Integrand size = 10, antiderivative size = 10 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^{10}} \, dx=-\frac {b^2}{504 x^7}+\frac {b^6 \pi ^2}{5184 x^3}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{504 x^7}+\frac {187 b^6 \pi ^2 \cos \left (b^2 \pi x^2\right )}{181440 x^3}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{36 x^8}+\frac {b^5 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{864 x^4}-\frac {\operatorname {FresnelC}(b x)^2}{9 x^9}+\frac {853 b^9 \pi ^4 \operatorname {FresnelC}\left (\sqrt {2} b x\right )}{181440 \sqrt {2}}+\frac {b^3 \pi \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{216 x^6}-\frac {b^7 \pi ^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{1728 x^2}+\frac {19 b^4 \pi \sin \left (b^2 \pi x^2\right )}{15120 x^5}-\frac {853 b^8 \pi ^3 \sin \left (b^2 \pi x^2\right )}{362880 x}+\frac {b^9 \pi ^4 \text {Int}\left (\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x},x\right )}{1728} \]
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Not integrable
Time = 0.23 (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 = {} \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^{10}} \, dx=\int \frac {\operatorname {FresnelC}(b x)^2}{x^{10}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = -\frac {\operatorname {FresnelC}(b x)^2}{9 x^9}+\frac {1}{9} (2 b) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^9} \, dx \\ & = -\frac {b^2}{504 x^7}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{36 x^8}-\frac {\operatorname {FresnelC}(b x)^2}{9 x^9}+\frac {1}{72} b^2 \int \frac {\cos \left (b^2 \pi x^2\right )}{x^8} \, dx-\frac {1}{36} \left (b^3 \pi \right ) \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^7} \, dx \\ & = -\frac {b^2}{504 x^7}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{504 x^7}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{36 x^8}-\frac {\operatorname {FresnelC}(b x)^2}{9 x^9}+\frac {b^3 \pi \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{216 x^6}-\frac {1}{432} \left (b^4 \pi \right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^6} \, dx-\frac {1}{252} \left (b^4 \pi \right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^6} \, dx-\frac {1}{216} \left (b^5 \pi ^2\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^5} \, dx \\ & = -\frac {b^2}{504 x^7}+\frac {b^6 \pi ^2}{5184 x^3}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{504 x^7}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{36 x^8}+\frac {b^5 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{864 x^4}-\frac {\operatorname {FresnelC}(b x)^2}{9 x^9}+\frac {b^3 \pi \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{216 x^6}+\frac {19 b^4 \pi \sin \left (b^2 \pi x^2\right )}{15120 x^5}-\frac {\left (b^6 \pi ^2\right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^4} \, dx}{1728}-\frac {\left (b^6 \pi ^2\right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^4} \, dx}{1080}-\frac {1}{630} \left (b^6 \pi ^2\right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^4} \, dx+\frac {1}{864} \left (b^7 \pi ^3\right ) \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^3} \, dx \\ & = -\frac {b^2}{504 x^7}+\frac {b^6 \pi ^2}{5184 x^3}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{504 x^7}+\frac {187 b^6 \pi ^2 \cos \left (b^2 \pi x^2\right )}{181440 x^3}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{36 x^8}+\frac {b^5 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{864 x^4}-\frac {\operatorname {FresnelC}(b x)^2}{9 x^9}+\frac {b^3 \pi \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{216 x^6}-\frac {b^7 \pi ^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{1728 x^2}+\frac {19 b^4 \pi \sin \left (b^2 \pi x^2\right )}{15120 x^5}+\frac {\left (b^8 \pi ^3\right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2} \, dx}{3456}+\frac {\left (b^8 \pi ^3\right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2} \, dx}{2592}+\frac {\left (b^8 \pi ^3\right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2} \, dx}{1620}+\frac {1}{945} \left (b^8 \pi ^3\right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2} \, dx+\frac {\left (b^9 \pi ^4\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x} \, dx}{1728} \\ & = -\frac {b^2}{504 x^7}+\frac {b^6 \pi ^2}{5184 x^3}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{504 x^7}+\frac {187 b^6 \pi ^2 \cos \left (b^2 \pi x^2\right )}{181440 x^3}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{36 x^8}+\frac {b^5 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{864 x^4}-\frac {\operatorname {FresnelC}(b x)^2}{9 x^9}+\frac {b^3 \pi \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{216 x^6}-\frac {b^7 \pi ^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{1728 x^2}+\frac {19 b^4 \pi \sin \left (b^2 \pi x^2\right )}{15120 x^5}-\frac {853 b^8 \pi ^3 \sin \left (b^2 \pi x^2\right )}{362880 x}+\frac {\left (b^9 \pi ^4\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x} \, dx}{1728}+\frac {\left (b^{10} \pi ^4\right ) \int \cos \left (b^2 \pi x^2\right ) \, dx}{1728}+\frac {\left (b^{10} \pi ^4\right ) \int \cos \left (b^2 \pi x^2\right ) \, dx}{1296}+\frac {1}{810} \left (b^{10} \pi ^4\right ) \int \cos \left (b^2 \pi x^2\right ) \, dx+\frac {1}{945} \left (2 b^{10} \pi ^4\right ) \int \cos \left (b^2 \pi x^2\right ) \, dx \\ & = -\frac {b^2}{504 x^7}+\frac {b^6 \pi ^2}{5184 x^3}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{504 x^7}+\frac {187 b^6 \pi ^2 \cos \left (b^2 \pi x^2\right )}{181440 x^3}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{36 x^8}+\frac {b^5 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{864 x^4}-\frac {\operatorname {FresnelC}(b x)^2}{9 x^9}+\frac {67 b^9 \pi ^4 \operatorname {FresnelC}\left (\sqrt {2} b x\right )}{25920 \sqrt {2}}+\frac {1}{945} \sqrt {2} b^9 \pi ^4 \operatorname {FresnelC}\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 )}{216 x^6}-\frac {b^7 \pi ^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{1728 x^2}+\frac {19 b^4 \pi \sin \left (b^2 \pi x^2\right )}{15120 x^5}-\frac {853 b^8 \pi ^3 \sin \left (b^2 \pi x^2\right )}{362880 x}+\frac {\left (b^9 \pi ^4\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x} \, dx}{1728} \\ \end{align*}
Not integrable
Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^{10}} \, dx=\int \frac {\operatorname {FresnelC}(b x)^2}{x^{10}} \, dx \]
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Not integrable
Time = 0.06 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00
\[\int \frac {\operatorname {FresnelC}\left (b x \right )^{2}}{x^{10}}d x\]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^{10}} \, dx=\int { \frac {\operatorname {C}\left (b x\right )^{2}}{x^{10}} \,d x } \]
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Not integrable
Time = 2.18 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^{10}} \, dx=\int \frac {C^{2}\left (b x\right )}{x^{10}}\, dx \]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^{10}} \, dx=\int { \frac {\operatorname {C}\left (b x\right )^{2}}{x^{10}} \,d x } \]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^{10}} \, dx=\int { \frac {\operatorname {C}\left (b x\right )^{2}}{x^{10}} \,d x } \]
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Not integrable
Time = 4.72 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^{10}} \, dx=\int \frac {{\mathrm {FresnelC}\left (b\,x\right )}^2}{x^{10}} \,d x \]
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