Optimal. Leaf size=90 \[ \frac {e^{-2 b^2 x^2} x}{4 b^3 \sqrt {\pi }}-\frac {5 \text {Erf}\left (\sqrt {2} b x\right )}{8 \sqrt {2} b^4}-\frac {e^{-b^2 x^2} \text {Erfc}(b x)}{2 b^4}-\frac {e^{-b^2 x^2} x^2 \text {Erfc}(b x)}{2 b^2} \]
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Rubi [A]
time = 0.07, 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 = {6521, 6518,
2236, 2243} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 2236
Rule 2243
Rule 6518
Rule 6521
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 = 0.07, size = 69, normalized size = 0.77 \begin {gather*} \frac {-5 \sqrt {2} \text {Erf}\left (\sqrt {2} b x\right )+4 e^{-2 b^2 x^2} \left (\frac {b x}{\sqrt {\pi }}-2 e^{b^2 x^2} \left (1+b^2 x^2\right ) \text {Erfc}(b x)\right )}{16 b^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.58, size = 118, normalized size = 1.31
method | result | size |
default | \(\frac {\frac {-\frac {b^{2} x^{2} {\mathrm e}^{-b^{2} x^{2}}}{2}-\frac {{\mathrm e}^{-b^{2} x^{2}}}{2}}{b^{3}}-\frac {\erf \left (b x \right ) \left (-\frac {b^{2} x^{2} {\mathrm e}^{-b^{2} x^{2}}}{2}-\frac {{\mathrm e}^{-b^{2} x^{2}}}{2}\right )}{b^{3}}+\frac {-\frac {5 \sqrt {2}\, \sqrt {\pi }\, \erf \left (b x \sqrt {2}\right )}{16}+\frac {{\mathrm e}^{-2 b^{2} x^{2}} b x}{4}}{\sqrt {\pi }\, b^{3}}}{b}\) | \(118\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 90, normalized size = 1.00 \begin {gather*} \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}} \end {gather*}
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 \begin {gather*} \int x^{3} e^{- b^{2} x^{2}} \operatorname {erfc}{\left (b x \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x^3\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erfc}\left (b\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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