3.3.10 \(\int \frac {\text {Erfi}(b x)}{x} \, dx\) [210]

Optimal. Leaf size=31 \[ \frac {2 b x \, _2F_2\left (\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};b^2 x^2\right )}{\sqrt {\pi }} \]

[Out]

2*b*x*hypergeom([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 = 31, 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 = {6495} \begin {gather*} \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 {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6495

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

Rubi steps

\begin {align*} \int \frac {\text {erfi}(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 = 0.01, size = 31, normalized size = 1.00 \begin {gather*} \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 {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Erfi[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 = 0.07, size = 22, normalized size = 0.71

method	result	size
meijerg	\(\frac {2 b x \hypergeom \left (\left [\frac {1}{2}, \frac {1}{2}\right ], \left [\frac {3}{2}, \frac {3}{2}\right ], b^{2} x^{2}\right )}{\sqrt {\pi }}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erfi(b*x)/x,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(erfi(b*x)/x, x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(erfi(b*x)/x, x)

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Sympy [A]
time = 0.29, size = 24, normalized size = 0.77 \begin {gather*} \frac {2 b x {{}_{2}F_{2}\left (\begin {matrix} \frac {1}{2}, \frac {1}{2} \\ \frac {3}{2}, \frac {3}{2} \end {matrix}\middle | {b^{2} x^{2}} \right )}}{\sqrt {\pi }} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b*x*hyper((1/2, 1/2), (3/2, 3/2), b**2*x**2)/sqrt(pi)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(erfi(b*x)/x, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\mathrm {erfi}\left (b\,x\right )}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(erfi(b*x)/x, x)

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