Optimal. Leaf size=20 \[ \frac{\sqrt{\pi } e^c \log (\text{Erfi}(b x))}{2 b} \]
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Rubi [A] time = 0.0287121, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {6375, 29} \[ \frac{\sqrt{\pi } e^c \log (\text{Erfi}(b x))}{2 b} \]
Antiderivative was successfully verified.
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Rule 6375
Rule 29
Rubi steps
\begin{align*} \int \frac{e^{c+b^2 x^2}}{\text{erfi}(b x)} \, dx &=\frac{\left (e^c \sqrt{\pi }\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\text{erfi}(b x)\right )}{2 b}\\ &=\frac{e^c \sqrt{\pi } \log (\text{erfi}(b x))}{2 b}\\ \end{align*}
Mathematica [A] time = 0.0105771, size = 20, normalized size = 1. \[ \frac{\sqrt{\pi } e^c \log (\text{Erfi}(b x))}{2 b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.061, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{{b}^{2}{x}^{2}+c}}}{{\it erfi} \left ( bx \right ) }}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left (b^{2} x^{2} + c\right )}}{\operatorname{erfi}\left (b x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.94376, size = 47, normalized size = 2.35 \begin{align*} \frac{\sqrt{\pi } e^{c} \log \left (\operatorname{erfi}\left (b x\right )\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.422322, size = 24, normalized size = 1.2 \begin{align*} \begin{cases} \frac{\sqrt{\pi } e^{c} \log{\left (\operatorname{erfi}{\left (b x \right )} \right )}}{2 b} & \text{for}\: b \neq 0 \\\tilde{\infty } x e^{c} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left (b^{2} x^{2} + c\right )}}{\operatorname{erfi}\left (b x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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