Optimal. Leaf size=18 \[ -\frac {\sqrt {\pi } \text {erfc}(b x)^2}{4 b} \]
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Rubi [A] time = 0.02, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {6374, 30} \[ -\frac {\sqrt {\pi } \text {Erfc}(b x)^2}{4 b} \]
Antiderivative was successfully verified.
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Rule 30
Rule 6374
Rubi steps
\begin {align*} \int e^{-b^2 x^2} \text {erfc}(b x) \, dx &=-\frac {\sqrt {\pi } \operatorname {Subst}(\int x \, dx,x,\text {erfc}(b x))}{2 b}\\ &=-\frac {\sqrt {\pi } \text {erfc}(b x)^2}{4 b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 18, normalized size = 1.00 \[ -\frac {\sqrt {\pi } \text {erfc}(b x)^2}{4 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 21, normalized size = 1.17 \[ -\frac {\sqrt {\pi } {\left (\operatorname {erf}\left (b x\right )^{2} - 2 \, \operatorname {erf}\left (b x\right )\right )}}{4 \, b} \]
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 \[ \int \operatorname {erfc}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 22, normalized size = 1.22 \[ \frac {\sqrt {\pi }\, \left (\erf \left (b x \right )-\frac {\erf \left (b x \right )^{2}}{2}\right )}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 14, normalized size = 0.78 \[ -\frac {\sqrt {\pi } \operatorname {erfc}\left (b x\right )^{2}}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 14, normalized size = 0.78 \[ -\frac {\sqrt {\pi }\,{\mathrm {erfc}\left (b\,x\right )}^2}{4\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.92, size = 17, normalized size = 0.94 \[ \begin {cases} - \frac {\sqrt {\pi } \operatorname {erfc}^{2}{\left (b x \right )}}{4 b} & \text {for}\: b \neq 0 \\x & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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