Optimal. Leaf size=112 \[ -\frac{x^3 e^{-b^2 x^2} \text{Erf}(b x)}{2 b^2}-\frac{3 x e^{-b^2 x^2} \text{Erf}(b x)}{4 b^4}+\frac{3 \sqrt{\pi } \text{Erf}(b x)^2}{16 b^5}-\frac{x^2 e^{-2 b^2 x^2}}{4 \sqrt{\pi } b^3}-\frac{e^{-2 b^2 x^2}}{2 \sqrt{\pi } b^5} \]
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Rubi [A] time = 0.146233, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {6385, 6373, 30, 2209, 2212} \[ -\frac{x^3 e^{-b^2 x^2} \text{Erf}(b x)}{2 b^2}-\frac{3 x e^{-b^2 x^2} \text{Erf}(b x)}{4 b^4}+\frac{3 \sqrt{\pi } \text{Erf}(b x)^2}{16 b^5}-\frac{x^2 e^{-2 b^2 x^2}}{4 \sqrt{\pi } b^3}-\frac{e^{-2 b^2 x^2}}{2 \sqrt{\pi } b^5} \]
Antiderivative was successfully verified.
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Rule 6385
Rule 6373
Rule 30
Rule 2209
Rule 2212
Rubi steps
\begin{align*} \int e^{-b^2 x^2} x^4 \text{erf}(b x) \, dx &=-\frac{e^{-b^2 x^2} x^3 \text{erf}(b x)}{2 b^2}+\frac{3 \int e^{-b^2 x^2} x^2 \text{erf}(b x) \, dx}{2 b^2}+\frac{\int e^{-2 b^2 x^2} x^3 \, dx}{b \sqrt{\pi }}\\ &=-\frac{e^{-2 b^2 x^2} x^2}{4 b^3 \sqrt{\pi }}-\frac{3 e^{-b^2 x^2} x \text{erf}(b x)}{4 b^4}-\frac{e^{-b^2 x^2} x^3 \text{erf}(b x)}{2 b^2}+\frac{3 \int e^{-b^2 x^2} \text{erf}(b x) \, dx}{4 b^4}+\frac{\int e^{-2 b^2 x^2} x \, dx}{2 b^3 \sqrt{\pi }}+\frac{3 \int e^{-2 b^2 x^2} x \, dx}{2 b^3 \sqrt{\pi }}\\ &=-\frac{e^{-2 b^2 x^2}}{2 b^5 \sqrt{\pi }}-\frac{e^{-2 b^2 x^2} x^2}{4 b^3 \sqrt{\pi }}-\frac{3 e^{-b^2 x^2} x \text{erf}(b x)}{4 b^4}-\frac{e^{-b^2 x^2} x^3 \text{erf}(b x)}{2 b^2}+\frac{\left (3 \sqrt{\pi }\right ) \operatorname{Subst}(\int x \, dx,x,\text{erf}(b x))}{8 b^5}\\ &=-\frac{e^{-2 b^2 x^2}}{2 b^5 \sqrt{\pi }}-\frac{e^{-2 b^2 x^2} x^2}{4 b^3 \sqrt{\pi }}-\frac{3 e^{-b^2 x^2} x \text{erf}(b x)}{4 b^4}-\frac{e^{-b^2 x^2} x^3 \text{erf}(b x)}{2 b^2}+\frac{3 \sqrt{\pi } \text{erf}(b x)^2}{16 b^5}\\ \end{align*}
Mathematica [A] time = 0.0285525, size = 85, normalized size = 0.76 \[ \frac{e^{-2 b^2 x^2} \left (3 \pi e^{2 b^2 x^2} \text{Erf}(b x)^2-4 \sqrt{\pi } b x e^{b^2 x^2} \left (2 b^2 x^2+3\right ) \text{Erf}(b x)-4 \left (b^2 x^2+2\right )\right )}{16 \sqrt{\pi } b^5} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.254, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{4}{\it Erf} \left ( bx \right ) }{{{\rm e}^{{b}^{2}{x}^{2}}}}}\, 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*} \frac{-\frac{{\left (2 \, b^{2} x^{2} + 1\right )} e^{\left (-2 \, b^{2} x^{2}\right )}}{4 \, b^{2}} - \frac{3 \, e^{\left (-2 \, b^{2} x^{2}\right )}}{4 \, b^{2}}}{2 \, \sqrt{\pi } b^{3}} - \frac{4 \,{\left (2 \, b^{3} x^{3} + 3 \, b x\right )} \operatorname{erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} - 3 \, \sqrt{\pi } \operatorname{erf}\left (b x\right )^{2}}{16 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 3.08856, size = 176, normalized size = 1.57 \begin{align*} -\frac{4 \,{\left (2 \, \pi b^{3} x^{3} + 3 \, \pi b x\right )} \operatorname{erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} - \sqrt{\pi }{\left (3 \, \pi \operatorname{erf}\left (b x\right )^{2} - 4 \,{\left (b^{2} x^{2} + 2\right )} e^{\left (-2 \, b^{2} x^{2}\right )}\right )}}{16 \, \pi b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 98.6659, size = 109, normalized size = 0.97 \begin{align*} \begin{cases} - \frac{x^{3} e^{- b^{2} x^{2}} \operatorname{erf}{\left (b x \right )}}{2 b^{2}} - \frac{x^{2} e^{- 2 b^{2} x^{2}}}{4 \sqrt{\pi } b^{3}} - \frac{3 x e^{- b^{2} x^{2}} \operatorname{erf}{\left (b x \right )}}{4 b^{4}} + \frac{3 \sqrt{\pi } \operatorname{erf}^{2}{\left (b x \right )}}{16 b^{5}} - \frac{e^{- 2 b^{2} x^{2}}}{2 \sqrt{\pi } b^{5}} & \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^{4} \operatorname{erf}\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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