3.89 \(\int \frac{S(b x) \sin (\frac{1}{2} b^2 \pi x^2)}{x^{10}} \, dx\)

Optimal. Leaf size=262 \[ \frac{1}{945} \pi ^4 b^8 \text{Unintegrable}\left (\frac{S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{x^2},x\right )-\frac{5 \pi ^4 b^9 \text{CosIntegral}\left (\pi b^2 x^2\right )}{2016}+\frac{\pi ^2 b^4 S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{315 x^5}-\frac{S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{9 x^9}+\frac{\pi ^3 b^6 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{945 x^3}-\frac{\pi b^2 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{63 x^7}+\frac{\pi ^2 b^5}{2520 x^4}+\frac{5 \pi ^3 b^7 \sin \left (\pi b^2 x^2\right )}{2016 x^2}-\frac{11 \pi b^3 \sin \left (\pi b^2 x^2\right )}{3024 x^6}-\frac{67 \pi ^2 b^5 \cos \left (\pi b^2 x^2\right )}{30240 x^4}+\frac{b \cos \left (\pi b^2 x^2\right )}{144 x^8}-\frac{b}{144 x^8} \]

[Out]

-b/(144*x^8) + (b^5*Pi^2)/(2520*x^4) + (b*Cos[b^2*Pi*x^2])/(144*x^8) - (67*b^5*Pi^2*Cos[b^2*Pi*x^2])/(30240*x^
4) - (5*b^9*Pi^4*CosIntegral[b^2*Pi*x^2])/2016 - (b^2*Pi*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(63*x^7) + (b^6*Pi
^3*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(945*x^3) - (FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/(9*x^9) + (b^4*Pi^2*Fres
nelS[b*x]*Sin[(b^2*Pi*x^2)/2])/(315*x^5) - (11*b^3*Pi*Sin[b^2*Pi*x^2])/(3024*x^6) + (5*b^7*Pi^3*Sin[b^2*Pi*x^2
])/(2016*x^2) + (b^8*Pi^4*Unintegrable[(FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/x^2, x])/945

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Rubi [A]  time = 0.476823, 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 = {} \[ \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^{10}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/x^10,x]

[Out]

-b/(144*x^8) + (b^5*Pi^2)/(2520*x^4) + (b*Cos[b^2*Pi*x^2])/(144*x^8) - (67*b^5*Pi^2*Cos[b^2*Pi*x^2])/(30240*x^
4) - (5*b^9*Pi^4*CosIntegral[b^2*Pi*x^2])/2016 - (b^2*Pi*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(63*x^7) + (b^6*Pi
^3*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(945*x^3) - (FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/(9*x^9) + (b^4*Pi^2*Fres
nelS[b*x]*Sin[(b^2*Pi*x^2)/2])/(315*x^5) - (11*b^3*Pi*Sin[b^2*Pi*x^2])/(3024*x^6) + (5*b^7*Pi^3*Sin[b^2*Pi*x^2
])/(2016*x^2) + (b^8*Pi^4*Defer[Int][(FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/x^2, x])/945

Rubi steps

\begin{align*} \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^{10}} \, dx &=-\frac{b}{144 x^8}-\frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{9 x^9}-\frac{1}{18} b \int \frac{\cos \left (b^2 \pi x^2\right )}{x^9} \, dx+\frac{1}{9} \left (b^2 \pi \right ) \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{x^8} \, dx\\ &=-\frac{b}{144 x^8}-\frac{b^2 \pi \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{63 x^7}-\frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{9 x^9}-\frac{1}{36} b \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x^5} \, dx,x,x^2\right )+\frac{1}{126} \left (b^3 \pi \right ) \int \frac{\sin \left (b^2 \pi x^2\right )}{x^7} \, dx-\frac{1}{63} \left (b^4 \pi ^2\right ) \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^6} \, dx\\ &=-\frac{b}{144 x^8}+\frac{b^5 \pi ^2}{2520 x^4}+\frac{b \cos \left (b^2 \pi x^2\right )}{144 x^8}-\frac{b^2 \pi \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{63 x^7}-\frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{9 x^9}+\frac{b^4 \pi ^2 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{315 x^5}+\frac{1}{252} \left (b^3 \pi \right ) \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x^4} \, dx,x,x^2\right )+\frac{1}{144} \left (b^3 \pi \right ) \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x^4} \, dx,x,x^2\right )+\frac{1}{630} \left (b^5 \pi ^2\right ) \int \frac{\cos \left (b^2 \pi x^2\right )}{x^5} \, dx-\frac{1}{315} \left (b^6 \pi ^3\right ) \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{x^4} \, dx\\ &=-\frac{b}{144 x^8}+\frac{b^5 \pi ^2}{2520 x^4}+\frac{b \cos \left (b^2 \pi x^2\right )}{144 x^8}-\frac{b^2 \pi \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{63 x^7}+\frac{b^6 \pi ^3 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{945 x^3}-\frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{9 x^9}+\frac{b^4 \pi ^2 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{315 x^5}-\frac{11 b^3 \pi \sin \left (b^2 \pi x^2\right )}{3024 x^6}+\frac{\left (b^5 \pi ^2\right ) \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )}{1260}+\frac{1}{756} \left (b^5 \pi ^2\right ) \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )+\frac{1}{432} \left (b^5 \pi ^2\right ) \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )-\frac{\left (b^7 \pi ^3\right ) \int \frac{\sin \left (b^2 \pi x^2\right )}{x^3} \, dx}{1890}+\frac{1}{945} \left (b^8 \pi ^4\right ) \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^2} \, dx\\ &=-\frac{b}{144 x^8}+\frac{b^5 \pi ^2}{2520 x^4}+\frac{b \cos \left (b^2 \pi x^2\right )}{144 x^8}-\frac{67 b^5 \pi ^2 \cos \left (b^2 \pi x^2\right )}{30240 x^4}-\frac{b^2 \pi \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{63 x^7}+\frac{b^6 \pi ^3 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{945 x^3}-\frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{9 x^9}+\frac{b^4 \pi ^2 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{315 x^5}-\frac{11 b^3 \pi \sin \left (b^2 \pi x^2\right )}{3024 x^6}-\frac{\left (b^7 \pi ^3\right ) \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )}{3780}-\frac{\left (b^7 \pi ^3\right ) \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )}{2520}-\frac{\left (b^7 \pi ^3\right ) \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )}{1512}-\frac{1}{864} \left (b^7 \pi ^3\right ) \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )+\frac{1}{945} \left (b^8 \pi ^4\right ) \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^2} \, dx\\ &=-\frac{b}{144 x^8}+\frac{b^5 \pi ^2}{2520 x^4}+\frac{b \cos \left (b^2 \pi x^2\right )}{144 x^8}-\frac{67 b^5 \pi ^2 \cos \left (b^2 \pi x^2\right )}{30240 x^4}-\frac{b^2 \pi \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{63 x^7}+\frac{b^6 \pi ^3 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{945 x^3}-\frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{9 x^9}+\frac{b^4 \pi ^2 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{315 x^5}-\frac{11 b^3 \pi \sin \left (b^2 \pi x^2\right )}{3024 x^6}+\frac{5 b^7 \pi ^3 \sin \left (b^2 \pi x^2\right )}{2016 x^2}+\frac{1}{945} \left (b^8 \pi ^4\right ) \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^2} \, dx-\frac{\left (b^9 \pi ^4\right ) \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )}{3780}-\frac{\left (b^9 \pi ^4\right ) \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )}{2520}-\frac{\left (b^9 \pi ^4\right ) \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )}{1512}-\frac{1}{864} \left (b^9 \pi ^4\right ) \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )\\ &=-\frac{b}{144 x^8}+\frac{b^5 \pi ^2}{2520 x^4}+\frac{b \cos \left (b^2 \pi x^2\right )}{144 x^8}-\frac{67 b^5 \pi ^2 \cos \left (b^2 \pi x^2\right )}{30240 x^4}-\frac{5 b^9 \pi ^4 \text{Ci}\left (b^2 \pi x^2\right )}{2016}-\frac{b^2 \pi \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{63 x^7}+\frac{b^6 \pi ^3 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{945 x^3}-\frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{9 x^9}+\frac{b^4 \pi ^2 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{315 x^5}-\frac{11 b^3 \pi \sin \left (b^2 \pi x^2\right )}{3024 x^6}+\frac{5 b^7 \pi ^3 \sin \left (b^2 \pi x^2\right )}{2016 x^2}+\frac{1}{945} \left (b^8 \pi ^4\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 = 0.0322328, size = 0, normalized size = 0. \[ \int \frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^{10}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/x^10,x]

[Out]

Integrate[(FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/x^10, x]

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Maple [A]  time = 0.053, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it FresnelS} \left ( bx \right ) }{{x}^{10}}\sin \left ({\frac{{b}^{2}\pi \,{x}^{2}}{2}} \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(FresnelS(b*x)*sin(1/2*b^2*Pi*x^2)/x^10,x)

[Out]

int(FresnelS(b*x)*sin(1/2*b^2*Pi*x^2)/x^10,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm fresnels}\left (b x\right ) \sin \left (\frac{1}{2} \, \pi b^{2} x^{2}\right )}{x^{10}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnels(b*x)*sin(1/2*b^2*pi*x^2)/x^10,x, algorithm="maxima")

[Out]

integrate(fresnels(b*x)*sin(1/2*pi*b^2*x^2)/x^10, x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\rm fresnels}\left (b x\right ) \sin \left (\frac{1}{2} \, \pi b^{2} x^{2}\right )}{x^{10}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnels(b*x)*sin(1/2*b^2*pi*x^2)/x^10,x, algorithm="fricas")

[Out]

integral(fresnels(b*x)*sin(1/2*pi*b^2*x^2)/x^10, 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(fresnels(b*x)*sin(1/2*b**2*pi*x**2)/x**10,x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm fresnels}\left (b x\right ) \sin \left (\frac{1}{2} \, \pi b^{2} x^{2}\right )}{x^{10}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnels(b*x)*sin(1/2*b^2*pi*x^2)/x^10,x, algorithm="giac")

[Out]

integrate(fresnels(b*x)*sin(1/2*pi*b^2*x^2)/x^10, x)