Optimal. Leaf size=135 \[ \frac {11 e^{-2 b^2 x^2} x}{16 b^5 \sqrt {\pi }}+\frac {e^{-2 b^2 x^2} x^3}{4 b^3 \sqrt {\pi }}-\frac {43 \text {Erf}\left (\sqrt {2} b x\right )}{32 \sqrt {2} b^6}-\frac {e^{-b^2 x^2} \text {Erfc}(b x)}{b^6}-\frac {e^{-b^2 x^2} x^2 \text {Erfc}(b x)}{b^4}-\frac {e^{-b^2 x^2} x^4 \text {Erfc}(b x)}{2 b^2} \]
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Rubi [A]
time = 0.13, antiderivative size = 135, normalized size of antiderivative = 1.00, 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 = {6521, 6518,
2236, 2243} \begin {gather*} -\frac {43 \text {Erf}\left (\sqrt {2} b x\right )}{32 \sqrt {2} b^6}-\frac {x^4 e^{-b^2 x^2} \text {Erfc}(b x)}{2 b^2}-\frac {e^{-b^2 x^2} \text {Erfc}(b x)}{b^6}+\frac {11 x e^{-2 b^2 x^2}}{16 \sqrt {\pi } b^5}-\frac {x^2 e^{-b^2 x^2} \text {Erfc}(b x)}{b^4}+\frac {x^3 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^5 \text {erfc}(b x) \, dx &=-\frac {e^{-b^2 x^2} x^4 \text {erfc}(b x)}{2 b^2}+\frac {2 \int e^{-b^2 x^2} x^3 \text {erfc}(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 {erfc}(b x)}{b^4}-\frac {e^{-b^2 x^2} x^4 \text {erfc}(b x)}{2 b^2}+\frac {2 \int e^{-b^2 x^2} x \text {erfc}(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 {erfc}(b x)}{b^6}-\frac {e^{-b^2 x^2} x^2 \text {erfc}(b x)}{b^4}-\frac {e^{-b^2 x^2} x^4 \text {erfc}(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 {43 \text {erf}\left (\sqrt {2} b x\right )}{32 \sqrt {2} b^6}-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{b^6}-\frac {e^{-b^2 x^2} x^2 \text {erfc}(b x)}{b^4}-\frac {e^{-b^2 x^2} x^4 \text {erfc}(b x)}{2 b^2}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 87, normalized size = 0.64 \begin {gather*} \frac {-43 \sqrt {2} \text {Erf}\left (\sqrt {2} b x\right )+4 e^{-2 b^2 x^2} \left (\frac {b x \left (11+4 b^2 x^2\right )}{\sqrt {\pi }}-8 e^{b^2 x^2} \left (2+2 b^2 x^2+b^4 x^4\right ) \text {Erfc}(b x)\right )}{64 b^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.25, size = 172, normalized size = 1.27
method | result | size |
default | \(\frac {\frac {-\frac {b^{4} x^{4} {\mathrm e}^{-b^{2} x^{2}}}{2}-b^{2} x^{2} {\mathrm e}^{-b^{2} x^{2}}-{\mathrm e}^{-b^{2} x^{2}}}{b^{5}}-\frac {\erf \left (b x \right ) \left (-\frac {b^{4} x^{4} {\mathrm e}^{-b^{2} x^{2}}}{2}-b^{2} x^{2} {\mathrm e}^{-b^{2} x^{2}}-{\mathrm e}^{-b^{2} x^{2}}\right )}{b^{5}}+\frac {-\frac {43 \sqrt {2}\, \sqrt {\pi }\, \erf \left (b x \sqrt {2}\right )}{64}+\frac {{\mathrm e}^{-2 b^{2} x^{2}} b^{3} x^{3}}{4}+\frac {11 \,{\mathrm e}^{-2 b^{2} x^{2}} b x}{16}}{\sqrt {\pi }\, b^{5}}}{b}\) | \(172\) |
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 = 0.35, size = 121, normalized size = 0.90 \begin {gather*} -\frac {43 \, \sqrt {2} \pi \sqrt {b^{2}} \operatorname {erf}\left (\sqrt {2} \sqrt {b^{2}} x\right ) - 4 \, \sqrt {\pi } {\left (4 \, b^{4} x^{3} + 11 \, b^{2} x\right )} e^{\left (-2 \, b^{2} x^{2}\right )} + 32 \, {\left (\pi b^{5} x^{4} + 2 \, \pi b^{3} x^{2} + 2 \, \pi b - {\left (\pi b^{5} x^{4} + 2 \, \pi b^{3} x^{2} + 2 \, \pi b\right )} \operatorname {erf}\left (b x\right )\right )} e^{\left (-b^{2} x^{2}\right )}}{64 \, \pi b^{7}} \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^{5} 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^5\,{\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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