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

Optimal. Leaf size=202 \[ \frac {1}{105} \pi ^3 b^6 \text {Int}\left (\frac {S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{x^2},x\right )+\frac {\pi b^3}{280 x^4}-\frac {S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}+\frac {\pi b^2 S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{35 x^5}-\frac {b \sin \left (\pi b^2 x^2\right )}{84 x^6}-\frac {1}{84} \pi ^3 b^7 \text {Ci}\left (b^2 \pi x^2\right )+\frac {\pi ^2 b^5 \sin \left (\pi b^2 x^2\right )}{84 x^2}+\frac {\pi ^2 b^4 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{105 x^3}-\frac {\pi b^3 \cos \left (\pi b^2 x^2\right )}{105 x^4} \]

[Out]

1/280*b^3*Pi/x^4-1/84*b^7*Pi^3*Ci(b^2*Pi*x^2)-1/105*b^3*Pi*cos(b^2*Pi*x^2)/x^4-1/7*cos(1/2*b^2*Pi*x^2)*Fresnel
S(b*x)/x^7+1/105*b^4*Pi^2*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/x^3+1/35*b^2*Pi*FresnelS(b*x)*sin(1/2*b^2*Pi*x^2)/
x^5-1/84*b*sin(b^2*Pi*x^2)/x^6+1/84*b^5*Pi^2*sin(b^2*Pi*x^2)/x^2+1/105*b^6*Pi^3*Unintegrable(FresnelS(b*x)*sin
(1/2*b^2*Pi*x^2)/x^2,x)

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Rubi [A]  time = 0.32, 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 = {} \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{x^8} \, dx \]

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*}

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Mathematica [A]  time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{x^8} \, dx \]

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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fricas [A]  time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) {\rm fresnels}\left (b x\right )}{x^{8}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) {\rm fresnels}\left (b x\right )}{x^{8}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.03, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \mathrm {S}\left (b x \right )}{x^{8}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/x^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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) {\rm fresnels}\left (b x\right )}{x^{8}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\mathrm {FresnelS}\left (b\,x\right )\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right )}{x^8} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((FresnelS(b*x)*cos((Pi*b^2*x^2)/2))/x^8,x)

[Out]

int((FresnelS(b*x)*cos((Pi*b^2*x^2)/2))/x^8, x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )}{x^{8}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cos(pi*b**2*x**2/2)*fresnels(b*x)/x**8, x)

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