Optimal. Leaf size=37 \[ -\frac {e^{-(a+b x)^2}}{b \sqrt {\pi }}+\frac {(a+b x) \text {Erfc}(a+b x)}{b} \]
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Rubi [A]
time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6485}
\begin {gather*} \frac {(a+b x) \text {Erfc}(a+b x)}{b}-\frac {e^{-(a+b x)^2}}{\sqrt {\pi } b} \end {gather*}
Antiderivative was successfully verified.
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Rule 6485
Rubi steps
\begin {align*} \int \text {erfc}(a+b x) \, dx &=-\frac {e^{-(a+b x)^2}}{b \sqrt {\pi }}+\frac {(a+b x) \text {erfc}(a+b x)}{b}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 42, normalized size = 1.14 \begin {gather*} -\frac {e^{-(a+b x)^2}}{b \sqrt {\pi }}-\frac {a \text {Erf}(a+b x)}{b}+x \text {Erfc}(a+b x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 33, normalized size = 0.89
method | result | size |
derivativedivides | \(\frac {\left (b x +a \right ) \mathrm {erfc}\left (b x +a \right )-\frac {{\mathrm e}^{-\left (b x +a \right )^{2}}}{\sqrt {\pi }}}{b}\) | \(33\) |
default | \(\frac {\left (b x +a \right ) \mathrm {erfc}\left (b x +a \right )-\frac {{\mathrm e}^{-\left (b x +a \right )^{2}}}{\sqrt {\pi }}}{b}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 32, normalized size = 0.86 \begin {gather*} \frac {{\left (b x + a\right )} \operatorname {erfc}\left (b x + a\right ) - \frac {e^{\left (-{\left (b x + a\right )}^{2}\right )}}{\sqrt {\pi }}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 53, normalized size = 1.43 \begin {gather*} \frac {\pi b x - {\left (\pi b x + \pi a\right )} \operatorname {erf}\left (b x + a\right ) - \sqrt {\pi } e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{\pi b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.30, size = 53, normalized size = 1.43 \begin {gather*} \begin {cases} \frac {a \operatorname {erfc}{\left (a + b x \right )}}{b} + x \operatorname {erfc}{\left (a + b x \right )} - \frac {e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{\sqrt {\pi } b} & \text {for}\: b \neq 0 \\x \operatorname {erfc}{\left (a \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 60, normalized size = 1.62 \begin {gather*} -x \operatorname {erf}\left (b x + a\right ) + x + \frac {\frac {\sqrt {\pi } a \operatorname {erf}\left (-b {\left (x + \frac {a}{b}\right )}\right )}{b} - \frac {e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{b}}{\sqrt {\pi }} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.11, size = 49, normalized size = 1.32 \begin {gather*} x\,\mathrm {erfc}\left (a+b\,x\right )+\frac {a\,\mathrm {erfc}\left (a+b\,x\right )}{b}-\frac {{\mathrm {e}}^{-b^2\,x^2}\,{\mathrm {e}}^{-a^2}\,{\mathrm {e}}^{-2\,a\,b\,x}}{b\,\sqrt {\pi }} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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