Optimal. Leaf size=53 \[ \frac{\text{Erfi}(b x) e^{c+d x^2}}{2 d}-\frac{b e^c \text{Erfi}\left (x \sqrt{b^2+d}\right )}{2 d \sqrt{b^2+d}} \]
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Rubi [A] time = 0.0391289, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {6384, 2204} \[ \frac{\text{Erfi}(b x) e^{c+d x^2}}{2 d}-\frac{b e^c \text{Erfi}\left (x \sqrt{b^2+d}\right )}{2 d \sqrt{b^2+d}} \]
Antiderivative was successfully verified.
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Rule 6384
Rule 2204
Rubi steps
\begin{align*} \int e^{c+d x^2} x \text{erfi}(b x) \, dx &=\frac{e^{c+d x^2} \text{erfi}(b x)}{2 d}-\frac{b \int e^{c+\left (b^2+d\right ) x^2} \, dx}{d \sqrt{\pi }}\\ &=\frac{e^{c+d x^2} \text{erfi}(b x)}{2 d}-\frac{b e^c \text{erfi}\left (\sqrt{b^2+d} x\right )}{2 d \sqrt{b^2+d}}\\ \end{align*}
Mathematica [A] time = 0.017985, size = 47, normalized size = 0.89 \[ \frac{e^c \left (e^{d x^2} \text{Erfi}(b x)-\frac{b \text{Erfi}\left (x \sqrt{b^2+d}\right )}{\sqrt{b^2+d}}\right )}{2 d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.237, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{d{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 (d 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 = 2.70942, size = 135, normalized size = 2.55 \begin{align*} \frac{\sqrt{-b^{2} - d} b \operatorname{erf}\left (\sqrt{-b^{2} - d} x\right ) e^{c} +{\left (b^{2} + d\right )} \operatorname{erfi}\left (b x\right ) e^{\left (d x^{2} + c\right )}}{2 \,{\left (b^{2} d + d^{2}\right )}} \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^{d 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 (d x^{2} + c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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