Optimal. Leaf size=43 \[ -\frac {e^{-b^2 x^2} \text {Erf}(b x)}{2 b^2}+\frac {\text {Erf}\left (\sqrt {2} b x\right )}{2 \sqrt {2} b^2} \]
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Rubi [A]
time = 0.02, 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 = {6517, 2236}
\begin {gather*} \frac {\text {Erf}\left (\sqrt {2} b x\right )}{2 \sqrt {2} b^2}-\frac {e^{-b^2 x^2} \text {Erf}(b x)}{2 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2236
Rule 6517
Rubi steps
\begin {align*} \int e^{-b^2 x^2} x \text {erf}(b x) \, dx &=-\frac {e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}+\frac {\int e^{-2 b^2 x^2} \, dx}{b \sqrt {\pi }}\\ &=-\frac {e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}+\frac {\text {erf}\left (\sqrt {2} b x\right )}{2 \sqrt {2} b^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 39, normalized size = 0.91 \begin {gather*} \frac {-2 e^{-b^2 x^2} \text {Erf}(b x)+\sqrt {2} \text {Erf}\left (\sqrt {2} b x\right )}{4 b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.32, size = 39, normalized size = 0.91
method | result | size |
default | \(\frac {-\frac {\erf \left (b x \right ) {\mathrm e}^{-b^{2} x^{2}}}{2 b}+\frac {\sqrt {2}\, \erf \left (b x \sqrt {2}\right )}{4 b}}{b}\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 34, normalized size = 0.79 \begin {gather*} -\frac {\operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{2 \, b^{2}} + \frac {\sqrt {2} \operatorname {erf}\left (\sqrt {2} b x\right )}{4 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 43, normalized size = 1.00 \begin {gather*} -\frac {2 \, b \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} - \sqrt {2} \sqrt {b^{2}} \operatorname {erf}\left (\sqrt {2} \sqrt {b^{2}} x\right )}{4 \, b^{3}} \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 e^{- b^{2} x^{2}} \operatorname {erf}{\left (b x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 35, normalized size = 0.81 \begin {gather*} -\frac {\operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{2 \, b^{2}} - \frac {\sqrt {2} \operatorname {erf}\left (-\sqrt {2} b x\right )}{4 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.18, size = 43, normalized size = 1.00 \begin {gather*} \frac {\sqrt {2}\,\mathrm {erf}\left (\sqrt {2}\,x\,\sqrt {b^2}\right )}{4\,b\,\sqrt {b^2}}-\frac {{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erf}\left (b\,x\right )}{2\,b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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