Optimal. Leaf size=21 \[ -\frac {\sqrt {\pi } e^c \text {erfc}(b x)^2}{4 b} \]
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Rubi [A] time = 0.02, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {6374, 30} \[ -\frac {\sqrt {\pi } e^c \text {Erfc}(b x)^2}{4 b} \]
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) \, dx &=-\frac {\left (e^c \sqrt {\pi }\right ) \operatorname {Subst}(\int x \, dx,x,\text {erfc}(b x))}{2 b}\\ &=-\frac {e^c \sqrt {\pi } \text {erfc}(b x)^2}{4 b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 21, normalized size = 1.00 \[ -\frac {\sqrt {\pi } e^c \text {erfc}(b x)^2}{4 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 23, normalized size = 1.10 \[ -\frac {\sqrt {\pi } {\left (\operatorname {erf}\left (b x\right )^{2} - 2 \, \operatorname {erf}\left (b x\right )\right )} e^{c}}{4 \, b} \]
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 \[ \int \operatorname {erfc}\left (b x\right ) e^{\left (-b^{2} x^{2} + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 30, normalized size = 1.43 \[ \frac {\frac {{\mathrm e}^{c} \sqrt {\pi }\, \erf \left (b x \right )}{2}-\frac {{\mathrm e}^{c} \sqrt {\pi }\, \erf \left (b x \right )^{2}}{4}}{b} \]
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 \[ \int \operatorname {erfc}\left (b x\right ) e^{\left (-b^{2} x^{2} + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 16, normalized size = 0.76 \[ -\frac {\sqrt {\pi }\,{\mathrm {e}}^c\,{\mathrm {erfc}\left (b\,x\right )}^2}{4\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.62, size = 24, normalized size = 1.14 \[ \begin {cases} - \frac {\sqrt {\pi } e^{c} \operatorname {erfc}^{2}{\left (b x \right )}}{4 b} & \text {for}\: b \neq 0 \\x e^{c} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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