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

Optimal. Leaf size=32 \[ \frac {2 b x \, _2F_2\left (\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};-b^2 x^2\right )}{\sqrt {\pi }} \]

[Out]

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 = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.08, size = 32, normalized size = 1.00 \[ \frac {2 b x \, _2F_2\left (\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};-b^2 x^2\right )}{\sqrt {\pi }} \]

Antiderivative was successfully verified.

[In]

Integrate[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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fricas [F]  time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {erf}\left (b x\right )}{x}, x\right ) \]

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

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)

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maple [A]  time = 0.21, size = 23, normalized size = 0.72 \[ \frac {2 b x \hypergeom \left (\left [\frac {1}{2}, \frac {1}{2}\right ], \left [\frac {3}{2}, \frac {3}{2}\right ], -b^{2} x^{2}\right )}{\sqrt {\pi }} \]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: AttributeError

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