Optimal. Leaf size=21 \[ -\frac{\sqrt{\pi } e^c \text{Erfc}(b x)^2}{4 b} \]
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Rubi [A] time = 0.0177382, antiderivative size = 21, normalized size of antiderivative = 1., 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 = {6374, 30} \[ -\frac{\sqrt{\pi } e^c \text{Erfc}(b x)^2}{4 b} \]
Antiderivative was successfully verified.
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Rule 6374
Rule 30
Rubi steps
\begin{align*} \int e^{c-b^2 x^2} \text{erfc}(b x) \, dx &=-\frac{\left (e^c \sqrt{\pi }\right ) \operatorname{Subst}(\int x \, dx,x,\text{erfc}(b x))}{2 b}\\ &=-\frac{e^c \sqrt{\pi } \text{erfc}(b x)^2}{4 b}\\ \end{align*}
Mathematica [A] time = 0.0053808, size = 21, normalized size = 1. \[ -\frac{\sqrt{\pi } e^c \text{Erfc}(b x)^2}{4 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.107, size = 30, normalized size = 1.4 \begin{align*}{\frac{1}{b} \left ({\frac{{{\rm e}^{c}}\sqrt{\pi }{\it Erf} \left ( bx \right ) }{2}}-{\frac{{{\rm e}^{c}}\sqrt{\pi } \left ({\it Erf} \left ( bx \right ) \right ) ^{2}}{4}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{erfc}\left (b x\right ) e^{\left (-b^{2} x^{2} + c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.16173, size = 63, normalized size = 3. \begin{align*} -\frac{\sqrt{\pi }{\left (\operatorname{erf}\left (b x\right )^{2} - 2 \, \operatorname{erf}\left (b x\right )\right )} e^{c}}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.691919, size = 24, normalized size = 1.14 \begin{align*} \begin{cases} - \frac{\sqrt{\pi } e^{c} \operatorname{erfc}^{2}{\left (b x \right )}}{4 b} & \text{for}\: b \neq 0 \\x e^{c} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{erfc}\left (b x\right ) e^{\left (-b^{2} x^{2} + c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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