3.119 \(\int \frac{\text{FresnelC}(b x)}{x^2} \, dx\)

Optimal. Leaf size=27 \[ \frac{1}{2} b \text{CosIntegral}\left (\frac{1}{2} \pi b^2 x^2\right )-\frac{\text{FresnelC}(b x)}{x} \]

[Out]

(b*CosIntegral[(b^2*Pi*x^2)/2])/2 - FresnelC[b*x]/x

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Rubi [A]  time = 0.0212742, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6427, 3376} \[ \frac{1}{2} b \text{CosIntegral}\left (\frac{1}{2} \pi b^2 x^2\right )-\frac{\text{FresnelC}(b x)}{x} \]

Antiderivative was successfully verified.

[In]

Int[FresnelC[b*x]/x^2,x]

[Out]

(b*CosIntegral[(b^2*Pi*x^2)/2])/2 - FresnelC[b*x]/x

Rule 6427

Int[FresnelC[(b_.)*(x_)]*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*FresnelC[b*x])/(d*(m + 1)), x] -
Dist[b/(d*(m + 1)), Int[(d*x)^(m + 1)*Cos[(Pi*b^2*x^2)/2], x], x] /; FreeQ[{b, d, m}, x] && NeQ[m, -1]

Rule 3376

Int[Cos[(d_.)*(x_)^(n_)]/(x_), x_Symbol] :> Simp[CosIntegral[d*x^n]/n, x] /; FreeQ[{d, n}, x]

Rubi steps

\begin{align*} \int \frac{C(b x)}{x^2} \, dx &=-\frac{C(b x)}{x}+b \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right )}{x} \, dx\\ &=\frac{1}{2} b \text{Ci}\left (\frac{1}{2} b^2 \pi x^2\right )-\frac{C(b x)}{x}\\ \end{align*}

Mathematica [A]  time = 0.0108917, size = 27, normalized size = 1. \[ \frac{1}{2} b \text{CosIntegral}\left (\frac{1}{2} \pi b^2 x^2\right )-\frac{\text{FresnelC}(b x)}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[FresnelC[b*x]/x^2,x]

[Out]

(b*CosIntegral[(b^2*Pi*x^2)/2])/2 - FresnelC[b*x]/x

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Maple [A]  time = 0.049, size = 28, normalized size = 1. \begin{align*} b \left ( -{\frac{{\it FresnelC} \left ( bx \right ) }{bx}}+{\frac{1}{2}{\it Ci} \left ({\frac{{b}^{2}\pi \,{x}^{2}}{2}} \right ) } \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(FresnelC(b*x)/x^2,x)

[Out]

b*(-FresnelC(b*x)/b/x+1/2*Ci(1/2*b^2*Pi*x^2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm fresnelc}\left (b x\right )}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnelc(b*x)/x^2,x, algorithm="maxima")

[Out]

integrate(fresnelc(b*x)/x^2, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\rm fresnelc}\left (b x\right )}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnelc(b*x)/x^2,x, algorithm="fricas")

[Out]

integral(fresnelc(b*x)/x^2, x)

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Sympy [B]  time = 1.13432, size = 53, normalized size = 1.96 \begin{align*} - \frac{\pi ^{2} b^{5} x^{4} \Gamma \left (\frac{5}{4}\right ){{}_{3}F_{4}\left (\begin{matrix} 1, 1, \frac{5}{4} \\ \frac{3}{2}, 2, 2, \frac{9}{4} \end{matrix}\middle |{- \frac{\pi ^{2} b^{4} x^{4}}{16}} \right )}}{128 \Gamma \left (\frac{9}{4}\right )} + \frac{b \log{\left (b^{4} x^{4} \right )}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnelc(b*x)/x**2,x)

[Out]

-pi**2*b**5*x**4*gamma(5/4)*hyper((1, 1, 5/4), (3/2, 2, 2, 9/4), -pi**2*b**4*x**4/16)/(128*gamma(9/4)) + b*log
(b**4*x**4)/4

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm fresnelc}\left (b x\right )}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnelc(b*x)/x^2,x, algorithm="giac")

[Out]

integrate(fresnelc(b*x)/x^2, x)