Integrand size = 8, antiderivative size = 102 \[ \int \frac {\operatorname {FresnelC}(b x)}{x^8} \, dx=-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{42 x^6}+\frac {b^5 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{336 x^2}-\frac {\operatorname {FresnelC}(b x)}{7 x^7}+\frac {b^3 \pi \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{168 x^4}+\frac {1}{672} b^7 \pi ^3 \text {Si}\left (\frac {1}{2} b^2 \pi x^2\right ) \]
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Time = 0.09 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6562, 3461, 3378, 3380} \[ \int \frac {\operatorname {FresnelC}(b x)}{x^8} \, dx=-\frac {b \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{42 x^6}+\frac {1}{672} \pi ^3 b^7 \text {Si}\left (\frac {1}{2} b^2 \pi x^2\right )+\frac {\pi ^2 b^5 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{336 x^2}+\frac {\pi b^3 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{168 x^4}-\frac {\operatorname {FresnelC}(b x)}{7 x^7} \]
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Rule 3378
Rule 3380
Rule 3461
Rule 6562
Rubi steps \begin{align*} \text {integral}& = -\frac {\operatorname {FresnelC}(b x)}{7 x^7}+\frac {1}{7} b \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right )}{x^7} \, dx \\ & = -\frac {\operatorname {FresnelC}(b x)}{7 x^7}+\frac {1}{14} b \text {Subst}\left (\int \frac {\cos \left (\frac {1}{2} b^2 \pi x\right )}{x^4} \, dx,x,x^2\right ) \\ & = -\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{42 x^6}-\frac {\operatorname {FresnelC}(b x)}{7 x^7}-\frac {1}{84} \left (b^3 \pi \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 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{42 x^6}-\frac {\operatorname {FresnelC}(b x)}{7 x^7}+\frac {b^3 \pi \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{168 x^4}-\frac {1}{336} \left (b^5 \pi ^2\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {1}{2} b^2 \pi x\right )}{x^2} \, dx,x,x^2\right ) \\ & = -\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{42 x^6}+\frac {b^5 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{336 x^2}-\frac {\operatorname {FresnelC}(b x)}{7 x^7}+\frac {b^3 \pi \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{168 x^4}+\frac {1}{672} \left (b^7 \pi ^3\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {1}{2} b^2 \pi x\right )}{x} \, dx,x,x^2\right ) \\ & = -\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{42 x^6}+\frac {b^5 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{336 x^2}-\frac {\operatorname {FresnelC}(b x)}{7 x^7}+\frac {b^3 \pi \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{168 x^4}+\frac {1}{672} b^7 \pi ^3 \text {Si}\left (\frac {1}{2} b^2 \pi x^2\right ) \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.82 \[ \int \frac {\operatorname {FresnelC}(b x)}{x^8} \, dx=\frac {1}{672} \left (\frac {2 b \left (-8+b^4 \pi ^2 x^4\right ) \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{x^6}-\frac {96 \operatorname {FresnelC}(b x)}{x^7}+\frac {4 b^3 \pi \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4}+b^7 \pi ^3 \text {Si}\left (\frac {1}{2} b^2 \pi x^2\right )\right ) \]
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Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 0.42 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.25
method | result | size |
meijerg | \(-\frac {b \operatorname {hypergeom}\left (\left [-\frac {3}{2}, \frac {1}{4}\right ], \left [-\frac {1}{2}, \frac {1}{2}, \frac {5}{4}\right ], -\frac {x^{4} \pi ^{2} b^{4}}{16}\right )}{6 x^{6}}\) | \(26\) |
parts | \(-\frac {\operatorname {FresnelC}\left (b x \right )}{7 x^{7}}+\frac {b \left (-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{6 x^{6}}-\frac {b^{2} \pi \left (-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{4 x^{4}}+\frac {b^{2} \pi \left (-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{2 x^{2}}-\frac {b^{2} \pi \,\operatorname {Si}\left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{4}\right )}{4}\right )}{6}\right )}{7}\) | \(90\) |
derivativedivides | \(b^{7} \left (-\frac {\operatorname {FresnelC}\left (b x \right )}{7 b^{7} x^{7}}-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{42 b^{6} x^{6}}-\frac {\pi \left (-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{4 b^{4} x^{4}}+\frac {\pi \left (-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{2 b^{2} x^{2}}-\frac {\pi \,\operatorname {Si}\left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{4}\right )}{4}\right )}{42}\right )\) | \(93\) |
default | \(b^{7} \left (-\frac {\operatorname {FresnelC}\left (b x \right )}{7 b^{7} x^{7}}-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{42 b^{6} x^{6}}-\frac {\pi \left (-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{4 b^{4} x^{4}}+\frac {\pi \left (-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{2 b^{2} x^{2}}-\frac {\pi \,\operatorname {Si}\left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{4}\right )}{4}\right )}{42}\right )\) | \(93\) |
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Time = 0.27 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.76 \[ \int \frac {\operatorname {FresnelC}(b x)}{x^8} \, dx=\frac {\pi ^{3} b^{7} x^{7} \operatorname {Si}\left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + 4 \, \pi b^{3} x^{3} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + 2 \, {\left (\pi ^{2} b^{5} x^{5} - 8 \, b x\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - 96 \, \operatorname {C}\left (b x\right )}{672 \, x^{7}} \]
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Time = 0.89 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.43 \[ \int \frac {\operatorname {FresnelC}(b x)}{x^8} \, dx=- \frac {b \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{3}\left (\begin {matrix} - \frac {3}{2}, \frac {1}{4} \\ - \frac {1}{2}, \frac {1}{2}, \frac {5}{4} \end {matrix}\middle | {- \frac {\pi ^{2} b^{4} x^{4}}{16}} \right )}}{24 x^{6} \Gamma \left (\frac {5}{4}\right )} \]
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Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.47 \[ \int \frac {\operatorname {FresnelC}(b x)}{x^8} \, dx=-\frac {1}{224} \, {\left (-i \, \pi ^{3} \Gamma \left (-3, \frac {1}{2} i \, \pi b^{2} x^{2}\right ) + i \, \pi ^{3} \Gamma \left (-3, -\frac {1}{2} i \, \pi b^{2} x^{2}\right )\right )} b^{7} - \frac {\operatorname {C}\left (b x\right )}{7 \, x^{7}} \]
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\[ \int \frac {\operatorname {FresnelC}(b x)}{x^8} \, dx=\int { \frac {\operatorname {C}\left (b x\right )}{x^{8}} \,d x } \]
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Timed out. \[ \int \frac {\operatorname {FresnelC}(b x)}{x^8} \, dx=\int \frac {\mathrm {FresnelC}\left (b\,x\right )}{x^8} \,d x \]
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