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 = 0.03, antiderivative size = 28, normalized size of antiderivative = 1.00, 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 30
Rule 6374
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*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 1.00 \[ -\frac {\sqrt {\pi } e^c \text {erfc}(b x)^{n+1}}{2 b (n+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 30, normalized size = 1.07 \[ \frac {\sqrt {\pi } {\left (-\operatorname {erf}\left (b x\right ) + 1\right )}^{n} {\left (\operatorname {erf}\left (b x\right ) - 1\right )} e^{c}}{2 \, {\left (b n + b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {erfc}\left (b x\right )^{n} e^{\left (-b^{2} x^{2} + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{-b^{2} x^{2}+c} \mathrm {erfc}\left (b x \right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {erfc}\left (b x\right )^{n} e^{\left (-b^{2} x^{2} + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.20, size = 23, normalized size = 0.82 \[ -\frac {\sqrt {\pi }\,{\mathrm {e}}^c\,{\mathrm {erfc}\left (b\,x\right )}^{n+1}}{2\,b\,\left (n+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.13, size = 60, normalized size = 2.14 \[ \begin {cases} x e^{c} & \text {for}\: b = 0 \wedge \left (b = 0 \vee n = -1\right ) \\- \frac {\sqrt {\pi } e^{c} \log {\left (\operatorname {erfc}{\left (b x \right )} \right )}}{2 b} & \text {for}\: n = -1 \\- \frac {\sqrt {\pi } e^{c} \operatorname {erfc}{\left (b x \right )} \operatorname {erfc}^{n}{\left (b x \right )}}{2 b n + 2 b} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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