Optimal. Leaf size=35 \[ \log (x)-\frac{2 b x \text{HypergeometricPFQ}\left (\left \{\frac{1}{2},\frac{1}{2}\right \},\left \{\frac{3}{2},\frac{3}{2}\right \},-b^2 x^2\right )}{\sqrt{\pi }} \]
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Rubi [A] time = 0.0234426, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6359, 6358} \[ \log (x)-\frac{2 b x \, _2F_2\left (\frac{1}{2},\frac{1}{2};\frac{3}{2},\frac{3}{2};-b^2 x^2\right )}{\sqrt{\pi }} \]
Antiderivative was successfully verified.
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Rule 6359
Rule 6358
Rubi steps
\begin{align*} \int \frac{\text{erfc}(b x)}{x} \, dx &=\log (x)-\int \frac{\text{erf}(b x)}{x} \, dx\\ &=-\frac{2 b x \, _2F_2\left (\frac{1}{2},\frac{1}{2};\frac{3}{2},\frac{3}{2};-b^2 x^2\right )}{\sqrt{\pi }}+\log (x)\\ \end{align*}
Mathematica [A] time = 0.0205093, size = 45, normalized size = 1.29 \[ \log (x) (\text{Erf}(b x)+\text{Erfc}(b x))-\frac{2 b x \text{HypergeometricPFQ}\left (\left \{\frac{1}{2},\frac{1}{2}\right \},\left \{\frac{3}{2},\frac{3}{2}\right \},-b^2 x^2\right )}{\sqrt{\pi }} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.062, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it erfc} \left ( bx \right ) }{x}}\, 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 )}{x}\,{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{\operatorname{erf}\left (b x\right ) - 1}{x}, 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 )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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