Optimal. Leaf size=18 \[ -\frac{\sqrt{\pi } \text{Erfc}(b x)^2}{4 b} \]
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Rubi [A] time = 0.0183673, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {6374, 30} \[ -\frac{\sqrt{\pi } \text{Erfc}(b x)^2}{4 b} \]
Antiderivative was successfully verified.
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Rule 6374
Rule 30
Rubi steps
\begin{align*} \int e^{-b^2 x^2} \text{erfc}(b x) \, dx &=-\frac{\sqrt{\pi } \operatorname{Subst}(\int x \, dx,x,\text{erfc}(b x))}{2 b}\\ &=-\frac{\sqrt{\pi } \text{erfc}(b x)^2}{4 b}\\ \end{align*}
Mathematica [A] time = 0.0045196, size = 18, normalized size = 1. \[ -\frac{\sqrt{\pi } \text{Erfc}(b x)^2}{4 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.071, size = 22, normalized size = 1.2 \begin{align*}{\frac{\sqrt{\pi }}{2\,b} \left ( -{\frac{ \left ({\it Erf} \left ( bx \right ) \right ) ^{2}}{2}}+{\it Erf} \left ( bx \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.995533, size = 19, normalized size = 1.06 \begin{align*} -\frac{\sqrt{\pi } \operatorname{erfc}\left (b x\right )^{2}}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.02123, size = 58, normalized size = 3.22 \begin{align*} -\frac{\sqrt{\pi }{\left (\operatorname{erf}\left (b x\right )^{2} - 2 \, \operatorname{erf}\left (b x\right )\right )}}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.49723, size = 17, normalized size = 0.94 \begin{align*} \begin{cases} - \frac{\sqrt{\pi } \operatorname{erfc}^{2}{\left (b x \right )}}{4 b} & \text{for}\: b \neq 0 \\x & \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}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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