Optimal. Leaf size=32 \[ \frac{x}{\sqrt{\pi } b}-\frac{e^{-b^2 x^2} \text{Erfi}(b x)}{2 b^2} \]
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Rubi [A] time = 0.027782, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {6384, 8} \[ \frac{x}{\sqrt{\pi } b}-\frac{e^{-b^2 x^2} \text{Erfi}(b x)}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 6384
Rule 8
Rubi steps
\begin{align*} \int e^{-b^2 x^2} x \text{erfi}(b x) \, dx &=-\frac{e^{-b^2 x^2} \text{erfi}(b x)}{2 b^2}+\frac{\int 1 \, dx}{b \sqrt{\pi }}\\ &=\frac{x}{b \sqrt{\pi }}-\frac{e^{-b^2 x^2} \text{erfi}(b x)}{2 b^2}\\ \end{align*}
Mathematica [A] time = 0.0140244, size = 32, normalized size = 1. \[ \frac{x}{\sqrt{\pi } b}-\frac{e^{-b^2 x^2} \text{Erfi}(b x)}{2 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.063, size = 41, normalized size = 1.3 \begin{align*}{\frac{2\,{{\rm e}^{{b}^{2}{x}^{2}}}bx-\sqrt{\pi }{\it erfi} \left ( bx \right ) }{2\,{b}^{2}\sqrt{\pi }{{\rm e}^{{b}^{2}{x}^{2}}}}} \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}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.43269, size = 96, normalized size = 3. \begin{align*} \frac{{\left (2 \, \sqrt{\pi } b x e^{\left (b^{2} x^{2}\right )} - \pi \operatorname{erfi}\left (b x\right )\right )} e^{\left (-b^{2} x^{2}\right )}}{2 \, \pi b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 32.3085, size = 27, normalized size = 0.84 \begin{align*} \begin{cases} \frac{x}{\sqrt{\pi } b} - \frac{e^{- b^{2} x^{2}} \operatorname{erfi}{\left (b x \right )}}{2 b^{2}} & \text{for}\: b \neq 0 \\0 & \text{otherwise} \end{cases} \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}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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