Optimal. Leaf size=80 \[ \frac{x^2 e^{b^2 x^2+c} \text{Erfc}(b x)}{2 b^2}-\frac{e^{b^2 x^2+c} \text{Erfc}(b x)}{2 b^4}-\frac{e^c x}{\sqrt{\pi } b^3}+\frac{e^c x^3}{3 \sqrt{\pi } b} \]
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Rubi [A] time = 0.0892668, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {6386, 6383, 8, 12, 30} \[ \frac{x^2 e^{b^2 x^2+c} \text{Erfc}(b x)}{2 b^2}-\frac{e^{b^2 x^2+c} \text{Erfc}(b x)}{2 b^4}-\frac{e^c x}{\sqrt{\pi } b^3}+\frac{e^c x^3}{3 \sqrt{\pi } b} \]
Antiderivative was successfully verified.
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Rule 6386
Rule 6383
Rule 8
Rule 12
Rule 30
Rubi steps
\begin{align*} \int e^{c+b^2 x^2} x^3 \text{erfc}(b x) \, dx &=\frac{e^{c+b^2 x^2} x^2 \text{erfc}(b x)}{2 b^2}-\frac{\int e^{c+b^2 x^2} x \text{erfc}(b x) \, dx}{b^2}+\frac{\int e^c x^2 \, dx}{b \sqrt{\pi }}\\ &=-\frac{e^{c+b^2 x^2} \text{erfc}(b x)}{2 b^4}+\frac{e^{c+b^2 x^2} x^2 \text{erfc}(b x)}{2 b^2}-\frac{\int e^c \, dx}{b^3 \sqrt{\pi }}+\frac{e^c \int x^2 \, dx}{b \sqrt{\pi }}\\ &=-\frac{e^c x}{b^3 \sqrt{\pi }}+\frac{e^c x^3}{3 b \sqrt{\pi }}-\frac{e^{c+b^2 x^2} \text{erfc}(b x)}{2 b^4}+\frac{e^{c+b^2 x^2} x^2 \text{erfc}(b x)}{2 b^2}\\ \end{align*}
Mathematica [A] time = 0.0345227, size = 58, normalized size = 0.72 \[ \frac{e^c \left (3 \sqrt{\pi } e^{b^2 x^2} \left (b^2 x^2-1\right ) \text{Erfc}(b x)+2 b x \left (b^2 x^2-3\right )\right )}{6 \sqrt{\pi } b^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.209, size = 99, normalized size = 1.2 \begin{align*}{\frac{1}{b} \left ({\frac{{{\rm e}^{c}}}{{b}^{3}} \left ({\frac{{b}^{2}{x}^{2}{{\rm e}^{{b}^{2}{x}^{2}}}}{2}}-{\frac{{{\rm e}^{{b}^{2}{x}^{2}}}}{2}} \right ) }-{\frac{{\it Erf} \left ( bx \right ){{\rm e}^{c}}}{{b}^{3}} \left ({\frac{{b}^{2}{x}^{2}{{\rm e}^{{b}^{2}{x}^{2}}}}{2}}-{\frac{{{\rm e}^{{b}^{2}{x}^{2}}}}{2}} \right ) }+{\frac{{{\rm e}^{c}}}{{b}^{3}\sqrt{\pi }} \left ({\frac{{x}^{3}{b}^{3}}{3}}-bx \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*} \int x^{3} \operatorname{erfc}\left (b x\right ) e^{\left (b^{2} 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 = 2.14394, size = 157, normalized size = 1.96 \begin{align*} \frac{2 \, \sqrt{\pi }{\left (b^{3} x^{3} - 3 \, b x\right )} e^{c} - 3 \,{\left (\pi - \pi b^{2} x^{2} -{\left (\pi - \pi b^{2} x^{2}\right )} \operatorname{erf}\left (b x\right )\right )} e^{\left (b^{2} x^{2} + c\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{erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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