3.9 \(\int \frac{S(b x)}{x} \, dx\)

Optimal. Leaf size=73 \[ \frac{1}{2} i b x \text{HypergeometricPFQ}\left (\left \{\frac{1}{2},\frac{1}{2}\right \},\left \{\frac{3}{2},\frac{3}{2}\right \},-\frac{1}{2} i \pi b^2 x^2\right )-\frac{1}{2} i b x \text{HypergeometricPFQ}\left (\left \{\frac{1}{2},\frac{1}{2}\right \},\left \{\frac{3}{2},\frac{3}{2}\right \},\frac{1}{2} i \pi b^2 x^2\right ) \]

[Out]

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

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Rubi [A]  time = 0.0466434, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {6424, 6358, 6360} \[ \frac{1}{2} i b x \, _2F_2\left (\frac{1}{2},\frac{1}{2};\frac{3}{2},\frac{3}{2};-\frac{1}{2} i b^2 \pi x^2\right )-\frac{1}{2} i b x \, _2F_2\left (\frac{1}{2},\frac{1}{2};\frac{3}{2},\frac{3}{2};\frac{1}{2} i b^2 \pi x^2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6424

Int[FresnelS[(b_.)*(x_)]/(x_), x_Symbol] :> Dist[(1 + I)/4, Int[Erf[(Sqrt[Pi]*(1 + I)*b*x)/2]/x, x], x] + Dist
[(1 - I)/4, Int[Erf[(Sqrt[Pi]*(1 - I)*b*x)/2]/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]

Rule 6360

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

Rubi steps

\begin{align*} \int \frac{S(b x)}{x} \, dx &=\left (-\frac{1}{4}-\frac{i}{4}\right ) \int \frac{\text{erfi}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) b \sqrt{\pi } x\right )}{x} \, dx+\left (\frac{1}{4}+\frac{i}{4}\right ) \int \frac{\text{erf}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) b \sqrt{\pi } x\right )}{x} \, dx\\ &=\frac{1}{2} i b x \, _2F_2\left (\frac{1}{2},\frac{1}{2};\frac{3}{2},\frac{3}{2};-\frac{1}{2} i b^2 \pi x^2\right )-\frac{1}{2} i b x \, _2F_2\left (\frac{1}{2},\frac{1}{2};\frac{3}{2},\frac{3}{2};\frac{1}{2} i b^2 \pi x^2\right )\\ \end{align*}

Mathematica [F]  time = 0.0136888, size = 0, normalized size = 0. \[ \int \frac{S(b x)}{x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Integrate[FresnelS[b*x]/x, x]

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Maple [A]  time = 0.08, size = 29, normalized size = 0.4 \begin{align*}{\frac{\pi \,{x}^{3}{b}^{3}}{18}{\mbox{$_2$F$_3$}({\frac{3}{4}},{\frac{3}{4}};\,{\frac{3}{2}},{\frac{7}{4}},{\frac{7}{4}};\,-{\frac{{x}^{4}{\pi }^{2}{b}^{4}}{16}})}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*Pi*x^3*b^3*hypergeom([3/4,3/4],[3/2,7/4,7/4],-1/16*x^4*Pi^2*b^4)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(fresnels(b*x)/x, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\rm fresnels}\left (b x\right )}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(fresnels(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(fresnels(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{{\rm fresnels}\left (b x\right )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(fresnels(b*x)/x, x)