3.48 \(\int \frac {S(b x)^2}{x^{10}} \, dx\)

Optimal. Leaf size=286 \[ \frac {\pi ^4 b^9 \text {Int}\left (\frac {S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{x},x\right )}{1728}-\frac {853 \pi ^4 b^9 C\left (\sqrt {2} b x\right )}{181440 \sqrt {2}}+\frac {\pi ^2 b^6}{5184 x^3}-\frac {b S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{36 x^8}-\frac {b^2}{504 x^7}+\frac {b^2 \cos \left (\pi b^2 x^2\right )}{504 x^7}+\frac {853 \pi ^3 b^8 \sin \left (\pi b^2 x^2\right )}{362880 x}+\frac {\pi ^3 b^7 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{1728 x^2}-\frac {187 \pi ^2 b^6 \cos \left (\pi b^2 x^2\right )}{181440 x^3}+\frac {\pi ^2 b^5 S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{864 x^4}-\frac {19 \pi b^4 \sin \left (\pi b^2 x^2\right )}{15120 x^5}-\frac {\pi b^3 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{216 x^6}-\frac {S(b x)^2}{9 x^9} \]

[Out]

-1/504*b^2/x^7+1/5184*b^6*Pi^2/x^3+1/504*b^2*cos(b^2*Pi*x^2)/x^7-187/181440*b^6*Pi^2*cos(b^2*Pi*x^2)/x^3-1/216
*b^3*Pi*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/x^6+1/1728*b^7*Pi^3*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/x^2-1/9*Fresne
lS(b*x)^2/x^9-1/36*b*FresnelS(b*x)*sin(1/2*b^2*Pi*x^2)/x^8+1/864*b^5*Pi^2*FresnelS(b*x)*sin(1/2*b^2*Pi*x^2)/x^
4-19/15120*b^4*Pi*sin(b^2*Pi*x^2)/x^5+853/362880*b^8*Pi^3*sin(b^2*Pi*x^2)/x-853/362880*b^9*Pi^4*FresnelC(b*x*2
^(1/2))*2^(1/2)+1/1728*b^9*Pi^4*Unintegrable(FresnelS(b*x)*sin(1/2*b^2*Pi*x^2)/x,x)

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Rubi [A]  time = 0.36, 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 = {} \[ \int \frac {S(b x)^2}{x^{10}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[FresnelS[b*x]^2/x^10,x]

[Out]

-b^2/(504*x^7) + (b^6*Pi^2)/(5184*x^3) + (b^2*Cos[b^2*Pi*x^2])/(504*x^7) - (187*b^6*Pi^2*Cos[b^2*Pi*x^2])/(181
440*x^3) - (853*b^9*Pi^4*FresnelC[Sqrt[2]*b*x])/(181440*Sqrt[2]) - (b^3*Pi*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/
(216*x^6) + (b^7*Pi^3*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(1728*x^2) - FresnelS[b*x]^2/(9*x^9) - (b*FresnelS[b*
x]*Sin[(b^2*Pi*x^2)/2])/(36*x^8) + (b^5*Pi^2*FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/(864*x^4) - (19*b^4*Pi*Sin[b^2
*Pi*x^2])/(15120*x^5) + (853*b^8*Pi^3*Sin[b^2*Pi*x^2])/(362880*x) + (b^9*Pi^4*Defer[Int][(FresnelS[b*x]*Sin[(b
^2*Pi*x^2)/2])/x, x])/1728

Rubi steps

\begin {align*} \int \frac {S(b x)^2}{x^{10}} \, dx &=-\frac {S(b x)^2}{9 x^9}+\frac {1}{9} (2 b) \int \frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^9} \, dx\\ &=-\frac {b^2}{504 x^7}-\frac {S(b x)^2}{9 x^9}-\frac {b S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{36 x^8}-\frac {1}{72} b^2 \int \frac {\cos \left (b^2 \pi x^2\right )}{x^8} \, dx+\frac {1}{36} \left (b^3 \pi \right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{x^7} \, dx\\ &=-\frac {b^2}{504 x^7}+\frac {b^2 \cos \left (b^2 \pi x^2\right )}{504 x^7}-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{216 x^6}-\frac {S(b x)^2}{9 x^9}-\frac {b S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{36 x^8}+\frac {1}{432} \left (b^4 \pi \right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^6} \, dx+\frac {1}{252} \left (b^4 \pi \right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^6} \, dx-\frac {1}{216} \left (b^5 \pi ^2\right ) \int \frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5} \, dx\\ &=-\frac {b^2}{504 x^7}+\frac {b^6 \pi ^2}{5184 x^3}+\frac {b^2 \cos \left (b^2 \pi x^2\right )}{504 x^7}-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{216 x^6}-\frac {S(b x)^2}{9 x^9}-\frac {b S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{36 x^8}+\frac {b^5 \pi ^2 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{864 x^4}-\frac {19 b^4 \pi \sin \left (b^2 \pi x^2\right )}{15120 x^5}+\frac {\left (b^6 \pi ^2\right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^4} \, dx}{1728}+\frac {\left (b^6 \pi ^2\right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^4} \, dx}{1080}+\frac {1}{630} \left (b^6 \pi ^2\right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^4} \, dx-\frac {1}{864} \left (b^7 \pi ^3\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{x^3} \, dx\\ &=-\frac {b^2}{504 x^7}+\frac {b^6 \pi ^2}{5184 x^3}+\frac {b^2 \cos \left (b^2 \pi x^2\right )}{504 x^7}-\frac {187 b^6 \pi ^2 \cos \left (b^2 \pi x^2\right )}{181440 x^3}-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{216 x^6}+\frac {b^7 \pi ^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{1728 x^2}-\frac {S(b x)^2}{9 x^9}-\frac {b S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{36 x^8}+\frac {b^5 \pi ^2 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{864 x^4}-\frac {19 b^4 \pi \sin \left (b^2 \pi x^2\right )}{15120 x^5}-\frac {\left (b^8 \pi ^3\right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2} \, dx}{3456}-\frac {\left (b^8 \pi ^3\right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2} \, dx}{2592}-\frac {\left (b^8 \pi ^3\right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2} \, dx}{1620}-\frac {1}{945} \left (b^8 \pi ^3\right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2} \, dx+\frac {\left (b^9 \pi ^4\right ) \int \frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x} \, dx}{1728}\\ &=-\frac {b^2}{504 x^7}+\frac {b^6 \pi ^2}{5184 x^3}+\frac {b^2 \cos \left (b^2 \pi x^2\right )}{504 x^7}-\frac {187 b^6 \pi ^2 \cos \left (b^2 \pi x^2\right )}{181440 x^3}-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{216 x^6}+\frac {b^7 \pi ^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{1728 x^2}-\frac {S(b x)^2}{9 x^9}-\frac {b S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{36 x^8}+\frac {b^5 \pi ^2 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{864 x^4}-\frac {19 b^4 \pi \sin \left (b^2 \pi x^2\right )}{15120 x^5}+\frac {853 b^8 \pi ^3 \sin \left (b^2 \pi x^2\right )}{362880 x}+\frac {\left (b^9 \pi ^4\right ) \int \frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x} \, dx}{1728}-\frac {\left (b^{10} \pi ^4\right ) \int \cos \left (b^2 \pi x^2\right ) \, dx}{1728}-\frac {\left (b^{10} \pi ^4\right ) \int \cos \left (b^2 \pi x^2\right ) \, dx}{1296}-\frac {1}{810} \left (b^{10} \pi ^4\right ) \int \cos \left (b^2 \pi x^2\right ) \, dx-\frac {1}{945} \left (2 b^{10} \pi ^4\right ) \int \cos \left (b^2 \pi x^2\right ) \, dx\\ &=-\frac {b^2}{504 x^7}+\frac {b^6 \pi ^2}{5184 x^3}+\frac {b^2 \cos \left (b^2 \pi x^2\right )}{504 x^7}-\frac {187 b^6 \pi ^2 \cos \left (b^2 \pi x^2\right )}{181440 x^3}-\frac {67 b^9 \pi ^4 C\left (\sqrt {2} b x\right )}{25920 \sqrt {2}}-\frac {1}{945} \sqrt {2} b^9 \pi ^4 C\left (\sqrt {2} b x\right )-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{216 x^6}+\frac {b^7 \pi ^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{1728 x^2}-\frac {S(b x)^2}{9 x^9}-\frac {b S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{36 x^8}+\frac {b^5 \pi ^2 S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{864 x^4}-\frac {19 b^4 \pi \sin \left (b^2 \pi x^2\right )}{15120 x^5}+\frac {853 b^8 \pi ^3 \sin \left (b^2 \pi x^2\right )}{362880 x}+\frac {\left (b^9 \pi ^4\right ) \int \frac {S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x} \, dx}{1728}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 0, normalized size = 0.00 \[ \int \frac {S(b x)^2}{x^{10}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[FresnelS[b*x]^2/x^10,x]

[Out]

Integrate[FresnelS[b*x]^2/x^10, x]

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fricas [A]  time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\rm fresnels}\left (b x\right )^{2}}{x^{10}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(fresnels(b*x)^2/x^10, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(fresnels(b*x)^2/x^10, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(fresnels(b*x)^2/x^10, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(FresnelS(b*x)^2/x^10, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnels(b*x)**2/x**10,x)

[Out]

Integral(fresnels(b*x)**2/x**10, x)

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