Optimal. Leaf size=80 \[ -\frac{2 b^3 e^c x^2 \text{HypergeometricPFQ}\left (\{1,1\},\left \{\frac{3}{2},2\right \},b^2 x^2\right )}{\sqrt{\pi }}-\frac{e^{b^2 x^2+c} \text{Erfc}(b x)}{x}+\sqrt{\pi } b e^c \text{Erfi}(b x)-\frac{2 b e^c \log (x)}{\sqrt{\pi }} \]
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Rubi [A] time = 0.0787574, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {6392, 6377, 2204, 6376, 12, 29} \[ -\frac{2 b^3 e^c x^2 \, _2F_2\left (1,1;\frac{3}{2},2;b^2 x^2\right )}{\sqrt{\pi }}-\frac{e^{b^2 x^2+c} \text{Erfc}(b x)}{x}+\sqrt{\pi } b e^c \text{Erfi}(b x)-\frac{2 b e^c \log (x)}{\sqrt{\pi }} \]
Antiderivative was successfully verified.
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Rule 6392
Rule 6377
Rule 2204
Rule 6376
Rule 12
Rule 29
Rubi steps
\begin{align*} \int \frac{e^{c+b^2 x^2} \text{erfc}(b x)}{x^2} \, dx &=-\frac{e^{c+b^2 x^2} \text{erfc}(b x)}{x}+\left (2 b^2\right ) \int e^{c+b^2 x^2} \text{erfc}(b x) \, dx-\frac{(2 b) \int \frac{e^c}{x} \, dx}{\sqrt{\pi }}\\ &=-\frac{e^{c+b^2 x^2} \text{erfc}(b x)}{x}+\left (2 b^2\right ) \int e^{c+b^2 x^2} \, dx-\left (2 b^2\right ) \int e^{c+b^2 x^2} \text{erf}(b x) \, dx-\frac{\left (2 b e^c\right ) \int \frac{1}{x} \, dx}{\sqrt{\pi }}\\ &=-\frac{e^{c+b^2 x^2} \text{erfc}(b x)}{x}+b e^c \sqrt{\pi } \text{erfi}(b x)-\frac{2 b^3 e^c x^2 \, _2F_2\left (1,1;\frac{3}{2},2;b^2 x^2\right )}{\sqrt{\pi }}-\frac{2 b e^c \log (x)}{\sqrt{\pi }}\\ \end{align*}
Mathematica [A] time = 0.262034, size = 99, normalized size = 1.24 \[ -\frac{e^c \left (-2 b^3 x^3 \text{HypergeometricPFQ}\left (\{1,1\},\left \{\frac{3}{2},2\right \},-b^2 x^2\right )+\text{Erf}(b x) \left (\pi b x \text{Erfi}(b x)-\sqrt{\pi } e^{b^2 x^2}\right )+\sqrt{\pi } e^{b^2 x^2}-\pi b x \text{Erfi}(b x)+2 b x \log (x)\right )}{\sqrt{\pi } x} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.128, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{{b}^{2}{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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erfc}\left (b x\right ) e^{\left (b^{2} 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 [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (\operatorname{erf}\left (b x\right ) - 1\right )} e^{\left (b^{2} 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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \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{\operatorname{erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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