Optimal. Leaf size=18 \[ -\frac {\sqrt {\pi } \text {Erfc}(b x)^2}{4 b} \]
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Rubi [A]
time = 0.01, 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 = {6509, 30}
\begin {gather*} -\frac {\sqrt {\pi } \text {Erfc}(b x)^2}{4 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 6509
Rubi steps
\begin {align*} \int e^{-b^2 x^2} \text {erfc}(b x) \, dx &=-\frac {\sqrt {\pi } \text {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 = 0.00, size = 18, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {\pi } \text {Erfc}(b x)^2}{4 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 22, normalized size = 1.22
method | result | size |
default | \(\frac {\sqrt {\pi }\, \left (-\frac {\erf \left (b x \right )^{2}}{2}+\erf \left (b x \right )\right )}{2 b}\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 14, normalized size = 0.78 \begin {gather*} -\frac {\sqrt {\pi } \operatorname {erfc}\left (b x\right )^{2}}{4 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 21, normalized size = 1.17 \begin {gather*} -\frac {\sqrt {\pi } {\left (\operatorname {erf}\left (b x\right )^{2} - 2 \, \operatorname {erf}\left (b x\right )\right )}}{4 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.34, size = 17, normalized size = 0.94 \begin {gather*} \begin {cases} - \frac {\sqrt {\pi } \operatorname {erfc}^{2}{\left (b x \right )}}{4 b} & \text {for}\: b \neq 0 \\x & \text {otherwise} \end {cases} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} -\frac {\sqrt {\pi }\,{\mathrm {erfc}\left (b\,x\right )}^2}{4\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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