Integrand size = 20, antiderivative size = 20 \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^9} \, dx=-\frac {b}{112 x^7}+\frac {b^5 \pi ^2}{1152 x^3}+\frac {b \cos \left (b^2 \pi x^2\right )}{112 x^7}-\frac {187 b^5 \pi ^2 \cos \left (b^2 \pi x^2\right )}{40320 x^3}-\frac {853 b^8 \pi ^4 \operatorname {FresnelC}\left (\sqrt {2} b x\right )}{40320 \sqrt {2}}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{48 x^6}+\frac {b^6 \pi ^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{384 x^2}-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{8 x^8}+\frac {b^4 \pi ^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{192 x^4}-\frac {19 b^3 \pi \sin \left (b^2 \pi x^2\right )}{3360 x^5}+\frac {853 b^7 \pi ^3 \sin \left (b^2 \pi x^2\right )}{80640 x}+\frac {1}{384} b^8 \pi ^4 \text {Int}\left (\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x},x\right ) \]
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Not integrable
Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^9} \, dx=\int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^9} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = -\frac {b}{112 x^7}-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{8 x^8}-\frac {1}{16} b \int \frac {\cos \left (b^2 \pi x^2\right )}{x^8} \, dx+\frac {1}{8} \left (b^2 \pi \right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^7} \, dx \\ & = -\frac {b}{112 x^7}+\frac {b \cos \left (b^2 \pi x^2\right )}{112 x^7}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{48 x^6}-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{8 x^8}+\frac {1}{96} \left (b^3 \pi \right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^6} \, dx+\frac {1}{56} \left (b^3 \pi \right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^6} \, dx-\frac {1}{48} \left (b^4 \pi ^2\right ) \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5} \, dx \\ & = -\frac {b}{112 x^7}+\frac {b^5 \pi ^2}{1152 x^3}+\frac {b \cos \left (b^2 \pi x^2\right )}{112 x^7}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{48 x^6}-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{8 x^8}+\frac {b^4 \pi ^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{192 x^4}-\frac {19 b^3 \pi \sin \left (b^2 \pi x^2\right )}{3360 x^5}+\frac {1}{384} \left (b^5 \pi ^2\right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^4} \, dx+\frac {1}{240} \left (b^5 \pi ^2\right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^4} \, dx+\frac {1}{140} \left (b^5 \pi ^2\right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^4} \, dx-\frac {1}{192} \left (b^6 \pi ^3\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^3} \, dx \\ & = -\frac {b}{112 x^7}+\frac {b^5 \pi ^2}{1152 x^3}+\frac {b \cos \left (b^2 \pi x^2\right )}{112 x^7}-\frac {187 b^5 \pi ^2 \cos \left (b^2 \pi x^2\right )}{40320 x^3}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{48 x^6}+\frac {b^6 \pi ^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{384 x^2}-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{8 x^8}+\frac {b^4 \pi ^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{192 x^4}-\frac {19 b^3 \pi \sin \left (b^2 \pi x^2\right )}{3360 x^5}-\frac {1}{768} \left (b^7 \pi ^3\right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2} \, dx-\frac {1}{576} \left (b^7 \pi ^3\right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2} \, dx-\frac {1}{360} \left (b^7 \pi ^3\right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2} \, dx-\frac {1}{210} \left (b^7 \pi ^3\right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2} \, dx+\frac {1}{384} \left (b^8 \pi ^4\right ) \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x} \, dx \\ & = -\frac {b}{112 x^7}+\frac {b^5 \pi ^2}{1152 x^3}+\frac {b \cos \left (b^2 \pi x^2\right )}{112 x^7}-\frac {187 b^5 \pi ^2 \cos \left (b^2 \pi x^2\right )}{40320 x^3}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{48 x^6}+\frac {b^6 \pi ^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{384 x^2}-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{8 x^8}+\frac {b^4 \pi ^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{192 x^4}-\frac {19 b^3 \pi \sin \left (b^2 \pi x^2\right )}{3360 x^5}+\frac {853 b^7 \pi ^3 \sin \left (b^2 \pi x^2\right )}{80640 x}+\frac {1}{384} \left (b^8 \pi ^4\right ) \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x} \, dx-\frac {1}{384} \left (b^9 \pi ^4\right ) \int \cos \left (b^2 \pi x^2\right ) \, dx-\frac {1}{288} \left (b^9 \pi ^4\right ) \int \cos \left (b^2 \pi x^2\right ) \, dx-\frac {1}{180} \left (b^9 \pi ^4\right ) \int \cos \left (b^2 \pi x^2\right ) \, dx-\frac {1}{105} \left (b^9 \pi ^4\right ) \int \cos \left (b^2 \pi x^2\right ) \, dx \\ & = -\frac {b}{112 x^7}+\frac {b^5 \pi ^2}{1152 x^3}+\frac {b \cos \left (b^2 \pi x^2\right )}{112 x^7}-\frac {187 b^5 \pi ^2 \cos \left (b^2 \pi x^2\right )}{40320 x^3}-\frac {853 b^8 \pi ^4 \operatorname {FresnelC}\left (\sqrt {2} b x\right )}{40320 \sqrt {2}}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{48 x^6}+\frac {b^6 \pi ^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{384 x^2}-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{8 x^8}+\frac {b^4 \pi ^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{192 x^4}-\frac {19 b^3 \pi \sin \left (b^2 \pi x^2\right )}{3360 x^5}+\frac {853 b^7 \pi ^3 \sin \left (b^2 \pi x^2\right )}{80640 x}+\frac {1}{384} \left (b^8 \pi ^4\right ) \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x} \, dx \\ \end{align*}
Not integrable
Time = 0.03 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^9} \, dx=\int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^9} \, dx \]
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Not integrable
Time = 0.12 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90
\[\int \frac {\operatorname {FresnelS}\left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{x^{9}}d x\]
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Not integrable
Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^9} \, dx=\int { \frac {\operatorname {S}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{x^{9}} \,d x } \]
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Not integrable
Time = 36.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^9} \, dx=\int \frac {\sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )}{x^{9}}\, dx \]
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Not integrable
Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^9} \, dx=\int { \frac {\operatorname {S}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{x^{9}} \,d x } \]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^9} \, dx=\int { \frac {\operatorname {S}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{x^{9}} \,d x } \]
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Not integrable
Time = 4.98 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^9} \, dx=\int \frac {\mathrm {FresnelS}\left (b\,x\right )\,\sin \left (\frac {\Pi \,b^2\,x^2}{2}\right )}{x^9} \,d x \]
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