3.4 \(\int \frac{\text{Erf}(b x)}{x} \, dx\)

Optimal. Leaf size=32 \[ \frac{2 b x \text{HypergeometricPFQ}\left (\left \{\frac{1}{2},\frac{1}{2}\right \},\left \{\frac{3}{2},\frac{3}{2}\right \},-b^2 x^2\right )}{\sqrt{\pi }} \]

[Out]

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

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Rubi [A]  time = 0.0137938, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {6358} \[ \frac{2 b x \, _2F_2\left (\frac{1}{2},\frac{1}{2};\frac{3}{2},\frac{3}{2};-b^2 x^2\right )}{\sqrt{\pi }} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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{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 }}\\ \end{align*}

Mathematica [A]  time = 0.0137354, size = 32, normalized size = 1. \[ \frac{2 b x \text{HypergeometricPFQ}\left (\left \{\frac{1}{2},\frac{1}{2}\right \},\left \{\frac{3}{2},\frac{3}{2}\right \},-b^2 x^2\right )}{\sqrt{\pi }} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.092, size = 23, normalized size = 0.7 \begin{align*} 2\,{\frac{bx{\mbox{$_2$F$_2$}(1/2,1/2;\,3/2,3/2;\,-{b}^{2}{x}^{2})}}{\sqrt{\pi }}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erf}\left (b x\right )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(erf(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 )}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(erf(b*x)/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(erf(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{erf}\left (b x\right )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(erf(b*x)/x, x)