Optimal. Leaf size=21 \[ \frac{\sqrt{\pi } e^c}{2 b \text{Erfc}(b x)} \]
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Rubi [A] time = 0.0277962, 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} \[ \frac{\sqrt{\pi } e^c}{2 b \text{Erfc}(b x)} \]
Antiderivative was successfully verified.
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Rule 6374
Rule 30
Rubi steps
\begin{align*} \int \frac{e^{c-b^2 x^2}}{\text{erfc}(b x)^2} \, dx &=-\frac{\left (e^c \sqrt{\pi }\right ) \operatorname{Subst}\left (\int \frac{1}{x^2} \, dx,x,\text{erfc}(b x)\right )}{2 b}\\ &=\frac{e^c \sqrt{\pi }}{2 b \text{erfc}(b x)}\\ \end{align*}
Mathematica [A] time = 0.0060713, size = 21, normalized size = 1. \[ \frac{\sqrt{\pi } e^c}{2 b \text{Erfc}(b x)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.148, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{-{b}^{2}{x}^{2}+c}}}{ \left ({\it erfc} \left ( bx \right ) \right ) ^{2}}}\, dx \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 \frac{e^{\left (-b^{2} x^{2} + c\right )}}{\operatorname{erfc}\left (b x\right )^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.16569, size = 49, normalized size = 2.33 \begin{align*} -\frac{\sqrt{\pi } e^{c}}{2 \,{\left (b \operatorname{erf}\left (b x\right ) - b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.42661, size = 20, normalized size = 0.95 \begin{align*} \begin{cases} \frac{\sqrt{\pi } e^{c}}{2 b \operatorname{erfc}{\left (b x \right )}} & \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 \frac{e^{\left (-b^{2} x^{2} + c\right )}}{\operatorname{erfc}\left (b x\right )^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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