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

Optimal. Leaf size=230 \[ \frac{1}{48} \pi ^3 b^6 \text{Unintegrable}\left (\frac{S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{x},x\right )-\frac{1}{45} \sqrt{2} \pi ^3 b^6 \text{FresnelC}\left (\sqrt{2} b x\right )-\frac{7 \pi ^3 b^6 \text{FresnelC}\left (\sqrt{2} b x\right )}{144 \sqrt{2}}+\frac{\pi b^2 S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{24 x^4}+\frac{\pi ^2 b^4 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{48 x^2}-\frac{S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{6 x^6}+\frac{\pi b^3}{144 x^3}+\frac{67 \pi ^2 b^5 \sin \left (\pi b^2 x^2\right )}{1440 x}-\frac{b \sin \left (\pi b^2 x^2\right )}{60 x^5}-\frac{13 \pi b^3 \cos \left (\pi b^2 x^2\right )}{720 x^3} \]

[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*Unintegrable[(FresnelS[b*x]*Sin[(b^2*Pi*x^2)
/2])/x, x])/48

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Rubi [A]  time = 0.204738, 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 = {} \[ \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{x^7} \, dx \]

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*}

Mathematica [A]  time = 0.032583, size = 0, normalized size = 0. \[ \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{x^7} \, dx \]

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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Maple [A]  time = 0.065, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it FresnelS} \left ( bx \right ) }{{x}^{7}}\cos \left ({\frac{{b}^{2}\pi \,{x}^{2}}{2}} \right ) }\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (\frac{1}{2} \, \pi b^{2} x^{2}\right ){\rm fresnels}\left (b x\right )}{x^{7}}\,{d x} \end{align*}

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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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}

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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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos{\left (\frac{\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )}{x^{7}}\, dx \end{align*}

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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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (\frac{1}{2} \, \pi b^{2} x^{2}\right ){\rm fresnels}\left (b x\right )}{x^{7}}\,{d x} \end{align*}

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)