3.44 \(\int \frac{S(b x)^2}{x^6} \, dx\)

Optimal. Leaf size=170 \[ -\frac{1}{20} \pi ^2 b^5 \text{Unintegrable}\left (\frac{S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{x},x\right )+\frac{7 \pi ^2 b^5 \text{FresnelC}\left (\sqrt{2} b x\right )}{60 \sqrt{2}}-\frac{b S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{10 x^4}-\frac{\pi b^3 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{20 x^2}-\frac{b^2}{60 x^3}-\frac{7 \pi b^4 \sin \left (\pi b^2 x^2\right )}{120 x}+\frac{b^2 \cos \left (\pi b^2 x^2\right )}{60 x^3}-\frac{S(b x)^2}{5 x^5} \]

[Out]

-b^2/(60*x^3) + (b^2*Cos[b^2*Pi*x^2])/(60*x^3) + (7*b^5*Pi^2*FresnelC[Sqrt[2]*b*x])/(60*Sqrt[2]) - (b^3*Pi*Cos
[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(20*x^2) - FresnelS[b*x]^2/(5*x^5) - (b*FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/(10
*x^4) - (7*b^4*Pi*Sin[b^2*Pi*x^2])/(120*x) - (b^5*Pi^2*Unintegrable[(FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/x, x])
/20

________________________________________________________________________________________

Rubi [A]  time = 0.153846, 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 = {} \[ \int \frac{S(b x)^2}{x^6} \, dx \]

Verification is Not applicable to the result.

[In]

Int[FresnelS[b*x]^2/x^6,x]

[Out]

-b^2/(60*x^3) + (b^2*Cos[b^2*Pi*x^2])/(60*x^3) + (7*b^5*Pi^2*FresnelC[Sqrt[2]*b*x])/(60*Sqrt[2]) - (b^3*Pi*Cos
[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(20*x^2) - FresnelS[b*x]^2/(5*x^5) - (b*FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/(10
*x^4) - (7*b^4*Pi*Sin[b^2*Pi*x^2])/(120*x) - (b^5*Pi^2*Defer[Int][(FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/x, x])/2
0

Rubi steps

\begin{align*} \int \frac{S(b x)^2}{x^6} \, dx &=-\frac{S(b x)^2}{5 x^5}+\frac{1}{5} (2 b) \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^5} \, dx\\ &=-\frac{b^2}{60 x^3}-\frac{S(b x)^2}{5 x^5}-\frac{b S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{10 x^4}-\frac{1}{20} b^2 \int \frac{\cos \left (b^2 \pi x^2\right )}{x^4} \, dx+\frac{1}{10} \left (b^3 \pi \right ) \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{x^3} \, dx\\ &=-\frac{b^2}{60 x^3}+\frac{b^2 \cos \left (b^2 \pi x^2\right )}{60 x^3}-\frac{b^3 \pi \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{20 x^2}-\frac{S(b x)^2}{5 x^5}-\frac{b S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{10 x^4}+\frac{1}{40} \left (b^4 \pi \right ) \int \frac{\sin \left (b^2 \pi x^2\right )}{x^2} \, dx+\frac{1}{30} \left (b^4 \pi \right ) \int \frac{\sin \left (b^2 \pi x^2\right )}{x^2} \, dx-\frac{1}{20} \left (b^5 \pi ^2\right ) \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x} \, dx\\ &=-\frac{b^2}{60 x^3}+\frac{b^2 \cos \left (b^2 \pi x^2\right )}{60 x^3}-\frac{b^3 \pi \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{20 x^2}-\frac{S(b x)^2}{5 x^5}-\frac{b S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{10 x^4}-\frac{7 b^4 \pi \sin \left (b^2 \pi x^2\right )}{120 x}-\frac{1}{20} \left (b^5 \pi ^2\right ) \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x} \, dx+\frac{1}{20} \left (b^6 \pi ^2\right ) \int \cos \left (b^2 \pi x^2\right ) \, dx+\frac{1}{15} \left (b^6 \pi ^2\right ) \int \cos \left (b^2 \pi x^2\right ) \, dx\\ &=-\frac{b^2}{60 x^3}+\frac{b^2 \cos \left (b^2 \pi x^2\right )}{60 x^3}+\frac{7 b^5 \pi ^2 C\left (\sqrt{2} b x\right )}{60 \sqrt{2}}-\frac{b^3 \pi \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{20 x^2}-\frac{S(b x)^2}{5 x^5}-\frac{b S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{10 x^4}-\frac{7 b^4 \pi \sin \left (b^2 \pi x^2\right )}{120 x}-\frac{1}{20} \left (b^5 \pi ^2\right ) \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x} \, dx\\ \end{align*}

Mathematica [A]  time = 0.0267984, size = 0, normalized size = 0. \[ \int \frac{S(b x)^2}{x^6} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[FresnelS[b*x]^2/x^6,x]

[Out]

Integrate[FresnelS[b*x]^2/x^6, x]

________________________________________________________________________________________

Maple [A]  time = 0.056, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\it FresnelS} \left ( bx \right ) \right ) ^{2}}{{x}^{6}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(FresnelS(b*x)^2/x^6,x)

[Out]

int(FresnelS(b*x)^2/x^6,x)

________________________________________________________________________________________

Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm fresnels}\left (b x\right )^{2}}{x^{6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnels(b*x)^2/x^6,x, algorithm="maxima")

[Out]

integrate(fresnels(b*x)^2/x^6, x)

________________________________________________________________________________________

Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\rm fresnels}\left (b x\right )^{2}}{x^{6}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnels(b*x)^2/x^6,x, algorithm="fricas")

[Out]

integral(fresnels(b*x)^2/x^6, x)

________________________________________________________________________________________

Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{S^{2}\left (b x\right )}{x^{6}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnels(b*x)**2/x**6,x)

[Out]

Integral(fresnels(b*x)**2/x**6, x)

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm fresnels}\left (b x\right )^{2}}{x^{6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnels(b*x)^2/x^6,x, algorithm="giac")

[Out]

integrate(fresnels(b*x)^2/x^6, x)