3.48 \(\int \frac{S(b x)^2}{x^{10}} \, dx\)

Optimal. Leaf size=285 \[ \frac{\pi ^4 b^9 \text{Unintegrable}\left (\frac{S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{x},x\right )}{1728}-\frac{853 \pi ^4 b^9 \text{FresnelC}\left (\sqrt{2} b x\right )}{181440 \sqrt{2}}+\frac{\pi ^2 b^5 S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{864 x^4}-\frac{b S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{36 x^8}+\frac{\pi ^3 b^7 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{1728 x^2}-\frac{\pi b^3 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{216 x^6}+\frac{\pi ^2 b^6}{5184 x^3}-\frac{b^2}{504 x^7}+\frac{853 \pi ^3 b^8 \sin \left (\pi b^2 x^2\right )}{362880 x}-\frac{19 \pi b^4 \sin \left (\pi b^2 x^2\right )}{15120 x^5}-\frac{187 \pi ^2 b^6 \cos \left (\pi b^2 x^2\right )}{181440 x^3}+\frac{b^2 \cos \left (\pi b^2 x^2\right )}{504 x^7}-\frac{S(b x)^2}{9 x^9} \]

[Out]

-b^2/(504*x^7) + (b^6*Pi^2)/(5184*x^3) + (b^2*Cos[b^2*Pi*x^2])/(504*x^7) - (187*b^6*Pi^2*Cos[b^2*Pi*x^2])/(181
440*x^3) - (853*b^9*Pi^4*FresnelC[Sqrt[2]*b*x])/(181440*Sqrt[2]) - (b^3*Pi*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/
(216*x^6) + (b^7*Pi^3*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(1728*x^2) - FresnelS[b*x]^2/(9*x^9) - (b*FresnelS[b*
x]*Sin[(b^2*Pi*x^2)/2])/(36*x^8) + (b^5*Pi^2*FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/(864*x^4) - (19*b^4*Pi*Sin[b^2
*Pi*x^2])/(15120*x^5) + (853*b^8*Pi^3*Sin[b^2*Pi*x^2])/(362880*x) + (b^9*Pi^4*Unintegrable[(FresnelS[b*x]*Sin[
(b^2*Pi*x^2)/2])/x, x])/1728

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Rubi [A]  time = 0.358317, 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 = {} \[ \int \frac{S(b x)^2}{x^{10}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[FresnelS[b*x]^2/x^10,x]

[Out]

-b^2/(504*x^7) + (b^6*Pi^2)/(5184*x^3) + (b^2*Cos[b^2*Pi*x^2])/(504*x^7) - (187*b^6*Pi^2*Cos[b^2*Pi*x^2])/(181
440*x^3) - (853*b^9*Pi^4*FresnelC[Sqrt[2]*b*x])/(181440*Sqrt[2]) - (b^3*Pi*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/
(216*x^6) + (b^7*Pi^3*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(1728*x^2) - FresnelS[b*x]^2/(9*x^9) - (b*FresnelS[b*
x]*Sin[(b^2*Pi*x^2)/2])/(36*x^8) + (b^5*Pi^2*FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/(864*x^4) - (19*b^4*Pi*Sin[b^2
*Pi*x^2])/(15120*x^5) + (853*b^8*Pi^3*Sin[b^2*Pi*x^2])/(362880*x) + (b^9*Pi^4*Defer[Int][(FresnelS[b*x]*Sin[(b
^2*Pi*x^2)/2])/x, x])/1728

Rubi steps

\begin{align*} \int \frac{S(b x)^2}{x^{10}} \, dx &=-\frac{S(b x)^2}{9 x^9}+\frac{1}{9} (2 b) \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^9} \, dx\\ &=-\frac{b^2}{504 x^7}-\frac{S(b x)^2}{9 x^9}-\frac{b S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{36 x^8}-\frac{1}{72} b^2 \int \frac{\cos \left (b^2 \pi x^2\right )}{x^8} \, dx+\frac{1}{36} \left (b^3 \pi \right ) \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{x^7} \, dx\\ &=-\frac{b^2}{504 x^7}+\frac{b^2 \cos \left (b^2 \pi x^2\right )}{504 x^7}-\frac{b^3 \pi \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{216 x^6}-\frac{S(b x)^2}{9 x^9}-\frac{b S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{36 x^8}+\frac{1}{432} \left (b^4 \pi \right ) \int \frac{\sin \left (b^2 \pi x^2\right )}{x^6} \, dx+\frac{1}{252} \left (b^4 \pi \right ) \int \frac{\sin \left (b^2 \pi x^2\right )}{x^6} \, dx-\frac{1}{216} \left (b^5 \pi ^2\right ) \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^5} \, dx\\ &=-\frac{b^2}{504 x^7}+\frac{b^6 \pi ^2}{5184 x^3}+\frac{b^2 \cos \left (b^2 \pi x^2\right )}{504 x^7}-\frac{b^3 \pi \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{216 x^6}-\frac{S(b x)^2}{9 x^9}-\frac{b S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{36 x^8}+\frac{b^5 \pi ^2 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{864 x^4}-\frac{19 b^4 \pi \sin \left (b^2 \pi x^2\right )}{15120 x^5}+\frac{\left (b^6 \pi ^2\right ) \int \frac{\cos \left (b^2 \pi x^2\right )}{x^4} \, dx}{1728}+\frac{\left (b^6 \pi ^2\right ) \int \frac{\cos \left (b^2 \pi x^2\right )}{x^4} \, dx}{1080}+\frac{1}{630} \left (b^6 \pi ^2\right ) \int \frac{\cos \left (b^2 \pi x^2\right )}{x^4} \, dx-\frac{1}{864} \left (b^7 \pi ^3\right ) \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{x^3} \, dx\\ &=-\frac{b^2}{504 x^7}+\frac{b^6 \pi ^2}{5184 x^3}+\frac{b^2 \cos \left (b^2 \pi x^2\right )}{504 x^7}-\frac{187 b^6 \pi ^2 \cos \left (b^2 \pi x^2\right )}{181440 x^3}-\frac{b^3 \pi \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{216 x^6}+\frac{b^7 \pi ^3 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{1728 x^2}-\frac{S(b x)^2}{9 x^9}-\frac{b S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{36 x^8}+\frac{b^5 \pi ^2 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{864 x^4}-\frac{19 b^4 \pi \sin \left (b^2 \pi x^2\right )}{15120 x^5}-\frac{\left (b^8 \pi ^3\right ) \int \frac{\sin \left (b^2 \pi x^2\right )}{x^2} \, dx}{3456}-\frac{\left (b^8 \pi ^3\right ) \int \frac{\sin \left (b^2 \pi x^2\right )}{x^2} \, dx}{2592}-\frac{\left (b^8 \pi ^3\right ) \int \frac{\sin \left (b^2 \pi x^2\right )}{x^2} \, dx}{1620}-\frac{1}{945} \left (b^8 \pi ^3\right ) \int \frac{\sin \left (b^2 \pi x^2\right )}{x^2} \, dx+\frac{\left (b^9 \pi ^4\right ) \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x} \, dx}{1728}\\ &=-\frac{b^2}{504 x^7}+\frac{b^6 \pi ^2}{5184 x^3}+\frac{b^2 \cos \left (b^2 \pi x^2\right )}{504 x^7}-\frac{187 b^6 \pi ^2 \cos \left (b^2 \pi x^2\right )}{181440 x^3}-\frac{b^3 \pi \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{216 x^6}+\frac{b^7 \pi ^3 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{1728 x^2}-\frac{S(b x)^2}{9 x^9}-\frac{b S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{36 x^8}+\frac{b^5 \pi ^2 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{864 x^4}-\frac{19 b^4 \pi \sin \left (b^2 \pi x^2\right )}{15120 x^5}+\frac{853 b^8 \pi ^3 \sin \left (b^2 \pi x^2\right )}{362880 x}+\frac{\left (b^9 \pi ^4\right ) \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x} \, dx}{1728}-\frac{\left (b^{10} \pi ^4\right ) \int \cos \left (b^2 \pi x^2\right ) \, dx}{1728}-\frac{\left (b^{10} \pi ^4\right ) \int \cos \left (b^2 \pi x^2\right ) \, dx}{1296}-\frac{1}{810} \left (b^{10} \pi ^4\right ) \int \cos \left (b^2 \pi x^2\right ) \, dx-\frac{1}{945} \left (2 b^{10} \pi ^4\right ) \int \cos \left (b^2 \pi x^2\right ) \, dx\\ &=-\frac{b^2}{504 x^7}+\frac{b^6 \pi ^2}{5184 x^3}+\frac{b^2 \cos \left (b^2 \pi x^2\right )}{504 x^7}-\frac{187 b^6 \pi ^2 \cos \left (b^2 \pi x^2\right )}{181440 x^3}-\frac{67 b^9 \pi ^4 C\left (\sqrt{2} b x\right )}{25920 \sqrt{2}}-\frac{1}{945} \sqrt{2} b^9 \pi ^4 C\left (\sqrt{2} b x\right )-\frac{b^3 \pi \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{216 x^6}+\frac{b^7 \pi ^3 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{1728 x^2}-\frac{S(b x)^2}{9 x^9}-\frac{b S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{36 x^8}+\frac{b^5 \pi ^2 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{864 x^4}-\frac{19 b^4 \pi \sin \left (b^2 \pi x^2\right )}{15120 x^5}+\frac{853 b^8 \pi ^3 \sin \left (b^2 \pi x^2\right )}{362880 x}+\frac{\left (b^9 \pi ^4\right ) \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x} \, dx}{1728}\\ \end{align*}

Mathematica [A]  time = 0.0271738, size = 0, normalized size = 0. \[ \int \frac{S(b x)^2}{x^{10}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[FresnelS[b*x]^2/x^10,x]

[Out]

Integrate[FresnelS[b*x]^2/x^10, x]

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Maple [A]  time = 0.056, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\it FresnelS} \left ( bx \right ) \right ) ^{2}}{{x}^{10}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(FresnelS(b*x)^2/x^10,x)

[Out]

int(FresnelS(b*x)^2/x^10,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^{10}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnels(b*x)^2/x^10,x, algorithm="maxima")

[Out]

integrate(fresnels(b*x)^2/x^10, 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^{10}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnels(b*x)^2/x^10,x, algorithm="fricas")

[Out]

integral(fresnels(b*x)^2/x^10, 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^{10}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnels(b*x)**2/x**10,x)

[Out]

Integral(fresnels(b*x)**2/x**10, 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^{10}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnels(b*x)^2/x^10,x, algorithm="giac")

[Out]

integrate(fresnels(b*x)^2/x^10, x)