3.107 \(\int \frac{\cos (\frac{1}{2} b^2 \pi x^2) S(b x)}{x^8} \, dx\)

Optimal. Leaf size=201 \[ \frac{1}{105} \pi ^3 b^6 \text{Unintegrable}\left (\frac{S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{x^2},x\right )-\frac{1}{84} \pi ^3 b^7 \text{CosIntegral}\left (\pi b^2 x^2\right )+\frac{\pi b^2 S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{35 x^5}+\frac{\pi ^2 b^4 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{105 x^3}-\frac{S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{7 x^7}+\frac{\pi b^3}{280 x^4}+\frac{\pi ^2 b^5 \sin \left (\pi b^2 x^2\right )}{84 x^2}-\frac{b \sin \left (\pi b^2 x^2\right )}{84 x^6}-\frac{\pi b^3 \cos \left (\pi b^2 x^2\right )}{105 x^4} \]

[Out]

(b^3*Pi)/(280*x^4) - (b^3*Pi*Cos[b^2*Pi*x^2])/(105*x^4) - (b^7*Pi^3*CosIntegral[b^2*Pi*x^2])/84 - (Cos[(b^2*Pi
*x^2)/2]*FresnelS[b*x])/(7*x^7) + (b^4*Pi^2*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(105*x^3) + (b^2*Pi*FresnelS[b*
x]*Sin[(b^2*Pi*x^2)/2])/(35*x^5) - (b*Sin[b^2*Pi*x^2])/(84*x^6) + (b^5*Pi^2*Sin[b^2*Pi*x^2])/(84*x^2) + (b^6*P
i^3*Unintegrable[(FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/x^2, x])/105

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Rubi [A]  time = 0.319427, 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 = {} \[ \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{x^8} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/x^8,x]

[Out]

(b^3*Pi)/(280*x^4) - (b^3*Pi*Cos[b^2*Pi*x^2])/(105*x^4) - (b^7*Pi^3*CosIntegral[b^2*Pi*x^2])/84 - (Cos[(b^2*Pi
*x^2)/2]*FresnelS[b*x])/(7*x^7) + (b^4*Pi^2*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(105*x^3) + (b^2*Pi*FresnelS[b*
x]*Sin[(b^2*Pi*x^2)/2])/(35*x^5) - (b*Sin[b^2*Pi*x^2])/(84*x^6) + (b^5*Pi^2*Sin[b^2*Pi*x^2])/(84*x^2) + (b^6*P
i^3*Defer[Int][(FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/x^2, x])/105

Rubi steps

\begin{align*} \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{x^8} \, dx &=-\frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{7 x^7}+\frac{1}{14} b \int \frac{\sin \left (b^2 \pi x^2\right )}{x^7} \, dx-\frac{1}{7} \left (b^2 \pi \right ) \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^6} \, dx\\ &=\frac{b^3 \pi }{280 x^4}-\frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{7 x^7}+\frac{b^2 \pi S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{35 x^5}+\frac{1}{28} b \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x^4} \, dx,x,x^2\right )+\frac{1}{70} \left (b^3 \pi \right ) \int \frac{\cos \left (b^2 \pi x^2\right )}{x^5} \, dx-\frac{1}{35} \left (b^4 \pi ^2\right ) \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{x^4} \, dx\\ &=\frac{b^3 \pi }{280 x^4}-\frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{7 x^7}+\frac{b^4 \pi ^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{105 x^3}+\frac{b^2 \pi S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{35 x^5}-\frac{b \sin \left (b^2 \pi x^2\right )}{84 x^6}+\frac{1}{140} \left (b^3 \pi \right ) \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )+\frac{1}{84} \left (b^3 \pi \right ) \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )-\frac{1}{210} \left (b^5 \pi ^2\right ) \int \frac{\sin \left (b^2 \pi x^2\right )}{x^3} \, dx+\frac{1}{105} \left (b^6 \pi ^3\right ) \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^2} \, dx\\ &=\frac{b^3 \pi }{280 x^4}-\frac{b^3 \pi \cos \left (b^2 \pi x^2\right )}{105 x^4}-\frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{7 x^7}+\frac{b^4 \pi ^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{105 x^3}+\frac{b^2 \pi S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{35 x^5}-\frac{b \sin \left (b^2 \pi x^2\right )}{84 x^6}-\frac{1}{420} \left (b^5 \pi ^2\right ) \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )-\frac{1}{280} \left (b^5 \pi ^2\right ) \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )-\frac{1}{168} \left (b^5 \pi ^2\right ) \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )+\frac{1}{105} \left (b^6 \pi ^3\right ) \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^2} \, dx\\ &=\frac{b^3 \pi }{280 x^4}-\frac{b^3 \pi \cos \left (b^2 \pi x^2\right )}{105 x^4}-\frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{7 x^7}+\frac{b^4 \pi ^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{105 x^3}+\frac{b^2 \pi S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{35 x^5}-\frac{b \sin \left (b^2 \pi x^2\right )}{84 x^6}+\frac{b^5 \pi ^2 \sin \left (b^2 \pi x^2\right )}{84 x^2}+\frac{1}{105} \left (b^6 \pi ^3\right ) \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^2} \, dx-\frac{1}{420} \left (b^7 \pi ^3\right ) \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )-\frac{1}{280} \left (b^7 \pi ^3\right ) \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )-\frac{1}{168} \left (b^7 \pi ^3\right ) \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )\\ &=\frac{b^3 \pi }{280 x^4}-\frac{b^3 \pi \cos \left (b^2 \pi x^2\right )}{105 x^4}-\frac{1}{84} b^7 \pi ^3 \text{Ci}\left (b^2 \pi x^2\right )-\frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{7 x^7}+\frac{b^4 \pi ^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{105 x^3}+\frac{b^2 \pi S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{35 x^5}-\frac{b \sin \left (b^2 \pi x^2\right )}{84 x^6}+\frac{b^5 \pi ^2 \sin \left (b^2 \pi x^2\right )}{84 x^2}+\frac{1}{105} \left (b^6 \pi ^3\right ) \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^2} \, dx\\ \end{align*}

Mathematica [A]  time = 0.0334512, size = 0, normalized size = 0. \[ \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{x^8} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/x^8,x]

[Out]

Integrate[(Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/x^8, x]

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Maple [A]  time = 0.063, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it FresnelS} \left ( bx \right ) }{{x}^{8}}\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^8,x)

[Out]

int(cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/x^8,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^{8}}\,{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^8,x, algorithm="maxima")

[Out]

integrate(cos(1/2*pi*b^2*x^2)*fresnels(b*x)/x^8, 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^{8}}, 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^8,x, algorithm="fricas")

[Out]

integral(cos(1/2*pi*b^2*x^2)*fresnels(b*x)/x^8, 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(cos(1/2*b**2*pi*x**2)*fresnels(b*x)/x**8,x)

[Out]

Timed out

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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^{8}}\,{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^8,x, algorithm="giac")

[Out]

integrate(cos(1/2*pi*b^2*x^2)*fresnels(b*x)/x^8, x)