Optimal. Leaf size=71 \[ -\frac{x^2 e^{-b^2 x^2} \text{Erfi}(b x)}{2 b^2}-\frac{e^{-b^2 x^2} \text{Erfi}(b x)}{2 b^4}+\frac{x}{\sqrt{\pi } b^3}+\frac{x^3}{3 \sqrt{\pi } b} \]
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Rubi [A] time = 0.0685207, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {6387, 6384, 8, 30} \[ -\frac{x^2 e^{-b^2 x^2} \text{Erfi}(b x)}{2 b^2}-\frac{e^{-b^2 x^2} \text{Erfi}(b x)}{2 b^4}+\frac{x}{\sqrt{\pi } b^3}+\frac{x^3}{3 \sqrt{\pi } b} \]
Antiderivative was successfully verified.
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Rule 6387
Rule 6384
Rule 8
Rule 30
Rubi steps
\begin{align*} \int e^{-b^2 x^2} x^3 \text{erfi}(b x) \, dx &=-\frac{e^{-b^2 x^2} x^2 \text{erfi}(b x)}{2 b^2}+\frac{\int e^{-b^2 x^2} x \text{erfi}(b x) \, dx}{b^2}+\frac{\int x^2 \, dx}{b \sqrt{\pi }}\\ &=\frac{x^3}{3 b \sqrt{\pi }}-\frac{e^{-b^2 x^2} \text{erfi}(b x)}{2 b^4}-\frac{e^{-b^2 x^2} x^2 \text{erfi}(b x)}{2 b^2}+\frac{\int 1 \, dx}{b^3 \sqrt{\pi }}\\ &=\frac{x}{b^3 \sqrt{\pi }}+\frac{x^3}{3 b \sqrt{\pi }}-\frac{e^{-b^2 x^2} \text{erfi}(b x)}{2 b^4}-\frac{e^{-b^2 x^2} x^2 \text{erfi}(b x)}{2 b^2}\\ \end{align*}
Mathematica [A] time = 0.026179, size = 51, normalized size = 0.72 \[ \frac{\frac{2 b x \left (b^2 x^2+3\right )}{\sqrt{\pi }}-3 e^{-b^2 x^2} \left (b^2 x^2+1\right ) \text{Erfi}(b x)}{6 b^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.115, size = 72, normalized size = 1. \begin{align*}{\frac{2\,{{\rm e}^{{b}^{2}{x}^{2}}}{b}^{3}{x}^{3}-3\,\sqrt{\pi }{\it erfi} \left ( bx \right ){b}^{2}{x}^{2}+6\,{{\rm e}^{{b}^{2}{x}^{2}}}bx-3\,\sqrt{\pi }{\it erfi} \left ( bx \right ) }{6\,\sqrt{\pi }{b}^{4}{{\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^{3} \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 = 2.57583, size = 138, normalized size = 1.94 \begin{align*} \frac{{\left (2 \, \sqrt{\pi }{\left (b^{3} x^{3} + 3 \, b x\right )} e^{\left (b^{2} x^{2}\right )} - 3 \,{\left (\pi + \pi b^{2} x^{2}\right )} \operatorname{erfi}\left (b x\right )\right )} e^{\left (-b^{2} x^{2}\right )}}{6 \, \pi b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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^{3} \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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