3.82 \(\int \frac{S(b x) \sin (\frac{1}{2} b^2 \pi x^2)}{x^3} \, dx\)

Optimal. Leaf size=101 \[ \frac{1}{2} \pi b^2 \text{Unintegrable}\left (\frac{S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{x},x\right )-\frac{S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{2 x^2}+\frac{\pi b^2 S\left (\sqrt{2} b x\right )}{2 \sqrt{2}}+\frac{b \cos \left (\pi b^2 x^2\right )}{4 x}-\frac{b}{4 x} \]

[Out]

-b/(4*x) + (b*Cos[b^2*Pi*x^2])/(4*x) + (b^2*Pi*FresnelS[Sqrt[2]*b*x])/(2*Sqrt[2]) - (FresnelS[b*x]*Sin[(b^2*Pi
*x^2)/2])/(2*x^2) + (b^2*Pi*Unintegrable[(Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/x, x])/2

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Rubi [A]  time = 0.0584933, 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 = {} \[ \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/x^3,x]

[Out]

-b/(4*x) + (b*Cos[b^2*Pi*x^2])/(4*x) + (b^2*Pi*FresnelS[Sqrt[2]*b*x])/(2*Sqrt[2]) - (FresnelS[b*x]*Sin[(b^2*Pi
*x^2)/2])/(2*x^2) + (b^2*Pi*Defer[Int][(Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/x, x])/2

Rubi steps

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

Mathematica [A]  time = 0.0303193, size = 0, normalized size = 0. \[ \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/x^3,x]

[Out]

Integrate[(FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/x^3, x]

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Maple [A]  time = 0.067, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it FresnelS} \left ( bx \right ) }{{x}^{3}}\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(FresnelS(b*x)*sin(1/2*b^2*Pi*x^2)/x^3,x)

[Out]

int(FresnelS(b*x)*sin(1/2*b^2*Pi*x^2)/x^3,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm fresnels}\left (b x\right ) \sin \left (\frac{1}{2} \, \pi b^{2} x^{2}\right )}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnels(b*x)*sin(1/2*b^2*pi*x^2)/x^3,x, algorithm="maxima")

[Out]

integrate(fresnels(b*x)*sin(1/2*pi*b^2*x^2)/x^3, x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\rm fresnels}\left (b x\right ) \sin \left (\frac{1}{2} \, \pi b^{2} x^{2}\right )}{x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnels(b*x)*sin(1/2*b^2*pi*x^2)/x^3,x, algorithm="fricas")

[Out]

integral(fresnels(b*x)*sin(1/2*pi*b^2*x^2)/x^3, x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin{\left (\frac{\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )}{x^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnels(b*x)*sin(1/2*b**2*pi*x**2)/x**3,x)

[Out]

Integral(sin(pi*b**2*x**2/2)*fresnels(b*x)/x**3, x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm fresnels}\left (b x\right ) \sin \left (\frac{1}{2} \, \pi b^{2} x^{2}\right )}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnels(b*x)*sin(1/2*b^2*pi*x^2)/x^3,x, algorithm="giac")

[Out]

integrate(fresnels(b*x)*sin(1/2*pi*b^2*x^2)/x^3, x)