Optimal. Leaf size=135 \[ -\frac{x^4 e^{-b^2 x^2} \text{Erf}(b x)}{2 b^2}-\frac{x^2 e^{-b^2 x^2} \text{Erf}(b x)}{b^4}-\frac{e^{-b^2 x^2} \text{Erf}(b x)}{b^6}+\frac{43 \text{Erf}\left (\sqrt{2} b x\right )}{32 \sqrt{2} b^6}-\frac{x^3 e^{-2 b^2 x^2}}{4 \sqrt{\pi } b^3}-\frac{11 x e^{-2 b^2 x^2}}{16 \sqrt{\pi } b^5} \]
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Rubi [A] time = 0.195389, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {6385, 6382, 2205, 2212} \[ -\frac{x^4 e^{-b^2 x^2} \text{Erf}(b x)}{2 b^2}-\frac{x^2 e^{-b^2 x^2} \text{Erf}(b x)}{b^4}-\frac{e^{-b^2 x^2} \text{Erf}(b x)}{b^6}+\frac{43 \text{Erf}\left (\sqrt{2} b x\right )}{32 \sqrt{2} b^6}-\frac{x^3 e^{-2 b^2 x^2}}{4 \sqrt{\pi } b^3}-\frac{11 x e^{-2 b^2 x^2}}{16 \sqrt{\pi } b^5} \]
Antiderivative was successfully verified.
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Rule 6385
Rule 6382
Rule 2205
Rule 2212
Rubi steps
\begin{align*} \int e^{-b^2 x^2} x^5 \text{erf}(b x) \, dx &=-\frac{e^{-b^2 x^2} x^4 \text{erf}(b x)}{2 b^2}+\frac{2 \int e^{-b^2 x^2} x^3 \text{erf}(b x) \, dx}{b^2}+\frac{\int e^{-2 b^2 x^2} x^4 \, dx}{b \sqrt{\pi }}\\ &=-\frac{e^{-2 b^2 x^2} x^3}{4 b^3 \sqrt{\pi }}-\frac{e^{-b^2 x^2} x^2 \text{erf}(b x)}{b^4}-\frac{e^{-b^2 x^2} x^4 \text{erf}(b x)}{2 b^2}+\frac{2 \int e^{-b^2 x^2} x \text{erf}(b x) \, dx}{b^4}+\frac{3 \int e^{-2 b^2 x^2} x^2 \, dx}{4 b^3 \sqrt{\pi }}+\frac{2 \int e^{-2 b^2 x^2} x^2 \, dx}{b^3 \sqrt{\pi }}\\ &=-\frac{11 e^{-2 b^2 x^2} x}{16 b^5 \sqrt{\pi }}-\frac{e^{-2 b^2 x^2} x^3}{4 b^3 \sqrt{\pi }}-\frac{e^{-b^2 x^2} \text{erf}(b x)}{b^6}-\frac{e^{-b^2 x^2} x^2 \text{erf}(b x)}{b^4}-\frac{e^{-b^2 x^2} x^4 \text{erf}(b x)}{2 b^2}+\frac{3 \int e^{-2 b^2 x^2} \, dx}{16 b^5 \sqrt{\pi }}+\frac{\int e^{-2 b^2 x^2} \, dx}{2 b^5 \sqrt{\pi }}+\frac{2 \int e^{-2 b^2 x^2} \, dx}{b^5 \sqrt{\pi }}\\ &=-\frac{11 e^{-2 b^2 x^2} x}{16 b^5 \sqrt{\pi }}-\frac{e^{-2 b^2 x^2} x^3}{4 b^3 \sqrt{\pi }}-\frac{e^{-b^2 x^2} \text{erf}(b x)}{b^6}-\frac{e^{-b^2 x^2} x^2 \text{erf}(b x)}{b^4}-\frac{e^{-b^2 x^2} x^4 \text{erf}(b x)}{2 b^2}+\frac{43 \text{erf}\left (\sqrt{2} b x\right )}{32 \sqrt{2} b^6}\\ \end{align*}
Mathematica [A] time = 0.0770188, size = 86, normalized size = 0.64 \[ \frac{-32 e^{-b^2 x^2} \left (b^4 x^4+2 b^2 x^2+2\right ) \text{Erf}(b x)-\frac{4 b x e^{-2 b^2 x^2} \left (4 b^2 x^2+11\right )}{\sqrt{\pi }}+43 \sqrt{2} \text{Erf}\left (\sqrt{2} b x\right )}{64 b^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.122, size = 119, normalized size = 0.9 \begin{align*}{\frac{1}{b} \left ({\frac{{\it Erf} \left ( bx \right ) }{{b}^{5}} \left ( -{\frac{{b}^{4}{x}^{4}}{2\,{{\rm e}^{{b}^{2}{x}^{2}}}}}-{\frac{{b}^{2}{x}^{2}}{{{\rm e}^{{b}^{2}{x}^{2}}}}}- \left ({{\rm e}^{{b}^{2}{x}^{2}}} \right ) ^{-1} \right ) }-{\frac{1}{\sqrt{\pi }{b}^{5}} \left ( -{\frac{43\,\sqrt{2}\sqrt{\pi }{\it Erf} \left ( bx\sqrt{2} \right ) }{64}}+{\frac{11\,bx}{16\, \left ({{\rm e}^{{b}^{2}{x}^{2}}} \right ) ^{2}}}+{\frac{{x}^{3}{b}^{3}}{4\, \left ({{\rm e}^{{b}^{2}{x}^{2}}} \right ) ^{2}}} \right ) } \right ) } \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{{\left (b^{4} x^{4} + 2 \, b^{2} x^{2} + 2\right )} \operatorname{erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{2 \, b^{6}} + \frac{-\frac{1}{64} \, b^{4}{\left (\frac{4 \,{\left (4 \, b^{2} x^{3} + 3 \, x\right )} e^{\left (-2 \, b^{2} x^{2}\right )}}{b^{4}} - \frac{3 \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (\sqrt{2} b x\right )}{b^{5}}\right )} - \frac{1}{8} \, b^{2}{\left (\frac{4 \, x e^{\left (-2 \, b^{2} x^{2}\right )}}{b^{2}} - \frac{\sqrt{2} \sqrt{\pi } \operatorname{erf}\left (\sqrt{2} b x\right )}{b^{3}}\right )} + \frac{\sqrt{2} \sqrt{\pi } \operatorname{erf}\left (\sqrt{2} b x\right )}{2 \, b}}{\sqrt{\pi } b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 3.00524, size = 244, normalized size = 1.81 \begin{align*} \frac{43 \, \sqrt{2} \pi \sqrt{b^{2}} \operatorname{erf}\left (\sqrt{2} \sqrt{b^{2}} x\right ) - 32 \,{\left (\pi b^{5} x^{4} + 2 \, \pi b^{3} x^{2} + 2 \, \pi b\right )} \operatorname{erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} - 4 \, \sqrt{\pi }{\left (4 \, b^{4} x^{3} + 11 \, b^{2} x\right )} e^{\left (-2 \, b^{2} x^{2}\right )}}{64 \, \pi b^{7}} \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 [A] time = 1.27722, size = 212, normalized size = 1.57 \begin{align*} -\frac{{\left (b^{4} x^{4} + 2 \, b^{2} x^{2} + 2\right )} \operatorname{erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{2 \, b^{6}} - \frac{\sqrt{\pi } b^{4}{\left (\frac{4 \,{\left (4 \, b^{2} x^{3} + 3 \, x\right )} e^{\left (-2 \, b^{2} x^{2}\right )}}{b^{4}} + \frac{3 \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\sqrt{2} b x\right )}{b^{5}}\right )} + 8 \, \sqrt{\pi } b^{2}{\left (\frac{4 \, x e^{\left (-2 \, b^{2} x^{2}\right )}}{b^{2}} + \frac{\sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\sqrt{2} b x\right )}{b^{3}}\right )} + \frac{32 \, \sqrt{2} \pi \operatorname{erf}\left (-\sqrt{2} b x\right )}{b}}{64 \, \pi b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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