Integrand size = 20, antiderivative size = 20 \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^{10}} \, dx=-\frac {b}{144 x^8}+\frac {b^5 \pi ^2}{2520 x^4}+\frac {b \cos \left (b^2 \pi x^2\right )}{144 x^8}-\frac {67 b^5 \pi ^2 \cos \left (b^2 \pi x^2\right )}{30240 x^4}-\frac {5 b^9 \pi ^4 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )}{2016}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{63 x^7}+\frac {b^6 \pi ^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{945 x^3}-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{9 x^9}+\frac {b^4 \pi ^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{315 x^5}-\frac {11 b^3 \pi \sin \left (b^2 \pi x^2\right )}{3024 x^6}+\frac {5 b^7 \pi ^3 \sin \left (b^2 \pi x^2\right )}{2016 x^2}+\frac {1}{945} b^8 \pi ^4 \text {Int}\left (\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2},x\right ) \]
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Not integrable
Time = 0.34 (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 = {} \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^{10}} \, dx=\int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^{10}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = -\frac {b}{144 x^8}-\frac {\operatorname {FresnelS}(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 ) \operatorname {FresnelS}(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 ) \operatorname {FresnelS}(b x)}{63 x^7}-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{9 x^9}-\frac {1}{36} b \text {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 {\operatorname {FresnelS}(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 ) \operatorname {FresnelS}(b x)}{63 x^7}-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{9 x^9}+\frac {b^4 \pi ^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{315 x^5}+\frac {1}{252} \left (b^3 \pi \right ) \text {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 ) \text {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 ) \operatorname {FresnelS}(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 ) \operatorname {FresnelS}(b x)}{63 x^7}+\frac {b^6 \pi ^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{945 x^3}-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{9 x^9}+\frac {b^4 \pi ^2 \operatorname {FresnelS}(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 ) \text {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 ) \text {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 ) \text {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 {\operatorname {FresnelS}(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 ) \operatorname {FresnelS}(b x)}{63 x^7}+\frac {b^6 \pi ^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{945 x^3}-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{9 x^9}+\frac {b^4 \pi ^2 \operatorname {FresnelS}(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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 {\operatorname {FresnelS}(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 ) \operatorname {FresnelS}(b x)}{63 x^7}+\frac {b^6 \pi ^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{945 x^3}-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{9 x^9}+\frac {b^4 \pi ^2 \operatorname {FresnelS}(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 {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2} \, dx-\frac {\left (b^9 \pi ^4\right ) \text {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 ) \text {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 ) \text {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 ) \text {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 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )}{2016}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{63 x^7}+\frac {b^6 \pi ^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{945 x^3}-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{9 x^9}+\frac {b^4 \pi ^2 \operatorname {FresnelS}(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 {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2} \, dx \\ \end{align*}
Not integrable
Time = 0.03 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^{10}} \, dx=\int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^{10}} \, dx \]
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Not integrable
Time = 0.14 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90
\[\int \frac {\operatorname {FresnelS}\left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{x^{10}}d x\]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^{10}} \, dx=\int { \frac {\operatorname {S}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{x^{10}} \,d x } \]
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Not integrable
Time = 66.99 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^{10}} \, dx=\int \frac {\sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )}{x^{10}}\, dx \]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^{10}} \, dx=\int { \frac {\operatorname {S}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{x^{10}} \,d x } \]
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Not integrable
Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^{10}} \, dx=\int { \frac {\operatorname {S}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{x^{10}} \,d x } \]
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Not integrable
Time = 4.94 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^{10}} \, dx=\int \frac {\mathrm {FresnelS}\left (b\,x\right )\,\sin \left (\frac {\Pi \,b^2\,x^2}{2}\right )}{x^{10}} \,d x \]
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