Integrand size = 10, antiderivative size = 10 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^8} \, dx=-\frac {b^2}{210 x^5}+\frac {b^6 \pi ^2}{336 x}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{210 x^5}+\frac {67 b^6 \pi ^2 \cos \left (b^2 \pi x^2\right )}{5040 x}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{21 x^6}+\frac {b^5 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{168 x^2}-\frac {\operatorname {FresnelC}(b x)^2}{7 x^7}+\frac {b^7 \pi ^3 \operatorname {FresnelS}\left (\sqrt {2} b x\right )}{72 \sqrt {2}}+\frac {2}{315} \sqrt {2} b^7 \pi ^3 \operatorname {FresnelS}\left (\sqrt {2} b x\right )+\frac {b^3 \pi \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{84 x^4}+\frac {13 b^4 \pi \sin \left (b^2 \pi x^2\right )}{2520 x^3}+\frac {1}{168} b^7 \pi ^3 \text {Int}\left (\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x},x\right ) \]
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Not integrable
Time = 0.16 (sec) , antiderivative size = 10, 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 = {} \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^8} \, dx=\int \frac {\operatorname {FresnelC}(b x)^2}{x^8} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = -\frac {\operatorname {FresnelC}(b x)^2}{7 x^7}+\frac {1}{7} (2 b) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^7} \, dx \\ & = -\frac {b^2}{210 x^5}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{21 x^6}-\frac {\operatorname {FresnelC}(b x)^2}{7 x^7}+\frac {1}{42} b^2 \int \frac {\cos \left (b^2 \pi x^2\right )}{x^6} \, dx-\frac {1}{21} \left (b^3 \pi \right ) \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5} \, dx \\ & = -\frac {b^2}{210 x^5}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{210 x^5}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{21 x^6}-\frac {\operatorname {FresnelC}(b x)^2}{7 x^7}+\frac {b^3 \pi \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{84 x^4}-\frac {1}{168} \left (b^4 \pi \right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^4} \, dx-\frac {1}{105} \left (b^4 \pi \right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^4} \, dx-\frac {1}{84} \left (b^5 \pi ^2\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^3} \, dx \\ & = -\frac {b^2}{210 x^5}+\frac {b^6 \pi ^2}{336 x}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{210 x^5}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{21 x^6}+\frac {b^5 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{168 x^2}-\frac {\operatorname {FresnelC}(b x)^2}{7 x^7}+\frac {b^3 \pi \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{84 x^4}+\frac {13 b^4 \pi \sin \left (b^2 \pi x^2\right )}{2520 x^3}-\frac {1}{336} \left (b^6 \pi ^2\right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^2} \, dx-\frac {1}{252} \left (b^6 \pi ^2\right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^2} \, dx-\frac {1}{315} \left (2 b^6 \pi ^2\right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^2} \, dx+\frac {1}{168} \left (b^7 \pi ^3\right ) \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x} \, dx \\ & = -\frac {b^2}{210 x^5}+\frac {b^6 \pi ^2}{336 x}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{210 x^5}+\frac {67 b^6 \pi ^2 \cos \left (b^2 \pi x^2\right )}{5040 x}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{21 x^6}+\frac {b^5 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{168 x^2}-\frac {\operatorname {FresnelC}(b x)^2}{7 x^7}+\frac {b^3 \pi \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{84 x^4}+\frac {13 b^4 \pi \sin \left (b^2 \pi x^2\right )}{2520 x^3}+\frac {1}{168} \left (b^7 \pi ^3\right ) \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x} \, dx+\frac {1}{168} \left (b^8 \pi ^3\right ) \int \sin \left (b^2 \pi x^2\right ) \, dx+\frac {1}{126} \left (b^8 \pi ^3\right ) \int \sin \left (b^2 \pi x^2\right ) \, dx+\frac {1}{315} \left (4 b^8 \pi ^3\right ) \int \sin \left (b^2 \pi x^2\right ) \, dx \\ & = -\frac {b^2}{210 x^5}+\frac {b^6 \pi ^2}{336 x}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{210 x^5}+\frac {67 b^6 \pi ^2 \cos \left (b^2 \pi x^2\right )}{5040 x}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{21 x^6}+\frac {b^5 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{168 x^2}-\frac {\operatorname {FresnelC}(b x)^2}{7 x^7}+\frac {b^7 \pi ^3 \operatorname {FresnelS}\left (\sqrt {2} b x\right )}{72 \sqrt {2}}+\frac {2}{315} \sqrt {2} b^7 \pi ^3 \operatorname {FresnelS}\left (\sqrt {2} b x\right )+\frac {b^3 \pi \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{84 x^4}+\frac {13 b^4 \pi \sin \left (b^2 \pi x^2\right )}{2520 x^3}+\frac {1}{168} \left (b^7 \pi ^3\right ) \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x} \, dx \\ \end{align*}
Not integrable
Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^8} \, dx=\int \frac {\operatorname {FresnelC}(b x)^2}{x^8} \, dx \]
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Not integrable
Time = 0.06 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00
\[\int \frac {\operatorname {FresnelC}\left (b x \right )^{2}}{x^{8}}d x\]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^8} \, dx=\int { \frac {\operatorname {C}\left (b x\right )^{2}}{x^{8}} \,d x } \]
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Not integrable
Time = 1.49 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^8} \, dx=\int \frac {C^{2}\left (b x\right )}{x^{8}}\, dx \]
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Not integrable
Time = 0.23 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^8} \, dx=\int { \frac {\operatorname {C}\left (b x\right )^{2}}{x^{8}} \,d x } \]
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Not integrable
Time = 0.25 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^8} \, dx=\int { \frac {\operatorname {C}\left (b x\right )^{2}}{x^{8}} \,d x } \]
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Not integrable
Time = 5.11 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^8} \, dx=\int \frac {{\mathrm {FresnelC}\left (b\,x\right )}^2}{x^8} \,d x \]
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