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

Optimal. Leaf size=263 \[ \frac {1}{945} \pi ^4 b^8 \text {Int}\left (\frac {S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{x^2},x\right )+\frac {\pi ^2 b^5}{2520 x^4}-\frac {S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}-\frac {\pi b^2 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{63 x^7}+\frac {b \cos \left (\pi b^2 x^2\right )}{144 x^8}-\frac {5 \pi ^4 b^9 \text {Ci}\left (b^2 \pi x^2\right )}{2016}+\frac {5 \pi ^3 b^7 \sin \left (\pi b^2 x^2\right )}{2016 x^2}+\frac {\pi ^3 b^6 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{945 x^3}-\frac {67 \pi ^2 b^5 \cos \left (\pi b^2 x^2\right )}{30240 x^4}+\frac {\pi ^2 b^4 S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{315 x^5}-\frac {11 \pi b^3 \sin \left (\pi b^2 x^2\right )}{3024 x^6}-\frac {b}{144 x^8} \]

[Out]

-1/144*b/x^8+1/2520*b^5*Pi^2/x^4-5/2016*b^9*Pi^4*Ci(b^2*Pi*x^2)+1/144*b*cos(b^2*Pi*x^2)/x^8-67/30240*b^5*Pi^2*
cos(b^2*Pi*x^2)/x^4-1/63*b^2*Pi*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/x^7+1/945*b^6*Pi^3*cos(1/2*b^2*Pi*x^2)*Fresn
elS(b*x)/x^3-1/9*FresnelS(b*x)*sin(1/2*b^2*Pi*x^2)/x^9+1/315*b^4*Pi^2*FresnelS(b*x)*sin(1/2*b^2*Pi*x^2)/x^5-11
/3024*b^3*Pi*sin(b^2*Pi*x^2)/x^6+5/2016*b^7*Pi^3*sin(b^2*Pi*x^2)/x^2+1/945*b^8*Pi^4*Unintegrable(FresnelS(b*x)
*sin(1/2*b^2*Pi*x^2)/x^2,x)

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Rubi [A]  time = 0.48, 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 = {} \[ \int \frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^{10}} \, dx \]

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*}

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Mathematica [A]  time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^{10}} \, dx \]

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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fricas [A]  time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\rm fresnels}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{x^{10}}, x\right ) \]

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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm fresnels}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{x^{10}}\,{d x} \]

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)

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maple [A]  time = 0.02, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {S}\left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{x^{10}}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm fresnels}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{x^{10}}\,{d x} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\mathrm {FresnelS}\left (b\,x\right )\,\sin \left (\frac {\Pi \,b^2\,x^2}{2}\right )}{x^{10}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )}{x^{10}}\, dx \]

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]

Integral(sin(pi*b**2*x**2/2)*fresnels(b*x)/x**10, x)

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