3.42 \(\int \frac{S(b x)^2}{x^4} \, dx\)

Optimal. Leaf size=119 \[ \frac{1}{3} \pi b^3 \text{Unintegrable}\left (\frac{S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{x},x\right )-\frac{b S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{3 x^2}+\frac{\pi b^3 S\left (\sqrt{2} b x\right )}{3 \sqrt{2}}+\frac{b^2 \cos \left (\pi b^2 x^2\right )}{6 x}-\frac{b^2}{6 x}-\frac{S(b x)^2}{3 x^3} \]

[Out]

-b^2/(6*x) + (b^2*Cos[b^2*Pi*x^2])/(6*x) - FresnelS[b*x]^2/(3*x^3) + (b^3*Pi*FresnelS[Sqrt[2]*b*x])/(3*Sqrt[2]
) - (b*FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/(3*x^2) + (b^3*Pi*Unintegrable[(Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/x
, x])/3

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Rubi [A]  time = 0.0873615, 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 = {} \[ \int \frac{S(b x)^2}{x^4} \, dx \]

Verification is Not applicable to the result.

[In]

Int[FresnelS[b*x]^2/x^4,x]

[Out]

-b^2/(6*x) + (b^2*Cos[b^2*Pi*x^2])/(6*x) - FresnelS[b*x]^2/(3*x^3) + (b^3*Pi*FresnelS[Sqrt[2]*b*x])/(3*Sqrt[2]
) - (b*FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/(3*x^2) + (b^3*Pi*Defer[Int][(Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/x,
x])/3

Rubi steps

\begin{align*} \int \frac{S(b x)^2}{x^4} \, dx &=-\frac{S(b x)^2}{3 x^3}+\frac{1}{3} (2 b) \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^3} \, dx\\ &=-\frac{b^2}{6 x}-\frac{S(b x)^2}{3 x^3}-\frac{b S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{3 x^2}-\frac{1}{6} b^2 \int \frac{\cos \left (b^2 \pi x^2\right )}{x^2} \, dx+\frac{1}{3} \left (b^3 \pi \right ) \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{x} \, dx\\ &=-\frac{b^2}{6 x}+\frac{b^2 \cos \left (b^2 \pi x^2\right )}{6 x}-\frac{S(b x)^2}{3 x^3}-\frac{b S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{3 x^2}+\frac{1}{3} \left (b^3 \pi \right ) \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{x} \, dx+\frac{1}{3} \left (b^4 \pi \right ) \int \sin \left (b^2 \pi x^2\right ) \, dx\\ &=-\frac{b^2}{6 x}+\frac{b^2 \cos \left (b^2 \pi x^2\right )}{6 x}-\frac{S(b x)^2}{3 x^3}+\frac{b^3 \pi S\left (\sqrt{2} b x\right )}{3 \sqrt{2}}-\frac{b S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{3 x^2}+\frac{1}{3} \left (b^3 \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 = 0.0259727, size = 0, normalized size = 0. \[ \int \frac{S(b x)^2}{x^4} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[FresnelS[b*x]^2/x^4,x]

[Out]

Integrate[FresnelS[b*x]^2/x^4, x]

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Maple [A]  time = 0.055, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\it FresnelS} \left ( bx \right ) \right ) ^{2}}{{x}^{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm fresnels}\left (b x\right )^{2}}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnels(b*x)^2/x^4,x, algorithm="maxima")

[Out]

integrate(fresnels(b*x)^2/x^4, 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 )^{2}}{x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnels(b*x)^2/x^4,x, algorithm="fricas")

[Out]

integral(fresnels(b*x)^2/x^4, x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{S^{2}\left (b x\right )}{x^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnels(b*x)**2/x**4,x)

[Out]

Integral(fresnels(b*x)**2/x**4, x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm fresnels}\left (b x\right )^{2}}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnels(b*x)^2/x^4,x, algorithm="giac")

[Out]

integrate(fresnels(b*x)^2/x^4, x)