3.1.9 \(\int \frac {S(b x)}{x} \, dx\) [9]

Optimal. Leaf size=73 \[ \frac {1}{2} i b x \, _2F_2\left (\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};-\frac {1}{2} i b^2 \pi x^2\right )-\frac {1}{2} i b x \, _2F_2\left (\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};\frac {1}{2} i b^2 \pi x^2\right ) \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 73, normalized size of antiderivative = 1.00, 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 = {6559, 6493, 6495} \begin {gather*} \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 {gather*}

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}, (-1/2*I)*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 6493

Int[Erf[(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]

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]

Rule 6559

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

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Mathematica [F]
time = 0.01, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {S(b x)}{x} \, dx \end {gather*}

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 = 0.33, size = 29, normalized size = 0.40

method	result	size
meijerg	\(\frac {\pi \,x^{3} b^{3} \hypergeom \left (\left [\frac {3}{4}, \frac {3}{4}\right ], \left [\frac {3}{2}, \frac {7}{4}, \frac {7}{4}\right ], -\frac {x^{4} \pi ^{2} b^{4}}{16}\right )}{18}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.00, 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(fresnel_sin(b*x)/x,x, algorithm="maxima")

[Out]

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

[Out]

integral(fresnel_sin(b*x)/x, x)

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Sympy [A]
time = 0.30, size = 46, normalized size = 0.63 \begin {gather*} \frac {\pi b^{3} x^{3} \Gamma ^{2}\left (\frac {3}{4}\right ) {{}_{2}F_{3}\left (\begin {matrix} \frac {3}{4}, \frac {3}{4} \\ \frac {3}{2}, \frac {7}{4}, \frac {7}{4} \end {matrix}\middle | {- \frac {\pi ^{2} b^{4} x^{4}}{16}} \right )}}{32 \Gamma ^{2}\left (\frac {7}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

pi*b**3*x**3*gamma(3/4)**2*hyper((3/4, 3/4), (3/2, 7/4, 7/4), -pi**2*b**4*x**4/16)/(32*gamma(7/4)**2)

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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(fresnel_sin(b*x)/x,x, algorithm="giac")

[Out]

integrate(fresnel_sin(b*x)/x, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\mathrm {FresnelS}\left (b\,x\right )}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(FresnelS(b*x)/x, x)

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