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 = 0.0069472, antiderivative size = 36, normalized size of antiderivative = 1., 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*}
Mathematica [A] time = 0.0402724, 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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Maple [A] time = 0.047, size = 32, normalized size = 0.9 \begin{align*}{\frac{1}{b} \left ({\it Erf} \left ( bx+a \right ) \left ( bx+a \right ) +{\frac{{{\rm e}^{- \left ( bx+a \right ) ^{2}}}}{\sqrt{\pi }}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07452, size = 42, normalized size = 1.17 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.5475, size = 107, normalized size = 2.97 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.819628, size = 53, normalized size = 1.47 \begin{align*} \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}{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.30072, size = 80, normalized size = 2.22 \begin{align*} 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 }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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