3.101 \(\int \frac{\cos (\frac{1}{2} b^2 \pi x^2) S(b x)}{x^2} \, dx\)

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

[Out]

-((Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/x) - (b*Pi*FresnelS[b*x]^2)/2 + (b*SinIntegral[b^2*Pi*x^2])/4

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Rubi [A]  time = 0.0413899, 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 = {6464, 6440, 30, 3375} \[ -\frac{S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{x}+\frac{1}{4} b \text{Si}\left (b^2 \pi x^2\right )-\frac{1}{2} \pi b S(b x)^2 \]

Antiderivative was successfully verified.

[In]

Int[(Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/x^2,x]

[Out]

-((Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/x) - (b*Pi*FresnelS[b*x]^2)/2 + (b*SinIntegral[b^2*Pi*x^2])/4

Rule 6464

Int[Cos[(d_.)*(x_)^2]*FresnelS[(b_.)*(x_)]*(x_)^(m_), x_Symbol] :> Simp[(x^(m + 1)*Cos[d*x^2]*FresnelS[b*x])/(
m + 1), x] + (Dist[(2*d)/(m + 1), Int[x^(m + 2)*Sin[d*x^2]*FresnelS[b*x], x], x] - Dist[d/(Pi*b*(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 6440

Int[FresnelS[(b_.)*(x_)]^(n_.)*Sin[(d_.)*(x_)^2], x_Symbol] :> Dist[(Pi*b)/(2*d), Subst[Int[x^n, x], x, Fresne
lS[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{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{x^2} \, dx &=-\frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{x}+\frac{1}{2} b \int \frac{\sin \left (b^2 \pi x^2\right )}{x} \, dx-\left (b^2 \pi \right ) \int S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right ) \, dx\\ &=-\frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{x}+\frac{1}{4} b \text{Si}\left (b^2 \pi x^2\right )-(b \pi ) \operatorname{Subst}(\int x \, dx,x,S(b x))\\ &=-\frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{x}-\frac{1}{2} b \pi S(b x)^2+\frac{1}{4} b \text{Si}\left (b^2 \pi x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0045022, size = 48, normalized size = 1. \[ -\frac{S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{x}+\frac{1}{4} b \text{Si}\left (b^2 \pi x^2\right )-\frac{1}{2} \pi b S(b x)^2 \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/x^2,x]

[Out]

-((Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/x) - (b*Pi*FresnelS[b*x]^2)/2 + (b*SinIntegral[b^2*Pi*x^2])/4

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Maple [F]  time = 0.063, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it FresnelS} \left ( bx \right ) }{{x}^{2}}\cos \left ({\frac{{b}^{2}\pi \,{x}^{2}}{2}} \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/x^2,x)

[Out]

int(cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/x^2,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(cos(1/2*pi*b^2*x^2)*fresnels(b*x)/x^2, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\cos \left (\frac{1}{2} \, \pi b^{2} x^{2}\right ){\rm fresnels}\left (b x\right )}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(cos(1/2*pi*b^2*x^2)*fresnels(b*x)/x^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(1/2*b**2*pi*x**2)*fresnels(b*x)/x**2,x)

[Out]

Integral(cos(pi*b**2*x**2/2)*fresnels(b*x)/x**2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(cos(1/2*pi*b^2*x^2)*fresnels(b*x)/x^2, x)