Integrand size = 8, antiderivative size = 27 \[ \int \frac {\operatorname {FresnelS}(b x)}{x^2} \, dx=-\frac {\operatorname {FresnelS}(b x)}{x}+\frac {1}{2} b \text {Si}\left (\frac {1}{2} b^2 \pi x^2\right ) \]
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Time = 0.01 (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 = {6561, 3456} \[ \int \frac {\operatorname {FresnelS}(b x)}{x^2} \, dx=\frac {1}{2} b \text {Si}\left (\frac {1}{2} b^2 \pi x^2\right )-\frac {\operatorname {FresnelS}(b x)}{x} \]
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Rule 3456
Rule 6561
Rubi steps \begin{align*} \text {integral}& = -\frac {\operatorname {FresnelS}(b x)}{x}+b \int \frac {\sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x} \, dx \\ & = -\frac {\operatorname {FresnelS}(b x)}{x}+\frac {1}{2} b \text {Si}\left (\frac {1}{2} b^2 \pi x^2\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelS}(b x)}{x^2} \, dx=-\frac {\operatorname {FresnelS}(b x)}{x}+\frac {1}{2} b \text {Si}\left (\frac {1}{2} b^2 \pi x^2\right ) \]
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Time = 0.44 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89
method | result | size |
parts | \(-\frac {\operatorname {FresnelS}\left (b x \right )}{x}+\frac {b \,\operatorname {Si}\left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{2}\) | \(24\) |
derivativedivides | \(b \left (-\frac {\operatorname {FresnelS}\left (b x \right )}{b x}+\frac {\operatorname {Si}\left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{2}\right )\) | \(28\) |
default | \(b \left (-\frac {\operatorname {FresnelS}\left (b x \right )}{b x}+\frac {\operatorname {Si}\left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{2}\right )\) | \(28\) |
meijerg | \(\frac {b^{3} \pi \,x^{2} \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {3}{4}\right ], \left [\frac {3}{2}, \frac {3}{2}, \frac {7}{4}\right ], -\frac {x^{4} \pi ^{2} b^{4}}{16}\right )}{12}\) | \(29\) |
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none
Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {\operatorname {FresnelS}(b x)}{x^2} \, dx=\frac {b x \operatorname {Si}\left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - 2 \, \operatorname {S}\left (b x\right )}{2 \, x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (20) = 40\).
Time = 0.36 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.56 \[ \int \frac {\operatorname {FresnelS}(b x)}{x^2} \, dx=\frac {\pi b^{3} x^{2} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{3}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {3}{2}, \frac {3}{2}, \frac {7}{4} \end {matrix}\middle | {- \frac {\pi ^{2} b^{4} x^{4}}{16}} \right )}}{16 \Gamma \left (\frac {7}{4}\right )} \]
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.41 \[ \int \frac {\operatorname {FresnelS}(b x)}{x^2} \, dx=-\frac {1}{4} \, b {\left (i \, {\rm Ei}\left (\frac {1}{2} i \, \pi b^{2} x^{2}\right ) - i \, {\rm Ei}\left (-\frac {1}{2} i \, \pi b^{2} x^{2}\right )\right )} - \frac {\operatorname {S}\left (b x\right )}{x} \]
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\[ \int \frac {\operatorname {FresnelS}(b x)}{x^2} \, dx=\int { \frac {\operatorname {S}\left (b x\right )}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\operatorname {FresnelS}(b x)}{x^2} \, dx=\int \frac {\mathrm {FresnelS}\left (b\,x\right )}{x^2} \,d x \]
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