Integrand size = 8, antiderivative size = 69 \[ \int \frac {\operatorname {FresnelS}(b x)}{x^5} \, dx=-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{12 x}-\frac {1}{12} b^4 \pi ^2 \operatorname {FresnelS}(b x)-\frac {\operatorname {FresnelS}(b x)}{4 x^4}-\frac {b \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{12 x^3} \]
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Time = 0.03 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6561, 3468, 3469, 3432} \[ \int \frac {\operatorname {FresnelS}(b x)}{x^5} \, dx=-\frac {1}{12} \pi ^2 b^4 \operatorname {FresnelS}(b x)-\frac {b \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{12 x^3}-\frac {\pi b^3 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{12 x}-\frac {\operatorname {FresnelS}(b x)}{4 x^4} \]
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Rule 3432
Rule 3468
Rule 3469
Rule 6561
Rubi steps \begin{align*} \text {integral}& = -\frac {\operatorname {FresnelS}(b x)}{4 x^4}+\frac {1}{4} b \int \frac {\sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4} \, dx \\ & = -\frac {\operatorname {FresnelS}(b x)}{4 x^4}-\frac {b \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{12 x^3}+\frac {1}{12} \left (b^3 \pi \right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right )}{x^2} \, dx \\ & = -\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{12 x}-\frac {\operatorname {FresnelS}(b x)}{4 x^4}-\frac {b \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{12 x^3}-\frac {1}{12} \left (b^5 \pi ^2\right ) \int \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx \\ & = -\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{12 x}-\frac {1}{12} b^4 \pi ^2 \operatorname {FresnelS}(b x)-\frac {\operatorname {FresnelS}(b x)}{4 x^4}-\frac {b \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{12 x^3} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelS}(b x)}{x^5} \, dx=-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{12 x}-\frac {1}{12} b^4 \pi ^2 \operatorname {FresnelS}(b x)-\frac {\operatorname {FresnelS}(b x)}{4 x^4}-\frac {b \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{12 x^3} \]
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Time = 0.40 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.94
method | result | size |
derivativedivides | \(b^{4} \left (-\frac {\operatorname {FresnelS}\left (b x \right )}{4 b^{4} x^{4}}-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{12 b^{3} x^{3}}+\frac {\pi \left (-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b x}-\pi \,\operatorname {FresnelS}\left (b x \right )\right )}{12}\right )\) | \(65\) |
default | \(b^{4} \left (-\frac {\operatorname {FresnelS}\left (b x \right )}{4 b^{4} x^{4}}-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{12 b^{3} x^{3}}+\frac {\pi \left (-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b x}-\pi \,\operatorname {FresnelS}\left (b x \right )\right )}{12}\right )\) | \(65\) |
meijerg | \(\frac {\pi ^{2} b^{4} \left (-\frac {32 \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3 \pi x b}-\frac {32 \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3 \pi ^{2} x^{3} b^{3}}-\frac {32 \left (x^{4} \pi ^{2} b^{4}+3\right ) \operatorname {FresnelS}\left (b x \right )}{3 \pi ^{2} x^{4} b^{4}}\right )}{128}\) | \(79\) |
parts | \(-\frac {\operatorname {FresnelS}\left (b x \right )}{4 x^{4}}+\frac {b \left (-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3 x^{3}}+\frac {b^{2} \pi \left (-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{x}-\frac {b^{2} \pi ^{\frac {3}{2}} \operatorname {FresnelS}\left (\frac {\sqrt {\pi }\, b^{2} x}{\sqrt {b^{2} \pi }}\right )}{\sqrt {b^{2} \pi }}\right )}{3}\right )}{4}\) | \(83\) |
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none
Time = 0.27 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.78 \[ \int \frac {\operatorname {FresnelS}(b x)}{x^5} \, dx=-\frac {\pi b^{3} x^{3} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + b x \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + {\left (\pi ^{2} b^{4} x^{4} + 3\right )} \operatorname {S}\left (b x\right )}{12 \, x^{4}} \]
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Time = 0.67 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.59 \[ \int \frac {\operatorname {FresnelS}(b x)}{x^5} \, dx=\frac {\pi ^{2} b^{4} S\left (b x\right ) \Gamma \left (- \frac {1}{4}\right )}{64 \Gamma \left (\frac {7}{4}\right )} + \frac {\pi b^{3} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (- \frac {1}{4}\right )}{64 x \Gamma \left (\frac {7}{4}\right )} + \frac {b \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (- \frac {1}{4}\right )}{64 x^{3} \Gamma \left (\frac {7}{4}\right )} + \frac {3 S\left (b x\right ) \Gamma \left (- \frac {1}{4}\right )}{64 x^{4} \Gamma \left (\frac {7}{4}\right )} \]
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Result contains complex when optimal does not.
Time = 0.36 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.88 \[ \int \frac {\operatorname {FresnelS}(b x)}{x^5} \, dx=-\frac {\sqrt {\frac {1}{2}} \left (\pi x^{2}\right )^{\frac {3}{2}} {\left (-\left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, \frac {1}{2} i \, \pi b^{2} x^{2}\right ) + \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, -\frac {1}{2} i \, \pi b^{2} x^{2}\right )\right )} b^{4}}{64 \, x^{3}} - \frac {\operatorname {S}\left (b x\right )}{4 \, x^{4}} \]
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\[ \int \frac {\operatorname {FresnelS}(b x)}{x^5} \, dx=\int { \frac {\operatorname {S}\left (b x\right )}{x^{5}} \,d x } \]
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Timed out. \[ \int \frac {\operatorname {FresnelS}(b x)}{x^5} \, dx=\int \frac {\mathrm {FresnelS}\left (b\,x\right )}{x^5} \,d x \]
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