3.107 \(\int \frac{\text{Erfc}(b x)}{x} \, dx\)

Optimal. Leaf size=35 \[ \log (x)-\frac{2 b x \text{HypergeometricPFQ}\left (\left \{\frac{1}{2},\frac{1}{2}\right \},\left \{\frac{3}{2},\frac{3}{2}\right \},-b^2 x^2\right )}{\sqrt{\pi }} \]

[Out]

(-2*b*x*HypergeometricPFQ[{1/2, 1/2}, {3/2, 3/2}, -(b^2*x^2)])/Sqrt[Pi] + Log[x]

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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.

[In]

Int[Erfc[b*x]/x,x]

[Out]

(-2*b*x*HypergeometricPFQ[{1/2, 1/2}, {3/2, 3/2}, -(b^2*x^2)])/Sqrt[Pi] + Log[x]

Rule 6359

Int[Erfc[(b_.)*(x_)]/(x_), x_Symbol] :> Simp[Log[x], x] - Int[Erf[b*x]/x, x] /; FreeQ[b, x]

Rule 6358

Int[Erf[(b_.)*(x_)]/(x_), x_Symbol] :> Simp[(2*b*x*HypergeometricPFQ[{1/2, 1/2}, {3/2, 3/2}, -(b^2*x^2)])/Sqrt
[Pi], x] /; FreeQ[b, x]

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.

[In]

Integrate[Erfc[b*x]/x,x]

[Out]

(-2*b*x*HypergeometricPFQ[{1/2, 1/2}, {3/2, 3/2}, -(b^2*x^2)])/Sqrt[Pi] + (Erf[b*x] + Erfc[b*x])*Log[x]

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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.

[In]

int(erfc(b*x)/x,x)

[Out]

int(erfc(b*x)/x,x)

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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.

[In]

integrate(erfc(b*x)/x,x, algorithm="maxima")

[Out]

integrate(erfc(b*x)/x, x)

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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.

[In]

integrate(erfc(b*x)/x,x, algorithm="fricas")

[Out]

integral(-(erf(b*x) - 1)/x, x)

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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.

[In]

integrate(erfc(b*x)/x,x)

[Out]

Exception raised: AttributeError

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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.

[In]

integrate(erfc(b*x)/x,x, algorithm="giac")

[Out]

integrate(erfc(b*x)/x, x)