Optimal. Leaf size=21 \[ \frac{\sqrt{\pi } e^c \text{Erf}(b x)^3}{6 b} \]
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Rubi [A] time = 0.0297161, 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 = {6373, 30} \[ \frac{\sqrt{\pi } e^c \text{Erf}(b x)^3}{6 b} \]
Antiderivative was successfully verified.
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Rule 6373
Rule 30
Rubi steps
\begin{align*} \int e^{c-b^2 x^2} \text{erf}(b x)^2 \, dx &=\frac{\left (e^c \sqrt{\pi }\right ) \operatorname{Subst}\left (\int x^2 \, dx,x,\text{erf}(b x)\right )}{2 b}\\ &=\frac{e^c \sqrt{\pi } \text{erf}(b x)^3}{6 b}\\ \end{align*}
Mathematica [A] time = 0.0090464, size = 21, normalized size = 1. \[ \frac{\sqrt{\pi } e^c \text{Erf}(b x)^3}{6 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 17, normalized size = 0.8 \begin{align*}{\frac{{{\rm e}^{c}} \left ({\it Erf} \left ( bx \right ) \right ) ^{3}\sqrt{\pi }}{6\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02145, size = 22, normalized size = 1.05 \begin{align*} \frac{\sqrt{\pi } \operatorname{erf}\left (b x\right )^{3} e^{c}}{6 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.79198, size = 42, normalized size = 2. \begin{align*} \frac{\sqrt{\pi } \operatorname{erf}\left (b x\right )^{3} e^{c}}{6 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.68973, size = 19, normalized size = 0.9 \begin{align*} \begin{cases} \frac{\sqrt{\pi } e^{c} \operatorname{erf}^{3}{\left (b x \right )}}{6 b} & \text{for}\: b \neq 0 \\0 & \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{erf}\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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