Optimal. Leaf size=21 \[ -\frac{\sqrt{\pi } e^c \text{Erfc}(b x)^3}{6 b} \]
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Rubi [A] time = 0.0273624, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {6374, 30} \[ -\frac{\sqrt{\pi } e^c \text{Erfc}(b x)^3}{6 b} \]
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)^2 \, dx &=-\frac{\left (e^c \sqrt{\pi }\right ) \operatorname{Subst}\left (\int x^2 \, dx,x,\text{erfc}(b x)\right )}{2 b}\\ &=-\frac{e^c \sqrt{\pi } \text{erfc}(b x)^3}{6 b}\\ \end{align*}
Mathematica [A] time = 0.0093958, size = 21, normalized size = 1. \[ -\frac{\sqrt{\pi } e^c \text{Erfc}(b x)^3}{6 b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.226, size = 43, normalized size = 2.1 \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}}{2}}+{\frac{{{\rm e}^{c}}\sqrt{\pi } \left ({\it Erf} \left ( bx \right ) \right ) ^{3}}{6}} \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 )^{2} 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 = 2.07739, size = 82, normalized size = 3.9 \begin{align*} \frac{\sqrt{\pi }{\left (\operatorname{erf}\left (b x\right )^{3} - 3 \, \operatorname{erf}\left (b x\right )^{2} + 3 \, \operatorname{erf}\left (b x\right )\right )} e^{c}}{6 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.69231, size = 24, normalized size = 1.14 \begin{align*} \begin{cases} - \frac{\sqrt{\pi } e^{c} \operatorname{erfc}^{3}{\left (b x \right )}}{6 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 )^{2} 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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