Integrand size = 8, antiderivative size = 27 \[ \int \frac {\operatorname {FresnelC}(b x)}{x^2} \, dx=\frac {1}{2} b \operatorname {CosIntegral}\left (\frac {1}{2} b^2 \pi x^2\right )-\frac {\operatorname {FresnelC}(b x)}{x} \]
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Time = 0.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6562, 3457} \[ \int \frac {\operatorname {FresnelC}(b x)}{x^2} \, dx=\frac {1}{2} b \operatorname {CosIntegral}\left (\frac {1}{2} b^2 \pi x^2\right )-\frac {\operatorname {FresnelC}(b x)}{x} \]
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Rule 3457
Rule 6562
Rubi steps \begin{align*} \text {integral}& = -\frac {\operatorname {FresnelC}(b x)}{x}+b \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right )}{x} \, dx \\ & = \frac {1}{2} b \operatorname {CosIntegral}\left (\frac {1}{2} b^2 \pi x^2\right )-\frac {\operatorname {FresnelC}(b x)}{x} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelC}(b x)}{x^2} \, dx=\frac {1}{2} b \operatorname {CosIntegral}\left (\frac {1}{2} b^2 \pi x^2\right )-\frac {\operatorname {FresnelC}(b x)}{x} \]
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Time = 0.44 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89
method | result | size |
parts | \(\frac {b \,\operatorname {Ci}\left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{2}-\frac {\operatorname {FresnelC}\left (b x \right )}{x}\) | \(24\) |
derivativedivides | \(b \left (-\frac {\operatorname {FresnelC}\left (b x \right )}{b x}+\frac {\operatorname {Ci}\left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{2}\right )\) | \(28\) |
default | \(b \left (-\frac {\operatorname {FresnelC}\left (b x \right )}{b x}+\frac {\operatorname {Ci}\left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{2}\right )\) | \(28\) |
meijerg | \(\frac {b \sqrt {\pi }\, \left (-\frac {\pi ^{\frac {3}{2}} x^{4} b^{4} \operatorname {hypergeom}\left (\left [1, 1, \frac {5}{4}\right ], \left [\frac {3}{2}, 2, 2, \frac {9}{4}\right ], -\frac {x^{4} \pi ^{2} b^{4}}{16}\right )}{10}+\frac {8 \gamma -8 \ln \left (2\right )-16+16 \ln \left (x \right )+8 \ln \left (\pi \right )+16 \ln \left (b \right )}{\sqrt {\pi }}\right )}{16}\) | \(66\) |
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none
Time = 0.26 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {\operatorname {FresnelC}(b x)}{x^2} \, dx=\frac {b x \operatorname {Ci}\left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - 2 \, \operatorname {C}\left (b x\right )}{2 \, x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (20) = 40\).
Time = 0.55 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.96 \[ \int \frac {\operatorname {FresnelC}(b x)}{x^2} \, dx=- \frac {\pi ^{2} b^{5} x^{4} \Gamma \left (\frac {5}{4}\right ) {{}_{3}F_{4}\left (\begin {matrix} 1, 1, \frac {5}{4} \\ \frac {3}{2}, 2, 2, \frac {9}{4} \end {matrix}\middle | {- \frac {\pi ^{2} b^{4} x^{4}}{16}} \right )}}{128 \Gamma \left (\frac {9}{4}\right )} + \frac {b \log {\left (b^{4} x^{4} \right )}}{4} \]
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26 \[ \int \frac {\operatorname {FresnelC}(b x)}{x^2} \, dx=\frac {1}{4} \, b {\left ({\rm Ei}\left (\frac {1}{2} i \, \pi b^{2} x^{2}\right ) + {\rm Ei}\left (-\frac {1}{2} i \, \pi b^{2} x^{2}\right )\right )} - \frac {\operatorname {C}\left (b x\right )}{x} \]
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\[ \int \frac {\operatorname {FresnelC}(b x)}{x^2} \, dx=\int { \frac {\operatorname {C}\left (b x\right )}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\operatorname {FresnelC}(b x)}{x^2} \, dx=\int \frac {\mathrm {FresnelC}\left (b\,x\right )}{x^2} \,d x \]
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