Integrand size = 8, antiderivative size = 127 \[ \int \frac {\operatorname {FresnelS}(b x)}{x^{10}} \, dx=-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{432 x^6}+\frac {b^7 \pi ^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{3456 x^2}-\frac {\operatorname {FresnelS}(b x)}{9 x^9}-\frac {b \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{72 x^8}+\frac {b^5 \pi ^2 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{1728 x^4}+\frac {b^9 \pi ^4 \text {Si}\left (\frac {1}{2} b^2 \pi x^2\right )}{6912} \]
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Time = 0.11 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6561, 3460, 3378, 3380} \[ \int \frac {\operatorname {FresnelS}(b x)}{x^{10}} \, dx=-\frac {b \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{72 x^8}+\frac {\pi ^4 b^9 \text {Si}\left (\frac {1}{2} b^2 \pi x^2\right )}{6912}+\frac {\pi ^3 b^7 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{3456 x^2}+\frac {\pi ^2 b^5 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{1728 x^4}-\frac {\pi b^3 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{432 x^6}-\frac {\operatorname {FresnelS}(b x)}{9 x^9} \]
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Rule 3378
Rule 3380
Rule 3460
Rule 6561
Rubi steps \begin{align*} \text {integral}& = -\frac {\operatorname {FresnelS}(b x)}{9 x^9}+\frac {1}{9} b \int \frac {\sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^9} \, dx \\ & = -\frac {\operatorname {FresnelS}(b x)}{9 x^9}+\frac {1}{18} b \text {Subst}\left (\int \frac {\sin \left (\frac {1}{2} b^2 \pi x\right )}{x^5} \, dx,x,x^2\right ) \\ & = -\frac {\operatorname {FresnelS}(b x)}{9 x^9}-\frac {b \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{72 x^8}+\frac {1}{144} \left (b^3 \pi \right ) \text {Subst}\left (\int \frac {\cos \left (\frac {1}{2} b^2 \pi x\right )}{x^4} \, dx,x,x^2\right ) \\ & = -\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{432 x^6}-\frac {\operatorname {FresnelS}(b x)}{9 x^9}-\frac {b \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{72 x^8}-\frac {1}{864} \left (b^5 \pi ^2\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {1}{2} b^2 \pi x\right )}{x^3} \, dx,x,x^2\right ) \\ & = -\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{432 x^6}-\frac {\operatorname {FresnelS}(b x)}{9 x^9}-\frac {b \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{72 x^8}+\frac {b^5 \pi ^2 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{1728 x^4}-\frac {\left (b^7 \pi ^3\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {1}{2} b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )}{3456} \\ & = -\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{432 x^6}+\frac {b^7 \pi ^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{3456 x^2}-\frac {\operatorname {FresnelS}(b x)}{9 x^9}-\frac {b \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{72 x^8}+\frac {b^5 \pi ^2 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{1728 x^4}+\frac {\left (b^9 \pi ^4\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {1}{2} b^2 \pi x\right )}{x} \, dx,x,x^2\right )}{6912} \\ & = -\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{432 x^6}+\frac {b^7 \pi ^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{3456 x^2}-\frac {\operatorname {FresnelS}(b x)}{9 x^9}-\frac {b \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{72 x^8}+\frac {b^5 \pi ^2 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{1728 x^4}+\frac {b^9 \pi ^4 \text {Si}\left (\frac {1}{2} b^2 \pi x^2\right )}{6912} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.76 \[ \int \frac {\operatorname {FresnelS}(b x)}{x^{10}} \, dx=\frac {\frac {2 b^3 \pi \left (-8+b^4 \pi ^2 x^4\right ) \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{x^6}-\frac {768 \operatorname {FresnelS}(b x)}{x^9}+\frac {4 b \left (-24+b^4 \pi ^2 x^4\right ) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8}+b^9 \pi ^4 \text {Si}\left (\frac {1}{2} b^2 \pi x^2\right )}{6912} \]
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Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 0.45 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.23
method | result | size |
meijerg | \(-\frac {\pi \,b^{3} \operatorname {hypergeom}\left (\left [-\frac {3}{2}, \frac {3}{4}\right ], \left [-\frac {1}{2}, \frac {3}{2}, \frac {7}{4}\right ], -\frac {x^{4} \pi ^{2} b^{4}}{16}\right )}{36 x^{6}}\) | \(29\) |
parts | \(-\frac {\operatorname {FresnelS}\left (b x \right )}{9 x^{9}}+\frac {b \left (-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{8 x^{8}}+\frac {b^{2} \pi \left (-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{6 x^{6}}-\frac {b^{2} \pi \left (-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{4 x^{4}}+\frac {b^{2} \pi \left (-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{2 x^{2}}-\frac {b^{2} \pi \,\operatorname {Si}\left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{4}\right )}{4}\right )}{6}\right )}{8}\right )}{9}\) | \(112\) |
derivativedivides | \(b^{9} \left (-\frac {\operatorname {FresnelS}\left (b x \right )}{9 b^{9} x^{9}}-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{72 b^{8} x^{8}}+\frac {\pi \left (-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{6 b^{6} x^{6}}-\frac {\pi \left (-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{4 b^{4} x^{4}}+\frac {\pi \left (-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{2 b^{2} x^{2}}-\frac {\pi \,\operatorname {Si}\left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{4}\right )}{4}\right )}{6}\right )}{72}\right )\) | \(115\) |
default | \(b^{9} \left (-\frac {\operatorname {FresnelS}\left (b x \right )}{9 b^{9} x^{9}}-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{72 b^{8} x^{8}}+\frac {\pi \left (-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{6 b^{6} x^{6}}-\frac {\pi \left (-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{4 b^{4} x^{4}}+\frac {\pi \left (-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{2 b^{2} x^{2}}-\frac {\pi \,\operatorname {Si}\left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{4}\right )}{4}\right )}{6}\right )}{72}\right )\) | \(115\) |
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Time = 0.28 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.72 \[ \int \frac {\operatorname {FresnelS}(b x)}{x^{10}} \, dx=\frac {\pi ^{4} b^{9} x^{9} \operatorname {Si}\left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + 2 \, {\left (\pi ^{3} b^{7} x^{7} - 8 \, \pi b^{3} x^{3}\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + 4 \, {\left (\pi ^{2} b^{5} x^{5} - 24 \, b x\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - 768 \, \operatorname {S}\left (b x\right )}{6912 \, x^{9}} \]
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Time = 1.58 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.38 \[ \int \frac {\operatorname {FresnelS}(b x)}{x^{10}} \, dx=- \frac {\pi b^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{3}\left (\begin {matrix} - \frac {3}{2}, \frac {3}{4} \\ - \frac {1}{2}, \frac {3}{2}, \frac {7}{4} \end {matrix}\middle | {- \frac {\pi ^{2} b^{4} x^{4}}{16}} \right )}}{48 x^{6} \Gamma \left (\frac {7}{4}\right )} \]
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.38 \[ \int \frac {\operatorname {FresnelS}(b x)}{x^{10}} \, dx=-\frac {1}{576} \, {\left (i \, \pi ^{4} \Gamma \left (-4, \frac {1}{2} i \, \pi b^{2} x^{2}\right ) - i \, \pi ^{4} \Gamma \left (-4, -\frac {1}{2} i \, \pi b^{2} x^{2}\right )\right )} b^{9} - \frac {\operatorname {S}\left (b x\right )}{9 \, x^{9}} \]
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\[ \int \frac {\operatorname {FresnelS}(b x)}{x^{10}} \, dx=\int { \frac {\operatorname {S}\left (b x\right )}{x^{10}} \,d x } \]
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Timed out. \[ \int \frac {\operatorname {FresnelS}(b x)}{x^{10}} \, dx=\int \frac {\mathrm {FresnelS}\left (b\,x\right )}{x^{10}} \,d x \]
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