Optimal. Leaf size=45 \[ \frac {\text {erfi}(b x)}{4 b^2}-\frac {x e^{b^2 x^2}}{2 \sqrt {\pi } b}+\frac {1}{2} x^2 \text {erfi}(b x) \]
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Rubi [A] time = 0.03, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6363, 2212, 2204} \[ \frac {\text {Erfi}(b x)}{4 b^2}-\frac {x e^{b^2 x^2}}{2 \sqrt {\pi } b}+\frac {1}{2} x^2 \text {Erfi}(b x) \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2212
Rule 6363
Rubi steps
\begin {align*} \int x \text {erfi}(b x) \, dx &=\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*}
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Mathematica [A] time = 0.04, size = 39, normalized size = 0.87 \[ \frac {1}{4} \left (\left (\frac {1}{b^2}+2 x^2\right ) \text {erfi}(b x)-\frac {2 x e^{b^2 x^2}}{\sqrt {\pi } b}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.18, size = 41, normalized size = 0.91 \[ -\frac {2 \, \sqrt {\pi } b x e^{\left (b^{2} x^{2}\right )} - {\left (\pi + 2 \, \pi b^{2} x^{2}\right )} \operatorname {erfi}\left (b x\right )}{4 \, \pi b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \operatorname {erfi}\left (b x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 45, normalized size = 1.00 \[ \frac {\frac {b^{2} x^{2} \erfi \left (b x \right )}{2}-\frac {\frac {{\mathrm e}^{b^{2} x^{2}} b x}{2}-\frac {\sqrt {\pi }\, \erfi \left (b x \right )}{4}}{\sqrt {\pi }}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.33, size = 44, normalized size = 0.98 \[ \frac {1}{2} \, x^{2} \operatorname {erfi}\left (b x\right ) - \frac {b {\left (\frac {2 \, x e^{\left (b^{2} x^{2}\right )}}{b^{2}} + \frac {i \, \sqrt {\pi } \operatorname {erf}\left (i \, b x\right )}{b^{3}}\right )}}{4 \, \sqrt {\pi }} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 43, normalized size = 0.96 \[ \frac {x^2\,\mathrm {erfi}\left (b\,x\right )}{2}+\frac {b\,\mathrm {erfi}\left (x\,\sqrt {b^2}\right )}{4\,{\left (b^2\right )}^{3/2}}-\frac {x\,{\mathrm {e}}^{b^2\,x^2}}{2\,b\,\sqrt {\pi }} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.19, size = 39, normalized size = 0.87 \[ \begin {cases} \frac {x^{2} \operatorname {erfi}{\left (b x \right )}}{2} - \frac {x e^{b^{2} x^{2}}}{2 \sqrt {\pi } b} + \frac {\operatorname {erfi}{\left (b x \right )}}{4 b^{2}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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