3.103 \(\int \frac{\cos (\frac{1}{2} b^2 \pi x^2) S(b x)}{x^4} \, dx\)

Optimal. Leaf size=88 \[ -\frac{1}{3} \pi b^2 \text{Unintegrable}\left (\frac{S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{x^2},x\right )+\frac{1}{12} \pi b^3 \text{CosIntegral}\left (\pi b^2 x^2\right )-\frac{S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{3 x^3}-\frac{b \sin \left (\pi b^2 x^2\right )}{12 x^2} \]

[Out]

(b^3*Pi*CosIntegral[b^2*Pi*x^2])/12 - (Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(3*x^3) - (b*Sin[b^2*Pi*x^2])/(12*x^
2) - (b^2*Pi*Unintegrable[(FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/x^2, x])/3

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Rubi [A]  time = 0.0878529, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{x^4} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/x^4,x]

[Out]

(b^3*Pi*CosIntegral[b^2*Pi*x^2])/12 - (Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(3*x^3) - (b*Sin[b^2*Pi*x^2])/(12*x^
2) - (b^2*Pi*Defer[Int][(FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/x^2, x])/3

Rubi steps

\begin{align*} \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{x^4} \, dx &=-\frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{3 x^3}+\frac{1}{6} b \int \frac{\sin \left (b^2 \pi x^2\right )}{x^3} \, dx-\frac{1}{3} \left (b^2 \pi \right ) \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^2} \, dx\\ &=-\frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{3 x^3}+\frac{1}{12} b \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )-\frac{1}{3} \left (b^2 \pi \right ) \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^2} \, dx\\ &=-\frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{3 x^3}-\frac{b \sin \left (b^2 \pi x^2\right )}{12 x^2}-\frac{1}{3} \left (b^2 \pi \right ) \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^2} \, dx+\frac{1}{12} \left (b^3 \pi \right ) \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )\\ &=\frac{1}{12} b^3 \pi \text{Ci}\left (b^2 \pi x^2\right )-\frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{3 x^3}-\frac{b \sin \left (b^2 \pi x^2\right )}{12 x^2}-\frac{1}{3} \left (b^2 \pi \right ) \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^2} \, dx\\ \end{align*}

Mathematica [A]  time = 0.0336194, size = 0, normalized size = 0. \[ \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{x^4} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/x^4,x]

[Out]

Integrate[(Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/x^4, x]

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Maple [A]  time = 0.061, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it FresnelS} \left ( bx \right ) }{{x}^{4}}\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^4,x)

[Out]

int(cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/x^4,x)

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Maxima [A]  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^{4}}\,{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^4,x, algorithm="maxima")

[Out]

integrate(cos(1/2*pi*b^2*x^2)*fresnels(b*x)/x^4, x)

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Fricas [A]  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^{4}}, 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^4,x, algorithm="fricas")

[Out]

integral(cos(1/2*pi*b^2*x^2)*fresnels(b*x)/x^4, x)

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Sympy [A]  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^{4}}\, 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**4,x)

[Out]

Integral(cos(pi*b**2*x**2/2)*fresnels(b*x)/x**4, x)

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Giac [A]  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^{4}}\,{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^4,x, algorithm="giac")

[Out]

integrate(cos(1/2*pi*b^2*x^2)*fresnels(b*x)/x^4, x)