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

Optimal. Leaf size=271 \[ \frac {1}{384} \pi ^4 b^8 \text {Int}\left (\frac {S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{x},x\right )+\frac {853 \pi ^4 b^8 S\left (\sqrt {2} b x\right )}{40320 \sqrt {2}}-\frac {\pi ^3 b^7}{768 x}+\frac {\pi b^3}{480 x^5}-\frac {S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{8 x^8}+\frac {\pi b^2 S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{48 x^6}-\frac {b \sin \left (\pi b^2 x^2\right )}{112 x^7}+\frac {853 \pi ^3 b^7 \cos \left (\pi b^2 x^2\right )}{80640 x}-\frac {\pi ^3 b^6 S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{384 x^2}+\frac {187 \pi ^2 b^5 \sin \left (\pi b^2 x^2\right )}{40320 x^3}+\frac {\pi ^2 b^4 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{192 x^4}-\frac {19 \pi b^3 \cos \left (\pi b^2 x^2\right )}{3360 x^5} \]

[Out]

1/480*b^3*Pi/x^5-1/768*b^7*Pi^3/x-19/3360*b^3*Pi*cos(b^2*Pi*x^2)/x^5+853/80640*b^7*Pi^3*cos(b^2*Pi*x^2)/x-1/8*
cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/x^8+1/192*b^4*Pi^2*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/x^4+1/48*b^2*Pi*Fresnel
S(b*x)*sin(1/2*b^2*Pi*x^2)/x^6-1/384*b^6*Pi^3*FresnelS(b*x)*sin(1/2*b^2*Pi*x^2)/x^2-1/112*b*sin(b^2*Pi*x^2)/x^
7+187/40320*b^5*Pi^2*sin(b^2*Pi*x^2)/x^3+853/80640*b^8*Pi^4*FresnelS(b*x*2^(1/2))*2^(1/2)+1/384*b^8*Pi^4*Unint
egrable(cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/x,x)

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Rubi [A]  time = 0.31, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{x^9} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(b^3*Pi)/(480*x^5) - (b^7*Pi^3)/(768*x) - (19*b^3*Pi*Cos[b^2*Pi*x^2])/(3360*x^5) + (853*b^7*Pi^3*Cos[b^2*Pi*x^
2])/(80640*x) - (Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(8*x^8) + (b^4*Pi^2*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(19
2*x^4) + (853*b^8*Pi^4*FresnelS[Sqrt[2]*b*x])/(40320*Sqrt[2]) + (b^2*Pi*FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/(48
*x^6) - (b^6*Pi^3*FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/(384*x^2) - (b*Sin[b^2*Pi*x^2])/(112*x^7) + (187*b^5*Pi^2
*Sin[b^2*Pi*x^2])/(40320*x^3) + (b^8*Pi^4*Defer[Int][(Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/x, x])/384

Rubi steps

\begin {align*} \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{x^9} \, dx &=-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{8 x^8}+\frac {1}{16} b \int \frac {\sin \left (b^2 \pi x^2\right )}{x^8} \, dx-\frac {1}{8} \left (b^2 \pi \right ) \int \frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^7} \, dx\\ &=\frac {b^3 \pi }{480 x^5}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{8 x^8}+\frac {b^2 \pi S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{48 x^6}-\frac {b \sin \left (b^2 \pi x^2\right )}{112 x^7}+\frac {1}{96} \left (b^3 \pi \right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^6} \, dx+\frac {1}{56} \left (b^3 \pi \right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^6} \, dx-\frac {1}{48} \left (b^4 \pi ^2\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{x^5} \, dx\\ &=\frac {b^3 \pi }{480 x^5}-\frac {19 b^3 \pi \cos \left (b^2 \pi x^2\right )}{3360 x^5}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{8 x^8}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{192 x^4}+\frac {b^2 \pi S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{48 x^6}-\frac {b \sin \left (b^2 \pi x^2\right )}{112 x^7}-\frac {1}{384} \left (b^5 \pi ^2\right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^4} \, dx-\frac {1}{240} \left (b^5 \pi ^2\right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^4} \, dx-\frac {1}{140} \left (b^5 \pi ^2\right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^4} \, dx+\frac {1}{192} \left (b^6 \pi ^3\right ) \int \frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^3} \, dx\\ &=\frac {b^3 \pi }{480 x^5}-\frac {b^7 \pi ^3}{768 x}-\frac {19 b^3 \pi \cos \left (b^2 \pi x^2\right )}{3360 x^5}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{8 x^8}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{192 x^4}+\frac {b^2 \pi S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{48 x^6}-\frac {b^6 \pi ^3 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{384 x^2}-\frac {b \sin \left (b^2 \pi x^2\right )}{112 x^7}+\frac {187 b^5 \pi ^2 \sin \left (b^2 \pi x^2\right )}{40320 x^3}-\frac {1}{768} \left (b^7 \pi ^3\right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^2} \, dx-\frac {1}{576} \left (b^7 \pi ^3\right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^2} \, dx-\frac {1}{360} \left (b^7 \pi ^3\right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^2} \, dx-\frac {1}{210} \left (b^7 \pi ^3\right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^2} \, dx+\frac {1}{384} \left (b^8 \pi ^4\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{x} \, dx\\ &=\frac {b^3 \pi }{480 x^5}-\frac {b^7 \pi ^3}{768 x}-\frac {19 b^3 \pi \cos \left (b^2 \pi x^2\right )}{3360 x^5}+\frac {853 b^7 \pi ^3 \cos \left (b^2 \pi x^2\right )}{80640 x}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{8 x^8}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{192 x^4}+\frac {b^2 \pi S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{48 x^6}-\frac {b^6 \pi ^3 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{384 x^2}-\frac {b \sin \left (b^2 \pi x^2\right )}{112 x^7}+\frac {187 b^5 \pi ^2 \sin \left (b^2 \pi x^2\right )}{40320 x^3}+\frac {1}{384} \left (b^8 \pi ^4\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{x} \, dx+\frac {1}{384} \left (b^9 \pi ^4\right ) \int \sin \left (b^2 \pi x^2\right ) \, dx+\frac {1}{288} \left (b^9 \pi ^4\right ) \int \sin \left (b^2 \pi x^2\right ) \, dx+\frac {1}{180} \left (b^9 \pi ^4\right ) \int \sin \left (b^2 \pi x^2\right ) \, dx+\frac {1}{105} \left (b^9 \pi ^4\right ) \int \sin \left (b^2 \pi x^2\right ) \, dx\\ &=\frac {b^3 \pi }{480 x^5}-\frac {b^7 \pi ^3}{768 x}-\frac {19 b^3 \pi \cos \left (b^2 \pi x^2\right )}{3360 x^5}+\frac {853 b^7 \pi ^3 \cos \left (b^2 \pi x^2\right )}{80640 x}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{8 x^8}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{192 x^4}+\frac {853 b^8 \pi ^4 S\left (\sqrt {2} b x\right )}{40320 \sqrt {2}}+\frac {b^2 \pi S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{48 x^6}-\frac {b^6 \pi ^3 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{384 x^2}-\frac {b \sin \left (b^2 \pi x^2\right )}{112 x^7}+\frac {187 b^5 \pi ^2 \sin \left (b^2 \pi x^2\right )}{40320 x^3}+\frac {1}{384} \left (b^8 \pi ^4\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{x} \, dx\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{x^9} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) {\rm fresnels}\left (b x\right )}{x^{9}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) {\rm fresnels}\left (b x\right )}{x^{9}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(1/2*b^2*pi*x^2)*fresnels(b*x)/x^9,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.03, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \mathrm {S}\left (b x \right )}{x^{9}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/x^9,x)

[Out]

int(cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/x^9,x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) {\rm fresnels}\left (b x\right )}{x^{9}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(1/2*b^2*pi*x^2)*fresnels(b*x)/x^9,x, algorithm="maxima")

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\mathrm {FresnelS}\left (b\,x\right )\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right )}{x^9} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((FresnelS(b*x)*cos((Pi*b^2*x^2)/2))/x^9,x)

[Out]

int((FresnelS(b*x)*cos((Pi*b^2*x^2)/2))/x^9, x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )}{x^{9}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cos(pi*b**2*x**2/2)*fresnels(b*x)/x**9, x)

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