Integrand size = 10, antiderivative size = 127 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^5} \, dx=-\frac {b^2}{24 x^2}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{24 x^2}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{6 x^3}-\frac {1}{12} b^4 \pi ^2 \operatorname {FresnelC}(b x)^2-\frac {\operatorname {FresnelC}(b x)^2}{4 x^4}+\frac {b^3 \pi \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{6 x}-\frac {1}{12} b^4 \pi \text {Si}\left (b^2 \pi x^2\right ) \]
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Time = 0.10 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {6566, 6592, 6600, 6576, 30, 3456, 3461, 3378, 3380} \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^5} \, dx=-\frac {1}{12} \pi ^2 b^4 \operatorname {FresnelC}(b x)^2-\frac {b \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^3}-\frac {b^2}{24 x^2}-\frac {b^2 \cos \left (\pi b^2 x^2\right )}{24 x^2}-\frac {1}{12} \pi b^4 \text {Si}\left (b^2 \pi x^2\right )+\frac {\pi b^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x}-\frac {\operatorname {FresnelC}(b x)^2}{4 x^4} \]
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Rule 30
Rule 3378
Rule 3380
Rule 3456
Rule 3461
Rule 6566
Rule 6576
Rule 6592
Rule 6600
Rubi steps \begin{align*} \text {integral}& = -\frac {\operatorname {FresnelC}(b x)^2}{4 x^4}+\frac {1}{2} b \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^4} \, dx \\ & = -\frac {b^2}{24 x^2}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{6 x^3}-\frac {\operatorname {FresnelC}(b x)^2}{4 x^4}+\frac {1}{12} b^2 \int \frac {\cos \left (b^2 \pi x^2\right )}{x^3} \, dx-\frac {1}{6} \left (b^3 \pi \right ) \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2} \, dx \\ & = -\frac {b^2}{24 x^2}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{6 x^3}-\frac {\operatorname {FresnelC}(b x)^2}{4 x^4}+\frac {b^3 \pi \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{6 x}+\frac {1}{24} b^2 \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )-\frac {1}{12} \left (b^4 \pi \right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x} \, dx-\frac {1}{6} \left (b^5 \pi ^2\right ) \int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx \\ & = -\frac {b^2}{24 x^2}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{24 x^2}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{6 x^3}-\frac {\operatorname {FresnelC}(b x)^2}{4 x^4}+\frac {b^3 \pi \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{6 x}-\frac {1}{24} b^4 \pi \text {Si}\left (b^2 \pi x^2\right )-\frac {1}{24} \left (b^4 \pi \right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )-\frac {1}{6} \left (b^4 \pi ^2\right ) \text {Subst}(\int x \, dx,x,\operatorname {FresnelC}(b x)) \\ & = -\frac {b^2}{24 x^2}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{24 x^2}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{6 x^3}-\frac {1}{12} b^4 \pi ^2 \operatorname {FresnelC}(b x)^2-\frac {\operatorname {FresnelC}(b x)^2}{4 x^4}+\frac {b^3 \pi \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{6 x}-\frac {1}{12} b^4 \pi \text {Si}\left (b^2 \pi x^2\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^5} \, dx=-\frac {b^2}{24 x^2}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{24 x^2}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{6 x^3}-\frac {1}{12} b^4 \pi ^2 \operatorname {FresnelC}(b x)^2-\frac {\operatorname {FresnelC}(b x)^2}{4 x^4}+\frac {b^3 \pi \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{6 x}-\frac {1}{12} b^4 \pi \text {Si}\left (b^2 \pi x^2\right ) \]
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\[\int \frac {\operatorname {FresnelC}\left (b x \right )^{2}}{x^{5}}d x\]
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none
Time = 0.26 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.80 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^5} \, dx=-\frac {\pi b^{4} x^{4} \operatorname {Si}\left (\pi b^{2} x^{2}\right ) - 2 \, \pi b^{3} x^{3} \operatorname {C}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + b^{2} x^{2} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} + 2 \, b x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right ) + {\left (\pi ^{2} b^{4} x^{4} + 3\right )} \operatorname {C}\left (b x\right )^{2}}{12 \, x^{4}} \]
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\[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^5} \, dx=\int \frac {C^{2}\left (b x\right )}{x^{5}}\, dx \]
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\[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^5} \, dx=\int { \frac {\operatorname {C}\left (b x\right )^{2}}{x^{5}} \,d x } \]
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\[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^5} \, dx=\int { \frac {\operatorname {C}\left (b x\right )^{2}}{x^{5}} \,d x } \]
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Timed out. \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^5} \, dx=\int \frac {{\mathrm {FresnelC}\left (b\,x\right )}^2}{x^5} \,d x \]
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