3.88 \(\int \frac{S(b x) \sin (\frac{1}{2} b^2 \pi x^2)}{x^9} \, dx\)

Optimal. Leaf size=267 \[ \frac{1}{384} \pi ^4 b^8 \text{Unintegrable}\left (\frac{S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{x},x\right )-\frac{853 \pi ^4 b^8 \text{FresnelC}\left (\sqrt{2} b x\right )}{40320 \sqrt{2}}+\frac{\pi ^2 b^4 S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{192 x^4}-\frac{S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{8 x^8}+\frac{\pi ^3 b^6 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{384 x^2}-\frac{\pi b^2 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{48 x^6}+\frac{\pi ^2 b^5}{1152 x^3}+\frac{853 \pi ^3 b^7 \sin \left (\pi b^2 x^2\right )}{80640 x}-\frac{19 \pi b^3 \sin \left (\pi b^2 x^2\right )}{3360 x^5}-\frac{187 \pi ^2 b^5 \cos \left (\pi b^2 x^2\right )}{40320 x^3}+\frac{b \cos \left (\pi b^2 x^2\right )}{112 x^7}-\frac{b}{112 x^7} \]

[Out]

-b/(112*x^7) + (b^5*Pi^2)/(1152*x^3) + (b*Cos[b^2*Pi*x^2])/(112*x^7) - (187*b^5*Pi^2*Cos[b^2*Pi*x^2])/(40320*x
^3) - (853*b^8*Pi^4*FresnelC[Sqrt[2]*b*x])/(40320*Sqrt[2]) - (b^2*Pi*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(48*x^
6) + (b^6*Pi^3*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(384*x^2) - (FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/(8*x^8) + (b
^4*Pi^2*FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/(192*x^4) - (19*b^3*Pi*Sin[b^2*Pi*x^2])/(3360*x^5) + (853*b^7*Pi^3*
Sin[b^2*Pi*x^2])/(80640*x) + (b^8*Pi^4*Unintegrable[(FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/x, x])/384

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Rubi [A]  time = 0.306247, 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 = {} \[ \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^9} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/x^9,x]

[Out]

-b/(112*x^7) + (b^5*Pi^2)/(1152*x^3) + (b*Cos[b^2*Pi*x^2])/(112*x^7) - (187*b^5*Pi^2*Cos[b^2*Pi*x^2])/(40320*x
^3) - (853*b^8*Pi^4*FresnelC[Sqrt[2]*b*x])/(40320*Sqrt[2]) - (b^2*Pi*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(48*x^
6) + (b^6*Pi^3*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(384*x^2) - (FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/(8*x^8) + (b
^4*Pi^2*FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/(192*x^4) - (19*b^3*Pi*Sin[b^2*Pi*x^2])/(3360*x^5) + (853*b^7*Pi^3*
Sin[b^2*Pi*x^2])/(80640*x) + (b^8*Pi^4*Defer[Int][(FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/x, x])/384

Rubi steps

\begin{align*} \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^9} \, dx &=-\frac{b}{112 x^7}-\frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{8 x^8}-\frac{1}{16} b \int \frac{\cos \left (b^2 \pi x^2\right )}{x^8} \, dx+\frac{1}{8} \left (b^2 \pi \right ) \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{x^7} \, dx\\ &=-\frac{b}{112 x^7}+\frac{b \cos \left (b^2 \pi x^2\right )}{112 x^7}-\frac{b^2 \pi \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{48 x^6}-\frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{8 x^8}+\frac{1}{96} \left (b^3 \pi \right ) \int \frac{\sin \left (b^2 \pi x^2\right )}{x^6} \, dx+\frac{1}{56} \left (b^3 \pi \right ) \int \frac{\sin \left (b^2 \pi x^2\right )}{x^6} \, dx-\frac{1}{48} \left (b^4 \pi ^2\right ) \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^5} \, dx\\ &=-\frac{b}{112 x^7}+\frac{b^5 \pi ^2}{1152 x^3}+\frac{b \cos \left (b^2 \pi x^2\right )}{112 x^7}-\frac{b^2 \pi \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{48 x^6}-\frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{8 x^8}+\frac{b^4 \pi ^2 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{192 x^4}-\frac{19 b^3 \pi \sin \left (b^2 \pi x^2\right )}{3360 x^5}+\frac{1}{384} \left (b^5 \pi ^2\right ) \int \frac{\cos \left (b^2 \pi x^2\right )}{x^4} \, dx+\frac{1}{240} \left (b^5 \pi ^2\right ) \int \frac{\cos \left (b^2 \pi x^2\right )}{x^4} \, dx+\frac{1}{140} \left (b^5 \pi ^2\right ) \int \frac{\cos \left (b^2 \pi x^2\right )}{x^4} \, dx-\frac{1}{192} \left (b^6 \pi ^3\right ) \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{x^3} \, dx\\ &=-\frac{b}{112 x^7}+\frac{b^5 \pi ^2}{1152 x^3}+\frac{b \cos \left (b^2 \pi x^2\right )}{112 x^7}-\frac{187 b^5 \pi ^2 \cos \left (b^2 \pi x^2\right )}{40320 x^3}-\frac{b^2 \pi \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{48 x^6}+\frac{b^6 \pi ^3 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{384 x^2}-\frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{8 x^8}+\frac{b^4 \pi ^2 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{192 x^4}-\frac{19 b^3 \pi \sin \left (b^2 \pi x^2\right )}{3360 x^5}-\frac{1}{768} \left (b^7 \pi ^3\right ) \int \frac{\sin \left (b^2 \pi x^2\right )}{x^2} \, dx-\frac{1}{576} \left (b^7 \pi ^3\right ) \int \frac{\sin \left (b^2 \pi x^2\right )}{x^2} \, dx-\frac{1}{360} \left (b^7 \pi ^3\right ) \int \frac{\sin \left (b^2 \pi x^2\right )}{x^2} \, dx-\frac{1}{210} \left (b^7 \pi ^3\right ) \int \frac{\sin \left (b^2 \pi x^2\right )}{x^2} \, dx+\frac{1}{384} \left (b^8 \pi ^4\right ) \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x} \, dx\\ &=-\frac{b}{112 x^7}+\frac{b^5 \pi ^2}{1152 x^3}+\frac{b \cos \left (b^2 \pi x^2\right )}{112 x^7}-\frac{187 b^5 \pi ^2 \cos \left (b^2 \pi x^2\right )}{40320 x^3}-\frac{b^2 \pi \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{48 x^6}+\frac{b^6 \pi ^3 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{384 x^2}-\frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{8 x^8}+\frac{b^4 \pi ^2 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{192 x^4}-\frac{19 b^3 \pi \sin \left (b^2 \pi x^2\right )}{3360 x^5}+\frac{853 b^7 \pi ^3 \sin \left (b^2 \pi x^2\right )}{80640 x}+\frac{1}{384} \left (b^8 \pi ^4\right ) \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x} \, dx-\frac{1}{384} \left (b^9 \pi ^4\right ) \int \cos \left (b^2 \pi x^2\right ) \, dx-\frac{1}{288} \left (b^9 \pi ^4\right ) \int \cos \left (b^2 \pi x^2\right ) \, dx-\frac{1}{180} \left (b^9 \pi ^4\right ) \int \cos \left (b^2 \pi x^2\right ) \, dx-\frac{1}{105} \left (b^9 \pi ^4\right ) \int \cos \left (b^2 \pi x^2\right ) \, dx\\ &=-\frac{b}{112 x^7}+\frac{b^5 \pi ^2}{1152 x^3}+\frac{b \cos \left (b^2 \pi x^2\right )}{112 x^7}-\frac{187 b^5 \pi ^2 \cos \left (b^2 \pi x^2\right )}{40320 x^3}-\frac{853 b^8 \pi ^4 C\left (\sqrt{2} b x\right )}{40320 \sqrt{2}}-\frac{b^2 \pi \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{48 x^6}+\frac{b^6 \pi ^3 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{384 x^2}-\frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{8 x^8}+\frac{b^4 \pi ^2 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{192 x^4}-\frac{19 b^3 \pi \sin \left (b^2 \pi x^2\right )}{3360 x^5}+\frac{853 b^7 \pi ^3 \sin \left (b^2 \pi x^2\right )}{80640 x}+\frac{1}{384} \left (b^8 \pi ^4\right ) \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x} \, dx\\ \end{align*}

Mathematica [A]  time = 0.0316223, size = 0, normalized size = 0. \[ \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^9} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/x^9,x]

[Out]

Integrate[(FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/x^9, x]

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Maple [A]  time = 0.062, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it FresnelS} \left ( bx \right ) }{{x}^{9}}\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^9,x)

[Out]

int(FresnelS(b*x)*sin(1/2*b^2*Pi*x^2)/x^9,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^{9}}\,{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^9,x, algorithm="maxima")

[Out]

integrate(fresnels(b*x)*sin(1/2*pi*b^2*x^2)/x^9, 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^{9}}, 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^9,x, algorithm="fricas")

[Out]

integral(fresnels(b*x)*sin(1/2*pi*b^2*x^2)/x^9, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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**9,x)

[Out]

Timed out

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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^{9}}\,{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^9,x, algorithm="giac")

[Out]

integrate(fresnels(b*x)*sin(1/2*pi*b^2*x^2)/x^9, x)