Optimal. Leaf size=62 \[ 2 d \text{Unintegrable}\left (\text{Erfc}(b x) e^{c+d x^2},x\right )-\frac{b e^c \text{ExpIntegralEi}\left (x^2 \left (-\left (b^2-d\right )\right )\right )}{\sqrt{\pi }}-\frac{\text{Erfc}(b x) e^{c+d x^2}}{x} \]
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Rubi [A] time = 0.114185, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{e^{c+d x^2} \text{Erfc}(b x)}{x^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{e^{c+d x^2} \text{erfc}(b x)}{x^2} \, dx &=-\frac{e^{c+d x^2} \text{erfc}(b x)}{x}+(2 d) \int e^{c+d x^2} \text{erfc}(b x) \, dx-\frac{(2 b) \int \frac{e^{c-\left (b^2-d\right ) x^2}}{x} \, dx}{\sqrt{\pi }}\\ &=-\frac{e^{c+d x^2} \text{erfc}(b x)}{x}-\frac{b e^c \text{Ei}\left (-\left (b^2-d\right ) x^2\right )}{\sqrt{\pi }}+(2 d) \int e^{c+d x^2} \text{erfc}(b x) \, dx\\ \end{align*}
Mathematica [A] time = 0.632476, size = 0, normalized size = 0. \[ \int \frac{e^{c+d x^2} \text{Erfc}(b x)}{x^2} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.134, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{d{x}^{2}+c}}{\it erfc} \left ( bx \right ) }{{x}^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erfc}\left (b x\right ) e^{\left (d x^{2} + c\right )}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (\operatorname{erf}\left (b x\right ) - 1\right )} e^{\left (d x^{2} + c\right )}}{x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} e^{c} \int \frac{e^{d x^{2}} \operatorname{erfc}{\left (b x \right )}}{x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erfc}\left (b x\right ) e^{\left (d x^{2} + c\right )}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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