Optimal. Leaf size=43 \[ -\frac {\text {erf}\left (\sqrt {2} b x\right )}{2 \sqrt {2} b^2}-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{2 b^2} \]
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Rubi [A] time = 0.03, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6383, 2205} \[ -\frac {\text {Erf}\left (\sqrt {2} b x\right )}{2 \sqrt {2} b^2}-\frac {e^{-b^2 x^2} \text {Erfc}(b x)}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 2205
Rule 6383
Rubi steps
\begin {align*} \int e^{-b^2 x^2} x \text {erfc}(b x) \, dx &=-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{2 b^2}-\frac {\int e^{-2 b^2 x^2} \, dx}{b \sqrt {\pi }}\\ &=-\frac {\text {erf}\left (\sqrt {2} b x\right )}{2 \sqrt {2} b^2}-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{2 b^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 39, normalized size = 0.91 \[ -\frac {2 e^{-b^2 x^2} \text {erfc}(b x)+\sqrt {2} \text {erf}\left (\sqrt {2} b x\right )}{4 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 47, normalized size = 1.09 \[ -\frac {\sqrt {2} \sqrt {b^{2}} \operatorname {erf}\left (\sqrt {2} \sqrt {b^{2}} x\right ) - 2 \, {\left (b \operatorname {erf}\left (b x\right ) - b\right )} e^{\left (-b^{2} x^{2}\right )}}{4 \, b^{3}} \]
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 \[ \int x \operatorname {erfc}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 53, normalized size = 1.23 \[ \frac {-\frac {{\mathrm e}^{-b^{2} x^{2}}}{2 b}+\frac {\erf \left (b x \right ) {\mathrm e}^{-b^{2} x^{2}}}{2 b}-\frac {\sqrt {2}\, \erf \left (b x \sqrt {2}\right )}{4 b}}{b} \]
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 \[ \int x \operatorname {erfc}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erfc}\left (b\,x\right ) \,d x \]
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 \[ \int x e^{- b^{2} x^{2}} \operatorname {erfc}{\left (b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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