Optimal. Leaf size=47 \[ \frac{e^{b^2 x^2+c} \text{Erfi}(b x)}{2 b^2}-\frac{e^c \text{Erfi}\left (\sqrt{2} b x\right )}{2 \sqrt{2} b^2} \]
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Rubi [A] time = 0.0380216, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {6384, 2204} \[ \frac{e^{b^2 x^2+c} \text{Erfi}(b x)}{2 b^2}-\frac{e^c \text{Erfi}\left (\sqrt{2} b x\right )}{2 \sqrt{2} b^2} \]
Antiderivative was successfully verified.
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Rule 6384
Rule 2204
Rubi steps
\begin{align*} \int e^{c+b^2 x^2} x \text{erfi}(b x) \, dx &=\frac{e^{c+b^2 x^2} \text{erfi}(b x)}{2 b^2}-\frac{\int e^{c+2 b^2 x^2} \, dx}{b \sqrt{\pi }}\\ &=\frac{e^{c+b^2 x^2} \text{erfi}(b x)}{2 b^2}-\frac{e^c \text{erfi}\left (\sqrt{2} b x\right )}{2 \sqrt{2} b^2}\\ \end{align*}
Mathematica [A] time = 0.009964, size = 42, normalized size = 0.89 \[ \frac{e^c \left (2 e^{b^2 x^2} \text{Erfi}(b x)-\sqrt{2} \text{Erfi}\left (\sqrt{2} b x\right )\right )}{4 b^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.227, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{{b}^{2}{x}^{2}+c}}x{\it erfi} \left ( bx \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \operatorname{erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.69679, size = 123, normalized size = 2.62 \begin{align*} \frac{2 \, b \operatorname{erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )} - \sqrt{2} \sqrt{b^{2}} \operatorname{erfi}\left (\sqrt{2} \sqrt{b^{2}} x\right ) e^{c}}{4 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} e^{c} \int x e^{b^{2} x^{2}} \operatorname{erfi}{\left (b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \operatorname{erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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