3.106 \(\int \frac {\cos (\frac {1}{2} b^2 \pi x^2) S(b x)}{x^7} \, dx\)

Optimal. Leaf size=231 \[ \frac {1}{48} \pi ^3 b^6 \text {Int}\left (\frac {S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{x},x\right )-\frac {1}{45} \sqrt {2} \pi ^3 b^6 C\left (\sqrt {2} b x\right )-\frac {7 \pi ^3 b^6 C\left (\sqrt {2} b x\right )}{144 \sqrt {2}}+\frac {\pi b^3}{144 x^3}-\frac {S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}+\frac {\pi b^2 S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{24 x^4}-\frac {b \sin \left (\pi b^2 x^2\right )}{60 x^5}+\frac {67 \pi ^2 b^5 \sin \left (\pi b^2 x^2\right )}{1440 x}+\frac {\pi ^2 b^4 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{48 x^2}-\frac {13 \pi b^3 \cos \left (\pi b^2 x^2\right )}{720 x^3} \]

[Out]

1/144*b^3*Pi/x^3-13/720*b^3*Pi*cos(b^2*Pi*x^2)/x^3-1/6*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/x^6+1/48*b^4*Pi^2*cos
(1/2*b^2*Pi*x^2)*FresnelS(b*x)/x^2+1/24*b^2*Pi*FresnelS(b*x)*sin(1/2*b^2*Pi*x^2)/x^4-1/60*b*sin(b^2*Pi*x^2)/x^
5+67/1440*b^5*Pi^2*sin(b^2*Pi*x^2)/x-67/1440*b^6*Pi^3*FresnelC(b*x*2^(1/2))*2^(1/2)+1/48*b^6*Pi^3*Unintegrable
(FresnelS(b*x)*sin(1/2*b^2*Pi*x^2)/x,x)

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Rubi [A]  time = 0.20, 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 = {} \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{x^7} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/x^7,x]

[Out]

(b^3*Pi)/(144*x^3) - (13*b^3*Pi*Cos[b^2*Pi*x^2])/(720*x^3) - (7*b^6*Pi^3*FresnelC[Sqrt[2]*b*x])/(144*Sqrt[2])
- (Sqrt[2]*b^6*Pi^3*FresnelC[Sqrt[2]*b*x])/45 - (Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(6*x^6) + (b^4*Pi^2*Cos[(b
^2*Pi*x^2)/2]*FresnelS[b*x])/(48*x^2) + (b^2*Pi*FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/(24*x^4) - (b*Sin[b^2*Pi*x^
2])/(60*x^5) + (67*b^5*Pi^2*Sin[b^2*Pi*x^2])/(1440*x) + (b^6*Pi^3*Defer[Int][(FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2
])/x, x])/48

Rubi steps

\begin {align*} \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{x^7} \, dx &=-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{6 x^6}+\frac {1}{12} b \int \frac {\sin \left (b^2 \pi x^2\right )}{x^6} \, dx-\frac {1}{6} \left (b^2 \pi \right ) \int \frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5} \, dx\\ &=\frac {b^3 \pi }{144 x^3}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{6 x^6}+\frac {b^2 \pi S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{24 x^4}-\frac {b \sin \left (b^2 \pi x^2\right )}{60 x^5}+\frac {1}{48} \left (b^3 \pi \right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^4} \, dx+\frac {1}{30} \left (b^3 \pi \right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^4} \, dx-\frac {1}{24} \left (b^4 \pi ^2\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{x^3} \, dx\\ &=\frac {b^3 \pi }{144 x^3}-\frac {13 b^3 \pi \cos \left (b^2 \pi x^2\right )}{720 x^3}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{6 x^6}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{48 x^2}+\frac {b^2 \pi S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{24 x^4}-\frac {b \sin \left (b^2 \pi x^2\right )}{60 x^5}-\frac {1}{96} \left (b^5 \pi ^2\right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2} \, dx-\frac {1}{72} \left (b^5 \pi ^2\right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2} \, dx-\frac {1}{45} \left (b^5 \pi ^2\right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2} \, dx+\frac {1}{48} \left (b^6 \pi ^3\right ) \int \frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x} \, dx\\ &=\frac {b^3 \pi }{144 x^3}-\frac {13 b^3 \pi \cos \left (b^2 \pi x^2\right )}{720 x^3}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{6 x^6}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{48 x^2}+\frac {b^2 \pi S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{24 x^4}-\frac {b \sin \left (b^2 \pi x^2\right )}{60 x^5}+\frac {67 b^5 \pi ^2 \sin \left (b^2 \pi x^2\right )}{1440 x}+\frac {1}{48} \left (b^6 \pi ^3\right ) \int \frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x} \, dx-\frac {1}{48} \left (b^7 \pi ^3\right ) \int \cos \left (b^2 \pi x^2\right ) \, dx-\frac {1}{36} \left (b^7 \pi ^3\right ) \int \cos \left (b^2 \pi x^2\right ) \, dx-\frac {1}{45} \left (2 b^7 \pi ^3\right ) \int \cos \left (b^2 \pi x^2\right ) \, dx\\ &=\frac {b^3 \pi }{144 x^3}-\frac {13 b^3 \pi \cos \left (b^2 \pi x^2\right )}{720 x^3}-\frac {7 b^6 \pi ^3 C\left (\sqrt {2} b x\right )}{144 \sqrt {2}}-\frac {1}{45} \sqrt {2} b^6 \pi ^3 C\left (\sqrt {2} b x\right )-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{6 x^6}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{48 x^2}+\frac {b^2 \pi S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{24 x^4}-\frac {b \sin \left (b^2 \pi x^2\right )}{60 x^5}+\frac {67 b^5 \pi ^2 \sin \left (b^2 \pi x^2\right )}{1440 x}+\frac {1}{48} \left (b^6 \pi ^3\right ) \int \frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x} \, dx\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{x^7} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/x^7,x]

[Out]

Integrate[(Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/x^7, x]

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fricas [A]  time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) {\rm fresnels}\left (b x\right )}{x^{7}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(1/2*b^2*pi*x^2)*fresnels(b*x)/x^7,x, algorithm="fricas")

[Out]

integral(cos(1/2*pi*b^2*x^2)*fresnels(b*x)/x^7, x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) {\rm fresnels}\left (b x\right )}{x^{7}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(1/2*b^2*pi*x^2)*fresnels(b*x)/x^7,x, algorithm="giac")

[Out]

integrate(cos(1/2*pi*b^2*x^2)*fresnels(b*x)/x^7, x)

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maple [A]  time = 0.03, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \mathrm {S}\left (b x \right )}{x^{7}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/x^7,x)

[Out]

int(cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/x^7,x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) {\rm fresnels}\left (b x\right )}{x^{7}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(1/2*b^2*pi*x^2)*fresnels(b*x)/x^7,x, algorithm="maxima")

[Out]

integrate(cos(1/2*pi*b^2*x^2)*fresnels(b*x)/x^7, x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\mathrm {FresnelS}\left (b\,x\right )\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right )}{x^7} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((FresnelS(b*x)*cos((Pi*b^2*x^2)/2))/x^7,x)

[Out]

int((FresnelS(b*x)*cos((Pi*b^2*x^2)/2))/x^7, x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )}{x^{7}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(1/2*b**2*pi*x**2)*fresnels(b*x)/x**7,x)

[Out]

Integral(cos(pi*b**2*x**2/2)*fresnels(b*x)/x**7, x)

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