Integrand size = 10, antiderivative size = 242 \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^9} \, dx=-\frac {b^2}{336 x^6}+\frac {b^6 \pi ^2}{1680 x^2}+\frac {b^2 \cos \left (b^2 \pi x^2\right )}{336 x^6}-\frac {b^6 \pi ^2 \cos \left (b^2 \pi x^2\right )}{336 x^2}-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{140 x^5}+\frac {b^7 \pi ^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{420 x}+\frac {1}{840} b^8 \pi ^4 \operatorname {FresnelS}(b x)^2-\frac {\operatorname {FresnelS}(b x)^2}{8 x^8}-\frac {b \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{28 x^7}+\frac {b^5 \pi ^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{420 x^3}-\frac {b^4 \pi \sin \left (b^2 \pi x^2\right )}{420 x^4}-\frac {1}{280} b^8 \pi ^3 \text {Si}\left (b^2 \pi x^2\right ) \]
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Time = 0.26 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6565, 6591, 6599, 6575, 30, 3456, 3461, 3378, 3380, 3460} \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^9} \, dx=\frac {1}{840} \pi ^4 b^8 \operatorname {FresnelS}(b x)^2+\frac {\pi ^2 b^6}{1680 x^2}-\frac {b \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{28 x^7}-\frac {b^2}{336 x^6}+\frac {b^2 \cos \left (\pi b^2 x^2\right )}{336 x^6}-\frac {1}{280} \pi ^3 b^8 \text {Si}\left (b^2 \pi x^2\right )+\frac {\pi ^3 b^7 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{420 x}-\frac {\pi ^2 b^6 \cos \left (\pi b^2 x^2\right )}{336 x^2}+\frac {\pi ^2 b^5 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{420 x^3}-\frac {\pi b^4 \sin \left (\pi b^2 x^2\right )}{420 x^4}-\frac {\pi b^3 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{140 x^5}-\frac {\operatorname {FresnelS}(b x)^2}{8 x^8} \]
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Rule 30
Rule 3378
Rule 3380
Rule 3456
Rule 3460
Rule 3461
Rule 6565
Rule 6575
Rule 6591
Rule 6599
Rubi steps \begin{align*} \text {integral}& = -\frac {\operatorname {FresnelS}(b x)^2}{8 x^8}+\frac {1}{4} b \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8} \, dx \\ & = -\frac {b^2}{336 x^6}-\frac {\operatorname {FresnelS}(b x)^2}{8 x^8}-\frac {b \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{28 x^7}-\frac {1}{56} b^2 \int \frac {\cos \left (b^2 \pi x^2\right )}{x^7} \, dx+\frac {1}{28} \left (b^3 \pi \right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^6} \, dx \\ & = -\frac {b^2}{336 x^6}-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{140 x^5}-\frac {\operatorname {FresnelS}(b x)^2}{8 x^8}-\frac {b \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{28 x^7}-\frac {1}{112} b^2 \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x^4} \, dx,x,x^2\right )+\frac {1}{280} \left (b^4 \pi \right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^5} \, dx-\frac {1}{140} \left (b^5 \pi ^2\right ) \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4} \, dx \\ & = -\frac {b^2}{336 x^6}+\frac {b^6 \pi ^2}{1680 x^2}+\frac {b^2 \cos \left (b^2 \pi x^2\right )}{336 x^6}-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{140 x^5}-\frac {\operatorname {FresnelS}(b x)^2}{8 x^8}-\frac {b \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{28 x^7}+\frac {b^5 \pi ^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{420 x^3}+\frac {1}{560} \left (b^4 \pi \right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )+\frac {1}{336} \left (b^4 \pi \right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )+\frac {1}{840} \left (b^6 \pi ^2\right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^3} \, dx-\frac {1}{420} \left (b^7 \pi ^3\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^2} \, dx \\ & = -\frac {b^2}{336 x^6}+\frac {b^6 \pi ^2}{1680 x^2}+\frac {b^2 \cos \left (b^2 \pi x^2\right )}{336 x^6}-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{140 x^5}+\frac {b^7 \pi ^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{420 x}-\frac {\operatorname {FresnelS}(b x)^2}{8 x^8}-\frac {b \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{28 x^7}+\frac {b^5 \pi ^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{420 x^3}-\frac {b^4 \pi \sin \left (b^2 \pi x^2\right )}{420 x^4}+\frac {\left (b^6 \pi ^2\right ) \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )}{1680}+\frac {\left (b^6 \pi ^2\right ) \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )}{1120}+\frac {1}{672} \left (b^6 \pi ^2\right ) \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )-\frac {1}{840} \left (b^8 \pi ^3\right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x} \, dx+\frac {1}{420} \left (b^9 \pi ^4\right ) \int \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx \\ & = -\frac {b^2}{336 x^6}+\frac {b^6 \pi ^2}{1680 x^2}+\frac {b^2 \cos \left (b^2 \pi x^2\right )}{336 x^6}-\frac {b^6 \pi ^2 \cos \left (b^2 \pi x^2\right )}{336 x^2}-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{140 x^5}+\frac {b^7 \pi ^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{420 x}-\frac {\operatorname {FresnelS}(b x)^2}{8 x^8}-\frac {b \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{28 x^7}+\frac {b^5 \pi ^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{420 x^3}-\frac {b^4 \pi \sin \left (b^2 \pi x^2\right )}{420 x^4}-\frac {b^8 \pi ^3 \text {Si}\left (b^2 \pi x^2\right )}{1680}-\frac {\left (b^8 \pi ^3\right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )}{1680}-\frac {\left (b^8 \pi ^3\right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )}{1120}-\frac {1}{672} \left (b^8 \pi ^3\right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )+\frac {1}{420} \left (b^8 \pi ^4\right ) \text {Subst}(\int x \, dx,x,\operatorname {FresnelS}(b x)) \\ & = -\frac {b^2}{336 x^6}+\frac {b^6 \pi ^2}{1680 x^2}+\frac {b^2 \cos \left (b^2 \pi x^2\right )}{336 x^6}-\frac {b^6 \pi ^2 \cos \left (b^2 \pi x^2\right )}{336 x^2}-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{140 x^5}+\frac {b^7 \pi ^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{420 x}+\frac {1}{840} b^8 \pi ^4 \operatorname {FresnelS}(b x)^2-\frac {\operatorname {FresnelS}(b x)^2}{8 x^8}-\frac {b \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{28 x^7}+\frac {b^5 \pi ^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{420 x^3}-\frac {b^4 \pi \sin \left (b^2 \pi x^2\right )}{420 x^4}-\frac {1}{280} b^8 \pi ^3 \text {Si}\left (b^2 \pi x^2\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^9} \, dx=-\frac {b^2}{336 x^6}+\frac {b^6 \pi ^2}{1680 x^2}+\frac {b^2 \cos \left (b^2 \pi x^2\right )}{336 x^6}-\frac {b^6 \pi ^2 \cos \left (b^2 \pi x^2\right )}{336 x^2}-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{140 x^5}+\frac {b^7 \pi ^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{420 x}+\frac {1}{840} b^8 \pi ^4 \operatorname {FresnelS}(b x)^2-\frac {\operatorname {FresnelS}(b x)^2}{8 x^8}-\frac {b \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{28 x^7}+\frac {b^5 \pi ^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{420 x^3}-\frac {b^4 \pi \sin \left (b^2 \pi x^2\right )}{420 x^4}-\frac {1}{280} b^8 \pi ^3 \text {Si}\left (b^2 \pi x^2\right ) \]
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\[\int \frac {\operatorname {FresnelS}\left (b x \right )^{2}}{x^{9}}d x\]
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Time = 0.26 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.77 \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^9} \, dx=-\frac {3 \, \pi ^{3} b^{8} x^{8} \operatorname {Si}\left (\pi b^{2} x^{2}\right ) - 3 \, \pi ^{2} b^{6} x^{6} + 5 \, b^{2} x^{2} + 5 \, {\left (\pi ^{2} b^{6} x^{6} - b^{2} x^{2}\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} - 2 \, {\left (\pi ^{3} b^{7} x^{7} - 3 \, \pi b^{3} x^{3}\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right ) - {\left (\pi ^{4} b^{8} x^{8} - 105\right )} \operatorname {S}\left (b x\right )^{2} + 2 \, {\left (2 \, \pi b^{4} x^{4} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - {\left (\pi ^{2} b^{5} x^{5} - 15 \, b x\right )} \operatorname {S}\left (b x\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{840 \, x^{8}} \]
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\[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^9} \, dx=\int \frac {S^{2}\left (b x\right )}{x^{9}}\, dx \]
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\[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^9} \, dx=\int { \frac {\operatorname {S}\left (b x\right )^{2}}{x^{9}} \,d x } \]
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\[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^9} \, dx=\int { \frac {\operatorname {S}\left (b x\right )^{2}}{x^{9}} \,d x } \]
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Timed out. \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^9} \, dx=\int \frac {{\mathrm {FresnelS}\left (b\,x\right )}^2}{x^9} \,d x \]
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