Optimal. Leaf size=36 \[ \frac {(a+b x) \text {erf}(a+b x)}{b}+\frac {e^{-(a+b x)^2}}{\sqrt {\pi } b} \]
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Rubi [A] time = 0.01, antiderivative size = 36, 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 = {6349} \[ \frac {(a+b x) \text {Erf}(a+b x)}{b}+\frac {e^{-(a+b x)^2}}{\sqrt {\pi } b} \]
Antiderivative was successfully verified.
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Rule 6349
Rubi steps
\begin {align*} \int \text {erf}(a+b x) \, dx &=\frac {e^{-(a+b x)^2}}{b \sqrt {\pi }}+\frac {(a+b x) \text {erf}(a+b x)}{b}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 35, normalized size = 0.97 \[ \left (\frac {a}{b}+x\right ) \text {erf}(a+b x)+\frac {e^{-(a+b x)^2}}{\sqrt {\pi } b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 47, normalized size = 1.31 \[ \frac {{\left (\pi b x + \pi a\right )} \operatorname {erf}\left (b x + a\right ) + \sqrt {\pi } e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{\pi b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 59, normalized size = 1.64 \[ x \operatorname {erf}\left (b x + a\right ) - \frac {\frac {\sqrt {\pi } a \operatorname {erf}\left (-b {\left (x + \frac {a}{b}\right )}\right )}{b} - \frac {e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{b}}{\sqrt {\pi }} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 32, normalized size = 0.89 \[ \frac {\left (b x +a \right ) \erf \left (b x +a \right )+\frac {{\mathrm e}^{-\left (b x +a \right )^{2}}}{\sqrt {\pi }}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 31, normalized size = 0.86 \[ \frac {{\left (b x + a\right )} \operatorname {erf}\left (b x + a\right ) + \frac {e^{\left (-{\left (b x + a\right )}^{2}\right )}}{\sqrt {\pi }}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.22, size = 48, normalized size = 1.33 \[ x\,\mathrm {erf}\left (a+b\,x\right )+\frac {a\,\mathrm {erf}\left (a+b\,x\right )}{b}+\frac {{\mathrm {e}}^{-b^2\,x^2}\,{\mathrm {e}}^{-a^2}\,{\mathrm {e}}^{-2\,a\,b\,x}}{b\,\sqrt {\pi }} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.62, size = 53, normalized size = 1.47 \[ \begin {cases} \frac {a \operatorname {erf}{\left (a + b x \right )}}{b} + x \operatorname {erf}{\left (a + b x \right )} + \frac {e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{\sqrt {\pi } b} & \text {for}\: b \neq 0 \\x \operatorname {erf}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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