Optimal. Leaf size=28 \[ -\frac{\sqrt{\pi } e^c \text{Erfc}(b x)^{n+1}}{2 b (n+1)} \]
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Rubi [A] time = 0.0326613, antiderivative size = 28, 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} \[ -\frac{\sqrt{\pi } e^c \text{Erfc}(b x)^{n+1}}{2 b (n+1)} \]
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)^n \, dx &=-\frac{\left (e^c \sqrt{\pi }\right ) \operatorname{Subst}\left (\int x^n \, dx,x,\text{erfc}(b x)\right )}{2 b}\\ &=-\frac{e^c \sqrt{\pi } \text{erfc}(b x)^{1+n}}{2 b (1+n)}\\ \end{align*}
Mathematica [A] time = 0.0106263, size = 28, normalized size = 1. \[ -\frac{\sqrt{\pi } e^c \text{Erfc}(b x)^{n+1}}{2 b (n+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.135, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{-{b}^{2}{x}^{2}+c}} \left ({\it erfc} \left ( bx \right ) \right ) ^{n}\, 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 \operatorname{erfc}\left (b x\right )^{n} 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 = 2.19796, size = 82, normalized size = 2.93 \begin{align*} \frac{\sqrt{\pi }{\left (-\operatorname{erf}\left (b x\right ) + 1\right )}^{n}{\left (\operatorname{erf}\left (b x\right ) - 1\right )} e^{c}}{2 \,{\left (b n + b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 11.366, size = 60, normalized size = 2.14 \begin{align*} \begin{cases} x e^{c} & \text{for}\: b = 0 \wedge \left (b = 0 \vee n = -1\right ) \\- \frac{\sqrt{\pi } e^{c} \log{\left (\operatorname{erfc}{\left (b x \right )} \right )}}{2 b} & \text{for}\: n = -1 \\- \frac{\sqrt{\pi } e^{c} \operatorname{erfc}{\left (b x \right )} \operatorname{erfc}^{n}{\left (b x \right )}}{2 b n + 2 b} & \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 )^{n} 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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