Integrand size = 20, antiderivative size = 20 \[ \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8} \, dx=-\frac {b^3 \pi }{280 x^4}-\frac {b^3 \pi \cos \left (b^2 \pi x^2\right )}{105 x^4}-\frac {1}{84} b^7 \pi ^3 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{35 x^5}-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 x^7}+\frac {b^4 \pi ^2 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{105 x^3}-\frac {b \sin \left (b^2 \pi x^2\right )}{84 x^6}+\frac {b^5 \pi ^2 \sin \left (b^2 \pi x^2\right )}{84 x^2}-\frac {1}{105} b^6 \pi ^3 \text {Int}\left (\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^2},x\right ) \]
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Not integrable
Time = 0.22 (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 = {} \[ \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8} \, dx=\int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = -\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 x^7}+\frac {1}{14} b \int \frac {\sin \left (b^2 \pi x^2\right )}{x^7} \, dx+\frac {1}{7} \left (b^2 \pi \right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^6} \, dx \\ & = -\frac {b^3 \pi }{280 x^4}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{35 x^5}-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 x^7}+\frac {1}{28} b \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x^4} \, dx,x,x^2\right )+\frac {1}{70} \left (b^3 \pi \right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^5} \, dx-\frac {1}{35} \left (b^4 \pi ^2\right ) \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4} \, dx \\ & = -\frac {b^3 \pi }{280 x^4}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{35 x^5}-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 x^7}+\frac {b^4 \pi ^2 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{105 x^3}-\frac {b \sin \left (b^2 \pi x^2\right )}{84 x^6}+\frac {1}{140} \left (b^3 \pi \right ) \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )+\frac {1}{84} \left (b^3 \pi \right ) \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )-\frac {1}{210} \left (b^5 \pi ^2\right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^3} \, dx-\frac {1}{105} \left (b^6 \pi ^3\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^2} \, dx \\ & = -\frac {b^3 \pi }{280 x^4}-\frac {b^3 \pi \cos \left (b^2 \pi x^2\right )}{105 x^4}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{35 x^5}-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 x^7}+\frac {b^4 \pi ^2 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{105 x^3}-\frac {b \sin \left (b^2 \pi x^2\right )}{84 x^6}-\frac {1}{420} \left (b^5 \pi ^2\right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )-\frac {1}{280} \left (b^5 \pi ^2\right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )-\frac {1}{168} \left (b^5 \pi ^2\right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )-\frac {1}{105} \left (b^6 \pi ^3\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^2} \, dx \\ & = -\frac {b^3 \pi }{280 x^4}-\frac {b^3 \pi \cos \left (b^2 \pi x^2\right )}{105 x^4}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{35 x^5}-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 x^7}+\frac {b^4 \pi ^2 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{105 x^3}-\frac {b \sin \left (b^2 \pi x^2\right )}{84 x^6}+\frac {b^5 \pi ^2 \sin \left (b^2 \pi x^2\right )}{84 x^2}-\frac {1}{105} \left (b^6 \pi ^3\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^2} \, dx-\frac {1}{420} \left (b^7 \pi ^3\right ) \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )-\frac {1}{280} \left (b^7 \pi ^3\right ) \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )-\frac {1}{168} \left (b^7 \pi ^3\right ) \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right ) \\ & = -\frac {b^3 \pi }{280 x^4}-\frac {b^3 \pi \cos \left (b^2 \pi x^2\right )}{105 x^4}-\frac {1}{84} b^7 \pi ^3 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{35 x^5}-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 x^7}+\frac {b^4 \pi ^2 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{105 x^3}-\frac {b \sin \left (b^2 \pi x^2\right )}{84 x^6}+\frac {b^5 \pi ^2 \sin \left (b^2 \pi x^2\right )}{84 x^2}-\frac {1}{105} \left (b^6 \pi ^3\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^2} \, dx \\ \end{align*}
Not integrable
Time = 0.03 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8} \, dx=\int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8} \, dx \]
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Not integrable
Time = 0.14 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90
\[\int \frac {\operatorname {FresnelC}\left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{x^{8}}d x\]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8} \, dx=\int { \frac {\operatorname {C}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{x^{8}} \,d x } \]
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Not integrable
Time = 20.38 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8} \, dx=\int \frac {\sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )}{x^{8}}\, dx \]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8} \, dx=\int { \frac {\operatorname {C}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{x^{8}} \,d x } \]
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Not integrable
Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8} \, dx=\int { \frac {\operatorname {C}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{x^{8}} \,d x } \]
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Not integrable
Time = 4.78 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8} \, dx=\int \frac {\mathrm {FresnelC}\left (b\,x\right )\,\sin \left (\frac {\Pi \,b^2\,x^2}{2}\right )}{x^8} \,d x \]
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