Optimal. Leaf size=36 \[ \frac {e^c x}{b \sqrt {\pi }}+\frac {e^{c+b^2 x^2} \text {Erfc}(b x)}{2 b^2} \]
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Rubi [A]
time = 0.02, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {6518, 8}
\begin {gather*} \frac {e^{b^2 x^2+c} \text {Erfc}(b x)}{2 b^2}+\frac {e^c x}{\sqrt {\pi } b} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 6518
Rubi steps
\begin {align*} \int e^{c+b^2 x^2} x \text {erfc}(b x) \, dx &=\frac {e^{c+b^2 x^2} \text {erfc}(b x)}{2 b^2}+\frac {\int e^c \, dx}{b \sqrt {\pi }}\\ &=\frac {e^c x}{b \sqrt {\pi }}+\frac {e^{c+b^2 x^2} \text {erfc}(b x)}{2 b^2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 36, normalized size = 1.00 \begin {gather*} \frac {e^c x}{b \sqrt {\pi }}+\frac {e^{c+b^2 x^2} \text {Erfc}(b x)}{2 b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 51, normalized size = 1.42
method | result | size |
default | \(\frac {2 \,{\mathrm e}^{b^{2} x^{2}+c} {\mathrm e}^{-b^{2} x^{2}} x b +{\mathrm e}^{b^{2} x^{2}+c} \mathrm {erfc}\left (b x \right ) \sqrt {\pi }}{2 \sqrt {\pi }\, b^{2}}\) | \(51\) |
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 = 0.39, size = 38, normalized size = 1.06 \begin {gather*} \frac {2 \, \sqrt {\pi } b x e^{c} + {\left (\pi - \pi \operatorname {erf}\left (b x\right )\right )} e^{\left (b^{2} x^{2} + c\right )}}{2 \, \pi b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.61, size = 41, normalized size = 1.14 \begin {gather*} \begin {cases} \frac {x e^{c}}{\sqrt {\pi } b} + \frac {e^{c} e^{b^{2} x^{2}} \operatorname {erfc}{\left (b x \right )}}{2 b^{2}} & \text {for}\: b \neq 0 \\\frac {x^{2} e^{c}}{2} & \text {otherwise} \end {cases} \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 [B]
time = 0.17, size = 30, normalized size = 0.83 \begin {gather*} \frac {x\,{\mathrm {e}}^c}{b\,\sqrt {\pi }}+\frac {{\mathrm {e}}^{b^2\,x^2}\,{\mathrm {e}}^c\,\mathrm {erfc}\left (b\,x\right )}{2\,b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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