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