3.2.18 \(\int \frac {\text {FresnelC}(b x)}{x} \, dx\) [118]

Optimal. Leaf size=69 \[ \frac {1}{2} b x \, _2F_2\left (\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};-\frac {1}{2} i b^2 \pi x^2\right )+\frac {1}{2} b x \, _2F_2\left (\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};\frac {1}{2} i b^2 \pi x^2\right ) \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 69, 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 = {6560, 6493, 6495} \begin {gather*} \frac {1}{2} 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} 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[FresnelC[b*x]/x,x]

[Out]

(b*x*HypergeometricPFQ[{1/2, 1/2}, {3/2, 3/2}, (-1/2*I)*b^2*Pi*x^2])/2 + (b*x*HypergeometricPFQ[{1/2, 1/2}, {3
/2, 3/2}, (I/2)*b^2*Pi*x^2])/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 6560

Int[FresnelC[(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 {C(b x)}{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+\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\\ &=\frac {1}{2} 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} 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 {\text {FresnelC}(b x)}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

Integrate[FresnelC[b*x]/x, x]

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Maple [A]
time = 0.38, size = 23, normalized size = 0.33

method	result	size
meijerg	\(b x \hypergeom \left (\left [\frac {1}{4}, \frac {1}{4}\right ], \left [\frac {1}{2}, \frac {5}{4}, \frac {5}{4}\right ], -\frac {x^{4} \pi ^{2} b^{4}}{16}\right )\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*x*hypergeom([1/4,1/4],[1/2,5/4,5/4],-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_cos(b*x)/x,x, algorithm="maxima")

[Out]

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

[Out]

integral(fresnel_cos(b*x)/x, x)

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Sympy [A]
time = 0.29, size = 41, normalized size = 0.59 \begin {gather*} \frac {b x \Gamma ^{2}\left (\frac {1}{4}\right ) {{}_{2}F_{3}\left (\begin {matrix} \frac {1}{4}, \frac {1}{4} \\ \frac {1}{2}, \frac {5}{4}, \frac {5}{4} \end {matrix}\middle | {- \frac {\pi ^{2} b^{4} x^{4}}{16}} \right )}}{16 \Gamma ^{2}\left (\frac {5}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*x*gamma(1/4)**2*hyper((1/4, 1/4), (1/2, 5/4, 5/4), -pi**2*b**4*x**4/16)/(16*gamma(5/4)**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_cos(b*x)/x,x, algorithm="giac")

[Out]

integrate(fresnel_cos(b*x)/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(FresnelC(b*x)/x, x)

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