Optimal. Leaf size=37 \[ \frac{e^{b^2 x^2+c} \text{Erf}(b x)}{2 b^2}-\frac{e^c x}{\sqrt{\pi } b} \]
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Rubi [A] time = 0.031491, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {6382, 8} \[ \frac{e^{b^2 x^2+c} \text{Erf}(b x)}{2 b^2}-\frac{e^c x}{\sqrt{\pi } b} \]
Antiderivative was successfully verified.
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Rule 6382
Rule 8
Rubi steps
\begin{align*} \int e^{c+b^2 x^2} x \text{erf}(b x) \, dx &=\frac{e^{c+b^2 x^2} \text{erf}(b x)}{2 b^2}-\frac{\int e^c \, dx}{b \sqrt{\pi }}\\ &=-\frac{e^c x}{b \sqrt{\pi }}+\frac{e^{c+b^2 x^2} \text{erf}(b x)}{2 b^2}\\ \end{align*}
Mathematica [A] time = 0.0201227, size = 34, normalized size = 0.92 \[ \frac{e^c \left (e^{b^2 x^2} \text{Erf}(b x)-\frac{2 b x}{\sqrt{\pi }}\right )}{2 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.136, size = 51, normalized size = 1.4 \begin{align*}{\frac{-2\,{{\rm e}^{{b}^{2}{x}^{2}+c}}{{\rm e}^{-{b}^{2}{x}^{2}}}xb+{{\rm e}^{{b}^{2}{x}^{2}+c}}{\it Erf} \left ( bx \right ) \sqrt{\pi }}{2\,{b}^{2}\sqrt{\pi }}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04374, size = 46, normalized size = 1.24 \begin{align*} -\frac{2 \, b x e^{c} - \sqrt{\pi } \operatorname{erf}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{2 \, \sqrt{\pi } b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 3.00517, size = 89, normalized size = 2.41 \begin{align*} -\frac{2 \, \sqrt{\pi } b x e^{c} - \pi \operatorname{erf}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{2 \, \pi b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 26.7039, size = 34, normalized size = 0.92 \begin{align*} \begin{cases} - \frac{x e^{c}}{\sqrt{\pi } b} + \frac{e^{c} e^{b^{2} x^{2}} \operatorname{erf}{\left (b x \right )}}{2 b^{2}} & \text{for}\: b \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.30347, size = 42, normalized size = 1.14 \begin{align*} -\frac{x e^{c}}{\sqrt{\pi } b} + \frac{\operatorname{erf}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{2 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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