Optimal. Leaf size=90 \[ -\frac {5 \text {erf}\left (\sqrt {2} b x\right )}{8 \sqrt {2} b^4}-\frac {x^2 e^{-b^2 x^2} \text {erfc}(b x)}{2 b^2}-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{2 b^4}+\frac {x e^{-2 b^2 x^2}}{4 \sqrt {\pi } b^3} \]
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Rubi [A] time = 0.10, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6386, 6383, 2205, 2212} \[ -\frac {5 \text {Erf}\left (\sqrt {2} b x\right )}{8 \sqrt {2} b^4}-\frac {x^2 e^{-b^2 x^2} \text {Erfc}(b x)}{2 b^2}-\frac {e^{-b^2 x^2} \text {Erfc}(b x)}{2 b^4}+\frac {x e^{-2 b^2 x^2}}{4 \sqrt {\pi } b^3} \]
Antiderivative was successfully verified.
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Rule 2205
Rule 2212
Rule 6383
Rule 6386
Rubi steps
\begin {align*} \int e^{-b^2 x^2} x^3 \text {erfc}(b x) \, dx &=-\frac {e^{-b^2 x^2} x^2 \text {erfc}(b x)}{2 b^2}+\frac {\int e^{-b^2 x^2} x \text {erfc}(b x) \, dx}{b^2}-\frac {\int e^{-2 b^2 x^2} x^2 \, dx}{b \sqrt {\pi }}\\ &=\frac {e^{-2 b^2 x^2} x}{4 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{2 b^4}-\frac {e^{-b^2 x^2} x^2 \text {erfc}(b x)}{2 b^2}-\frac {\int e^{-2 b^2 x^2} \, dx}{4 b^3 \sqrt {\pi }}-\frac {\int e^{-2 b^2 x^2} \, dx}{b^3 \sqrt {\pi }}\\ &=\frac {e^{-2 b^2 x^2} x}{4 b^3 \sqrt {\pi }}-\frac {5 \text {erf}\left (\sqrt {2} b x\right )}{8 \sqrt {2} b^4}-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{2 b^4}-\frac {e^{-b^2 x^2} x^2 \text {erfc}(b x)}{2 b^2}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 69, normalized size = 0.77 \[ \frac {4 e^{-2 b^2 x^2} \left (\frac {b x}{\sqrt {\pi }}-2 e^{b^2 x^2} \left (b^2 x^2+1\right ) \text {erfc}(b x)\right )-5 \sqrt {2} \text {erf}\left (\sqrt {2} b x\right )}{16 b^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 90, normalized size = 1.00 \[ \frac {4 \, \sqrt {\pi } b^{2} x e^{\left (-2 \, b^{2} x^{2}\right )} - 5 \, \sqrt {2} \pi \sqrt {b^{2}} \operatorname {erf}\left (\sqrt {2} \sqrt {b^{2}} x\right ) - 8 \, {\left (\pi b^{3} x^{2} + \pi b - {\left (\pi b^{3} x^{2} + \pi b\right )} \operatorname {erf}\left (b x\right )\right )} e^{\left (-b^{2} x^{2}\right )}}{16 \, \pi b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} \operatorname {erfc}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 118, normalized size = 1.31 \[ \frac {\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}} b^{2} x^{2}}{2}-\frac {{\mathrm e}^{-b^{2} x^{2}}}{2}}{b^{3}}-\frac {\erf \left (b x \right ) \left (-\frac {{\mathrm e}^{-b^{2} x^{2}} b^{2} x^{2}}{2}-\frac {{\mathrm e}^{-b^{2} x^{2}}}{2}\right )}{b^{3}}+\frac {-\frac {5 \sqrt {2}\, \sqrt {\pi }\, \erf \left (b x \sqrt {2}\right )}{16}+\frac {{\mathrm e}^{-2 b^{2} x^{2}} b x}{4}}{b^{3} \sqrt {\pi }}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} \operatorname {erfc}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^3\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erfc}\left (b\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} e^{- b^{2} x^{2}} \operatorname {erfc}{\left (b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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