Integrand size = 8, antiderivative size = 94 \[ \int \frac {\operatorname {FresnelS}(b x)}{x^7} \, dx=-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{90 x^3}-\frac {1}{90} b^6 \pi ^3 \operatorname {FresnelC}(b x)-\frac {\operatorname {FresnelS}(b x)}{6 x^6}-\frac {b \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{30 x^5}+\frac {b^5 \pi ^2 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{90 x} \]
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Time = 0.04 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6561, 3468, 3469, 3433} \[ \int \frac {\operatorname {FresnelS}(b x)}{x^7} \, dx=-\frac {1}{90} \pi ^3 b^6 \operatorname {FresnelC}(b x)-\frac {b \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{30 x^5}+\frac {\pi ^2 b^5 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{90 x}-\frac {\pi b^3 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{90 x^3}-\frac {\operatorname {FresnelS}(b x)}{6 x^6} \]
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Rule 3433
Rule 3468
Rule 3469
Rule 6561
Rubi steps \begin{align*} \text {integral}& = -\frac {\operatorname {FresnelS}(b x)}{6 x^6}+\frac {1}{6} b \int \frac {\sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^6} \, dx \\ & = -\frac {\operatorname {FresnelS}(b x)}{6 x^6}-\frac {b \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{30 x^5}+\frac {1}{30} \left (b^3 \pi \right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4} \, dx \\ & = -\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{90 x^3}-\frac {\operatorname {FresnelS}(b x)}{6 x^6}-\frac {b \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{30 x^5}-\frac {1}{90} \left (b^5 \pi ^2\right ) \int \frac {\sin \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 )}{90 x^3}-\frac {\operatorname {FresnelS}(b x)}{6 x^6}-\frac {b \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{30 x^5}+\frac {b^5 \pi ^2 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{90 x}-\frac {1}{90} \left (b^7 \pi ^3\right ) \int \cos \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 )}{90 x^3}-\frac {1}{90} b^6 \pi ^3 \operatorname {FresnelC}(b x)-\frac {\operatorname {FresnelS}(b x)}{6 x^6}-\frac {b \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{30 x^5}+\frac {b^5 \pi ^2 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{90 x} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.81 \[ \int \frac {\operatorname {FresnelS}(b x)}{x^7} \, dx=\frac {1}{90} \left (-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{x^3}-b^6 \pi ^3 \operatorname {FresnelC}(b x)-\frac {15 \operatorname {FresnelS}(b x)}{x^6}+\frac {b \left (-3+b^4 \pi ^2 x^4\right ) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5}\right ) \]
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Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 0.39 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.31
method | result | size |
meijerg | \(-\frac {\pi \,b^{3} \operatorname {hypergeom}\left (\left [-\frac {3}{4}, \frac {3}{4}\right ], \left [\frac {1}{4}, \frac {3}{2}, \frac {7}{4}\right ], -\frac {x^{4} \pi ^{2} b^{4}}{16}\right )}{18 x^{3}}\) | \(29\) |
derivativedivides | \(b^{6} \left (-\frac {\operatorname {FresnelS}\left (b x \right )}{6 b^{6} x^{6}}-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{30 b^{5} x^{5}}+\frac {\pi \left (-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3 b^{3} x^{3}}-\frac {\pi \left (-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b x}+\pi \,\operatorname {FresnelC}\left (b x \right )\right )}{3}\right )}{30}\right )\) | \(86\) |
default | \(b^{6} \left (-\frac {\operatorname {FresnelS}\left (b x \right )}{6 b^{6} x^{6}}-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{30 b^{5} x^{5}}+\frac {\pi \left (-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3 b^{3} x^{3}}-\frac {\pi \left (-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b x}+\pi \,\operatorname {FresnelC}\left (b x \right )\right )}{3}\right )}{30}\right )\) | \(86\) |
parts | \(-\frac {\operatorname {FresnelS}\left (b x \right )}{6 x^{6}}+\frac {b \left (-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{5 x^{5}}+\frac {b^{2} \pi \left (-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3 x^{3}}-\frac {b^{2} \pi \left (-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{x}+\frac {b^{2} \pi ^{\frac {3}{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\pi }\, b^{2} x}{\sqrt {b^{2} \pi }}\right )}{\sqrt {b^{2} \pi }}\right )}{3}\right )}{5}\right )}{6}\) | \(104\) |
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Time = 0.27 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.85 \[ \int \frac {\operatorname {FresnelS}(b x)}{x^7} \, dx=-\frac {\pi ^{3} \sqrt {b^{2}} b^{5} x^{6} \operatorname {C}\left (\sqrt {b^{2}} x\right ) + \pi b^{3} x^{3} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - {\left (\pi ^{2} b^{5} x^{5} - 3 \, b x\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + 15 \, \operatorname {S}\left (b x\right )}{90 \, x^{6}} \]
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Time = 0.75 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.60 \[ \int \frac {\operatorname {FresnelS}(b x)}{x^7} \, dx=\frac {\pi b^{3} \Gamma \left (- \frac {3}{4}\right ) \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{3}\left (\begin {matrix} - \frac {3}{4}, \frac {3}{4} \\ \frac {1}{4}, \frac {3}{2}, \frac {7}{4} \end {matrix}\middle | {- \frac {\pi ^{2} b^{4} x^{4}}{16}} \right )}}{32 x^{3} \Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {7}{4}\right )} \]
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Result contains complex when optimal does not.
Time = 0.36 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.65 \[ \int \frac {\operatorname {FresnelS}(b x)}{x^7} \, dx=-\frac {\sqrt {\frac {1}{2}} \left (\pi x^{2}\right )^{\frac {5}{2}} {\left (-\left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {5}{2}, \frac {1}{2} i \, \pi b^{2} x^{2}\right ) + \left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {5}{2}, -\frac {1}{2} i \, \pi b^{2} x^{2}\right )\right )} b^{6}}{192 \, x^{5}} - \frac {\operatorname {S}\left (b x\right )}{6 \, x^{6}} \]
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\[ \int \frac {\operatorname {FresnelS}(b x)}{x^7} \, dx=\int { \frac {\operatorname {S}\left (b x\right )}{x^{7}} \,d x } \]
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Timed out. \[ \int \frac {\operatorname {FresnelS}(b x)}{x^7} \, dx=\int \frac {\mathrm {FresnelS}\left (b\,x\right )}{x^7} \,d x \]
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