Optimal. Leaf size=69 \[ -\frac{3 \text{Erfi}(b x)}{16 b^4}-\frac{x^3 e^{b^2 x^2}}{4 \sqrt{\pi } b}+\frac{3 x e^{b^2 x^2}}{8 \sqrt{\pi } b^3}+\frac{1}{4} x^4 \text{Erfi}(b x) \]
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Rubi [A] time = 0.0578031, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {6363, 2212, 2204} \[ -\frac{3 \text{Erfi}(b x)}{16 b^4}-\frac{x^3 e^{b^2 x^2}}{4 \sqrt{\pi } b}+\frac{3 x e^{b^2 x^2}}{8 \sqrt{\pi } b^3}+\frac{1}{4} x^4 \text{Erfi}(b x) \]
Antiderivative was successfully verified.
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Rule 6363
Rule 2212
Rule 2204
Rubi steps
\begin{align*} \int x^3 \text{erfi}(b x) \, dx &=\frac{1}{4} x^4 \text{erfi}(b x)-\frac{b \int e^{b^2 x^2} x^4 \, dx}{2 \sqrt{\pi }}\\ &=-\frac{e^{b^2 x^2} x^3}{4 b \sqrt{\pi }}+\frac{1}{4} x^4 \text{erfi}(b x)+\frac{3 \int e^{b^2 x^2} x^2 \, dx}{4 b \sqrt{\pi }}\\ &=\frac{3 e^{b^2 x^2} x}{8 b^3 \sqrt{\pi }}-\frac{e^{b^2 x^2} x^3}{4 b \sqrt{\pi }}+\frac{1}{4} x^4 \text{erfi}(b x)-\frac{3 \int e^{b^2 x^2} \, dx}{8 b^3 \sqrt{\pi }}\\ &=\frac{3 e^{b^2 x^2} x}{8 b^3 \sqrt{\pi }}-\frac{e^{b^2 x^2} x^3}{4 b \sqrt{\pi }}-\frac{3 \text{erfi}(b x)}{16 b^4}+\frac{1}{4} x^4 \text{erfi}(b x)\\ \end{align*}
Mathematica [A] time = 0.0280703, size = 51, normalized size = 0.74 \[ \frac{\left (4 b^4 x^4-3\right ) \text{Erfi}(b x)-\frac{2 b x e^{b^2 x^2} \left (2 b^2 x^2-3\right )}{\sqrt{\pi }}}{16 b^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 61, normalized size = 0.9 \begin{align*}{\frac{1}{{b}^{4}} \left ({\frac{{b}^{4}{x}^{4}{\it erfi} \left ( bx \right ) }{4}}-{\frac{1}{2\,\sqrt{\pi }} \left ({\frac{{{\rm e}^{{b}^{2}{x}^{2}}}{b}^{3}{x}^{3}}{2}}-{\frac{3\,{{\rm e}^{{b}^{2}{x}^{2}}}bx}{4}}+{\frac{3\,\sqrt{\pi }{\it erfi} \left ( bx \right ) }{8}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 0.983883, size = 74, normalized size = 1.07 \begin{align*} \frac{1}{4} \, x^{4} \operatorname{erfi}\left (b x\right ) - \frac{b{\left (\frac{2 \,{\left (2 \, b^{2} x^{3} - 3 \, x\right )} e^{\left (b^{2} x^{2}\right )}}{b^{4}} - \frac{3 i \, \sqrt{\pi } \operatorname{erf}\left (i \, b x\right )}{b^{5}}\right )}}{16 \, \sqrt{\pi }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.12278, size = 128, normalized size = 1.86 \begin{align*} -\frac{2 \, \sqrt{\pi }{\left (2 \, b^{3} x^{3} - 3 \, b x\right )} e^{\left (b^{2} x^{2}\right )} +{\left (3 \, \pi - 4 \, \pi b^{4} x^{4}\right )} \operatorname{erfi}\left (b x\right )}{16 \, \pi b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.01405, size = 65, normalized size = 0.94 \begin{align*} \begin{cases} \frac{x^{4} \operatorname{erfi}{\left (b x \right )}}{4} - \frac{x^{3} e^{b^{2} x^{2}}}{4 \sqrt{\pi } b} + \frac{3 x e^{b^{2} x^{2}}}{8 \sqrt{\pi } b^{3}} - \frac{3 \operatorname{erfi}{\left (b x \right )}}{16 b^{4}} & \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^{3} \operatorname{erfi}\left (b x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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