Optimal. Leaf size=24 \[ (a+x) \text{Erf}(a+x)+\frac{e^{-(a+x)^2}}{\sqrt{\pi }} \]
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Rubi [A] time = 0.0076567, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6349} \[ (a+x) \text{Erf}(a+x)+\frac{e^{-(a+x)^2}}{\sqrt{\pi }} \]
Antiderivative was successfully verified.
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Rule 6349
Rubi steps
\begin{align*} \int \text{erf}(a+x) \, dx &=\frac{e^{-(a+x)^2}}{\sqrt{\pi }}+(a+x) \text{erf}(a+x)\\ \end{align*}
Mathematica [A] time = 0.0276084, size = 24, normalized size = 1. \[ (a+x) \text{Erf}(a+x)+\frac{e^{-(a+x)^2}}{\sqrt{\pi }} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 22, normalized size = 0.9 \begin{align*} \left ( a+x \right ){\it Erf} \left ( a+x \right ) +{\frac{{{\rm e}^{- \left ( a+x \right ) ^{2}}}}{\sqrt{\pi }}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.931253, size = 28, normalized size = 1.17 \begin{align*}{\left (a + x\right )} \operatorname{erf}\left (a + x\right ) + \frac{e^{\left (-{\left (a + x\right )}^{2}\right )}}{\sqrt{\pi }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.87621, size = 88, normalized size = 3.67 \begin{align*} \frac{{\left (\pi a + \pi x\right )} \operatorname{erf}\left (a + x\right ) + \sqrt{\pi } e^{\left (-a^{2} - 2 \, a x - x^{2}\right )}}{\pi } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.497251, size = 36, normalized size = 1.5 \begin{align*} a \operatorname{erf}{\left (a + x \right )} + x \operatorname{erf}{\left (a + x \right )} + \frac{e^{- a^{2}} e^{- x^{2}} e^{- 2 a x}}{\sqrt{\pi }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07223, size = 50, normalized size = 2.08 \begin{align*} x \operatorname{erf}\left (a + x\right ) + \frac{\sqrt{\pi } a \operatorname{erf}\left (a + x\right ) + e^{\left (-a^{2} - 2 \, a x - x^{2}\right )}}{\sqrt{\pi }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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