3.46 \(\int \frac{S(b x)^2}{x^8} \, dx\)

Optimal. Leaf size=258 \[ -\frac{1}{168} \pi ^3 b^7 \text{Unintegrable}\left (\frac{S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{x},x\right )+\frac{\pi ^2 b^5 S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{168 x^2}-\frac{b S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{21 x^6}-\frac{\pi b^3 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{84 x^4}-\frac{2}{315} \sqrt{2} \pi ^3 b^7 S\left (\sqrt{2} b x\right )-\frac{\pi ^3 b^7 S\left (\sqrt{2} b x\right )}{72 \sqrt{2}}-\frac{b^2}{210 x^5}-\frac{13 \pi b^4 \sin \left (\pi b^2 x^2\right )}{2520 x^3}-\frac{67 \pi ^2 b^6 \cos \left (\pi b^2 x^2\right )}{5040 x}+\frac{b^2 \cos \left (\pi b^2 x^2\right )}{210 x^5}+\frac{\pi ^2 b^6}{336 x}-\frac{S(b x)^2}{7 x^7} \]

[Out]

-b^2/(210*x^5) + (b^6*Pi^2)/(336*x) + (b^2*Cos[b^2*Pi*x^2])/(210*x^5) - (67*b^6*Pi^2*Cos[b^2*Pi*x^2])/(5040*x)
- (b^3*Pi*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(84*x^4) - FresnelS[b*x]^2/(7*x^7) - (b^7*Pi^3*FresnelS[Sqrt[2]*
b*x])/(72*Sqrt[2]) - (2*Sqrt[2]*b^7*Pi^3*FresnelS[Sqrt[2]*b*x])/315 - (b*FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/(2
1*x^6) + (b^5*Pi^2*FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/(168*x^2) - (13*b^4*Pi*Sin[b^2*Pi*x^2])/(2520*x^3) - (b^
7*Pi^3*Unintegrable[(Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/x, x])/168

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Rubi [A]  time = 0.244556, 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 = {} \[ \int \frac{S(b x)^2}{x^8} \, dx \]

Verification is Not applicable to the result.

[In]

Int[FresnelS[b*x]^2/x^8,x]

[Out]

-b^2/(210*x^5) + (b^6*Pi^2)/(336*x) + (b^2*Cos[b^2*Pi*x^2])/(210*x^5) - (67*b^6*Pi^2*Cos[b^2*Pi*x^2])/(5040*x)
- (b^3*Pi*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(84*x^4) - FresnelS[b*x]^2/(7*x^7) - (b^7*Pi^3*FresnelS[Sqrt[2]*
b*x])/(72*Sqrt[2]) - (2*Sqrt[2]*b^7*Pi^3*FresnelS[Sqrt[2]*b*x])/315 - (b*FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/(2
1*x^6) + (b^5*Pi^2*FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/(168*x^2) - (13*b^4*Pi*Sin[b^2*Pi*x^2])/(2520*x^3) - (b^
7*Pi^3*Defer[Int][(Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/x, x])/168

Rubi steps

\begin{align*} \int \frac{S(b x)^2}{x^8} \, dx &=-\frac{S(b x)^2}{7 x^7}+\frac{1}{7} (2 b) \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^7} \, dx\\ &=-\frac{b^2}{210 x^5}-\frac{S(b x)^2}{7 x^7}-\frac{b S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{21 x^6}-\frac{1}{42} b^2 \int \frac{\cos \left (b^2 \pi x^2\right )}{x^6} \, dx+\frac{1}{21} \left (b^3 \pi \right ) \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{x^5} \, dx\\ &=-\frac{b^2}{210 x^5}+\frac{b^2 \cos \left (b^2 \pi x^2\right )}{210 x^5}-\frac{b^3 \pi \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{84 x^4}-\frac{S(b x)^2}{7 x^7}-\frac{b S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{21 x^6}+\frac{1}{168} \left (b^4 \pi \right ) \int \frac{\sin \left (b^2 \pi x^2\right )}{x^4} \, dx+\frac{1}{105} \left (b^4 \pi \right ) \int \frac{\sin \left (b^2 \pi x^2\right )}{x^4} \, dx-\frac{1}{84} \left (b^5 \pi ^2\right ) \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^3} \, dx\\ &=-\frac{b^2}{210 x^5}+\frac{b^6 \pi ^2}{336 x}+\frac{b^2 \cos \left (b^2 \pi x^2\right )}{210 x^5}-\frac{b^3 \pi \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{84 x^4}-\frac{S(b x)^2}{7 x^7}-\frac{b S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{21 x^6}+\frac{b^5 \pi ^2 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{168 x^2}-\frac{13 b^4 \pi \sin \left (b^2 \pi x^2\right )}{2520 x^3}+\frac{1}{336} \left (b^6 \pi ^2\right ) \int \frac{\cos \left (b^2 \pi x^2\right )}{x^2} \, dx+\frac{1}{252} \left (b^6 \pi ^2\right ) \int \frac{\cos \left (b^2 \pi x^2\right )}{x^2} \, dx+\frac{1}{315} \left (2 b^6 \pi ^2\right ) \int \frac{\cos \left (b^2 \pi x^2\right )}{x^2} \, dx-\frac{1}{168} \left (b^7 \pi ^3\right ) \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{x} \, dx\\ &=-\frac{b^2}{210 x^5}+\frac{b^6 \pi ^2}{336 x}+\frac{b^2 \cos \left (b^2 \pi x^2\right )}{210 x^5}-\frac{67 b^6 \pi ^2 \cos \left (b^2 \pi x^2\right )}{5040 x}-\frac{b^3 \pi \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{84 x^4}-\frac{S(b x)^2}{7 x^7}-\frac{b S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{21 x^6}+\frac{b^5 \pi ^2 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{168 x^2}-\frac{13 b^4 \pi \sin \left (b^2 \pi x^2\right )}{2520 x^3}-\frac{1}{168} \left (b^7 \pi ^3\right ) \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{x} \, dx-\frac{1}{168} \left (b^8 \pi ^3\right ) \int \sin \left (b^2 \pi x^2\right ) \, dx-\frac{1}{126} \left (b^8 \pi ^3\right ) \int \sin \left (b^2 \pi x^2\right ) \, dx-\frac{1}{315} \left (4 b^8 \pi ^3\right ) \int \sin \left (b^2 \pi x^2\right ) \, dx\\ &=-\frac{b^2}{210 x^5}+\frac{b^6 \pi ^2}{336 x}+\frac{b^2 \cos \left (b^2 \pi x^2\right )}{210 x^5}-\frac{67 b^6 \pi ^2 \cos \left (b^2 \pi x^2\right )}{5040 x}-\frac{b^3 \pi \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{84 x^4}-\frac{S(b x)^2}{7 x^7}-\frac{b^7 \pi ^3 S\left (\sqrt{2} b x\right )}{72 \sqrt{2}}-\frac{2}{315} \sqrt{2} b^7 \pi ^3 S\left (\sqrt{2} b x\right )-\frac{b S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{21 x^6}+\frac{b^5 \pi ^2 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{168 x^2}-\frac{13 b^4 \pi \sin \left (b^2 \pi x^2\right )}{2520 x^3}-\frac{1}{168} \left (b^7 \pi ^3\right ) \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{x} \, dx\\ \end{align*}

Mathematica [A]  time = 0.0272621, size = 0, normalized size = 0. \[ \int \frac{S(b x)^2}{x^8} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[FresnelS[b*x]^2/x^8,x]

[Out]

Integrate[FresnelS[b*x]^2/x^8, x]

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Maple [A]  time = 0.057, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\it FresnelS} \left ( bx \right ) \right ) ^{2}}{{x}^{8}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(FresnelS(b*x)^2/x^8,x)

[Out]

int(FresnelS(b*x)^2/x^8,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^{8}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnels(b*x)^2/x^8,x, algorithm="maxima")

[Out]

integrate(fresnels(b*x)^2/x^8, 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^{8}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnels(b*x)^2/x^8,x, algorithm="fricas")

[Out]

integral(fresnels(b*x)^2/x^8, 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^{8}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnels(b*x)**2/x**8,x)

[Out]

Integral(fresnels(b*x)**2/x**8, 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^{8}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnels(b*x)^2/x^8,x, algorithm="giac")

[Out]

integrate(fresnels(b*x)^2/x^8, x)