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

Optimal. Leaf size=35 \[ \log (x)-\frac {2 b x \, _2F_2\left (\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};-b^2 x^2\right )}{\sqrt {\pi }} \]

[Out]

ln(x)-2*b*x*HypergeometricPFQ([1/2, 1/2],[3/2, 3/2],-b^2*x^2)/Pi^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 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]

Rule 6359

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

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Mathematica [A]  time = 0.03, size = 45, normalized size = 1.29 \[ \log (x) (\text {erf}(b x)+\text {erfc}(b x))-\frac {2 b x \, _2F_2\left (\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};-b^2 x^2\right )}{\sqrt {\pi }} \]

Antiderivative was successfully verified.

[In]

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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fricas [F]  time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\operatorname {erf}\left (b x\right ) - 1}{x}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {erfc}\left (b x\right )}{x}\,{d x} \]

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)

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maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {erfc}\left (b x \right )}{x}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {erfc}\left (b x\right )}{x}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\mathrm {erfc}\left (b\,x\right )}{x} \,d x \]

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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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: AttributeError} \]

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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