3.210 \(\int \frac{\text{FresnelC}(b x) \sin (\frac{1}{2} b^2 \pi x^2)}{x^2} \, dx\)

Optimal. Leaf size=48 \[ -\frac{\text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{x}+\frac{1}{4} b \text{Si}\left (b^2 \pi x^2\right )+\frac{1}{2} \pi b \text{FresnelC}(b x)^2 \]

[Out]

(b*Pi*FresnelC[b*x]^2)/2 - (FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/x + (b*SinIntegral[b^2*Pi*x^2])/4

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Rubi [A]  time = 0.0427923, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6465, 6441, 30, 3375} \[ -\frac{\text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{x}+\frac{1}{4} b \text{Si}\left (b^2 \pi x^2\right )+\frac{1}{2} \pi b \text{FresnelC}(b x)^2 \]

Antiderivative was successfully verified.

[In]

Int[(FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/x^2,x]

[Out]

(b*Pi*FresnelC[b*x]^2)/2 - (FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/x + (b*SinIntegral[b^2*Pi*x^2])/4

Rule 6465

Int[FresnelC[(b_.)*(x_)]*(x_)^(m_)*Sin[(d_.)*(x_)^2], x_Symbol] :> Simp[(x^(m + 1)*Sin[d*x^2]*FresnelC[b*x])/(
m + 1), x] + (-Dist[(2*d)/(m + 1), Int[x^(m + 2)*Cos[d*x^2]*FresnelC[b*x], x], x] - Dist[b/(2*(m + 1)), Int[x^
(m + 1)*Sin[2*d*x^2], x], x]) /; FreeQ[{b, d}, x] && EqQ[d^2, (Pi^2*b^4)/4] && ILtQ[m, -1]

Rule 6441

Int[Cos[(d_.)*(x_)^2]*FresnelC[(b_.)*(x_)]^(n_.), x_Symbol] :> Dist[(Pi*b)/(2*d), Subst[Int[x^n, x], x, Fresne
lC[b*x]], x] /; FreeQ[{b, d, n}, x] && EqQ[d^2, (Pi^2*b^4)/4]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 3375

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

Rubi steps

\begin{align*} \int \frac{C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^2} \, dx &=-\frac{C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x}+\frac{1}{2} b \int \frac{\sin \left (b^2 \pi x^2\right )}{x} \, dx+\left (b^2 \pi \right ) \int \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x) \, dx\\ &=-\frac{C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x}+\frac{1}{4} b \text{Si}\left (b^2 \pi x^2\right )+(b \pi ) \operatorname{Subst}(\int x \, dx,x,C(b x))\\ &=\frac{1}{2} b \pi C(b x)^2-\frac{C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x}+\frac{1}{4} b \text{Si}\left (b^2 \pi x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0050088, size = 48, normalized size = 1. \[ -\frac{\text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{x}+\frac{1}{4} b \text{Si}\left (b^2 \pi x^2\right )+\frac{1}{2} \pi b \text{FresnelC}(b x)^2 \]

Antiderivative was successfully verified.

[In]

Integrate[(FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/x^2,x]

[Out]

(b*Pi*FresnelC[b*x]^2)/2 - (FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/x + (b*SinIntegral[b^2*Pi*x^2])/4

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Maple [F]  time = 0.059, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it FresnelC} \left ( bx \right ) }{{x}^{2}}\sin \left ({\frac{{b}^{2}\pi \,{x}^{2}}{2}} \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/x^2,x)

[Out]

int(FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/x^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm fresnelc}\left (b x\right ) \sin \left (\frac{1}{2} \, \pi b^{2} x^{2}\right )}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnelc(b*x)*sin(1/2*b^2*pi*x^2)/x^2,x, algorithm="maxima")

[Out]

integrate(fresnelc(b*x)*sin(1/2*pi*b^2*x^2)/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 ) \sin \left (\frac{1}{2} \, \pi b^{2} x^{2}\right )}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnelc(b*x)*sin(1/2*b^2*pi*x^2)/x^2,x, algorithm="fricas")

[Out]

integral(fresnelc(b*x)*sin(1/2*pi*b^2*x^2)/x^2, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin{\left (\frac{\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )}{x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnelc(b*x)*sin(1/2*b**2*pi*x**2)/x**2,x)

[Out]

Integral(sin(pi*b**2*x**2/2)*fresnelc(b*x)/x**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm fresnelc}\left (b x\right ) \sin \left (\frac{1}{2} \, \pi b^{2} x^{2}\right )}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnelc(b*x)*sin(1/2*b^2*pi*x^2)/x^2,x, algorithm="giac")

[Out]

integrate(fresnelc(b*x)*sin(1/2*pi*b^2*x^2)/x^2, x)