt {\pi }} \\ & = -\frac {e^{b^2 x^2} x}{2 b \sqrt {\pi }}+\frac {1}{2} x^2 \text {erfi}(b x)+\frac {\int e^{b^2 x^2} \, dx}{2 b \sqrt {\pi }} \\ & = -\frac {e^{b^2 x^2} x}{2 b \sqrt {\pi }}+\frac {\text {erfi}(b x)}{4 b^2}+\frac {1}{2} x^2 \text {erfi}(b x) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.87 \[ \int x \text {erfi}(b x) \, dx=\frac {1}{4} \left (-\frac {2 e^{b^2 x^2} x}{b \sqrt {\pi }}+\left (\frac {1}{b^2}+2 x^2\right ) \text {erfi}(b x)\right ) \]
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Result contains complex when optimal does not.
Time = 0.14 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.98
method | result | size |
meijerg | \(\frac {i \left (i x b \,{\mathrm e}^{b^{2} x^{2}}-\frac {i \left (6 b^{2} x^{2}+3\right ) \operatorname {erfi}\left (b x \right ) \sqrt {\pi }}{6}\right )}{2 b^{2} \sqrt {\pi }}\) | \(44\) |
derivativedivides | \(\frac {\frac {b^{2} x^{2} \operatorname {erfi}\left (b x \right )}{2}-\frac {\frac {{\mathrm e}^{b^{2} x^{2}} b x}{2}-\frac {\operatorname {erfi}\left (b x \right ) \sqrt {\pi }}{4}}{\sqrt {\pi }}}{b^{2}}\) | \(45\) |
default | \(\frac {\frac {b^{2} x^{2} \operatorname {erfi}\left (b x \right )}{2}-\frac {\frac {{\mathrm e}^{b^{2} x^{2}} b x}{2}-\frac {\operatorname {erfi}\left (b x \right ) \sqrt {\pi }}{4}}{\sqrt {\pi }}}{b^{2}}\) | \(45\) |
parallelrisch | \(\frac {2 x^{2} \operatorname {erfi}\left (b x \right ) \sqrt {\pi }\, b^{2}-2 \,{\mathrm e}^{b^{2} x^{2}} b x +\operatorname {erfi}\left (b x \right ) \sqrt {\pi }}{4 \sqrt {\pi }\, b^{2}}\) | \(45\) |
parts | \(\frac {x^{2} \operatorname {erfi}\left (b x \right )}{2}-\frac {b \left (\frac {x \,{\mathrm e}^{b^{2} x^{2}}}{2 b^{2}}+\frac {i \sqrt {\pi }\, \operatorname {erf}\left (i b x \right )}{4 b^{3}}\right )}{\sqrt {\pi }}\) | \(47\) |
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none
Time = 0.25 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.91 \[ \int x \text {erfi}(b x) \, dx=-\frac {2 \, \sqrt {\pi } b x e^{\left (b^{2} x^{2}\right )} - {\left (\pi + 2 \, \pi b^{2} x^{2}\right )} \operatorname {erfi}\left (b x\right )}{4 \, \pi b^{2}} \]
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Time = 0.16 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.87 \[ \int x \text {erfi}(b x) \, dx=\begin {cases} \frac {x^{2} \operatorname {erfi}{\left (b x \right )}}{2} - \frac {x e^{b^{2} x^{2}}}{2 \sqrt {\pi } b} + \frac {\operatorname {erfi}{\left (b x \right )}}{4 b^{2}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]
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Result contains complex when optimal does not.
Time = 0.19 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.98 \[ \int x \text {erfi}(b x) \, dx=\frac {1}{2} \, x^{2} \operatorname {erfi}\left (b x\right ) - \frac {b {\left (\frac {2 \, x e^{\left (b^{2} x^{2}\right )}}{b^{2}} + \frac {i \, \sqrt {\pi } \operatorname {erf}\left (i \, b x\right )}{b^{3}}\right )}}{4 \, \sqrt {\pi }} \]
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\[ \int x \text {erfi}(b x) \, dx=\int { x \operatorname {erfi}\left (b x\right ) \,d x } \]
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Time = 5.09 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.96 \[ \int x \text {erfi}(b x) \, dx=\frac {x^2\,\mathrm {erfi}\left (b\,x\right )}{2}+\frac {b\,\mathrm {erfi}\left (x\,\sqrt {b^2}\right )}{4\,{\left (b^2\right )}^{3/2}}-\frac {x\,{\mathrm {e}}^{b^2\,x^2}}{2\,b\,\sqrt {\pi }} \]
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