Integrand size = 8, antiderivative size = 99 \[ \int x^5 \operatorname {FresnelC}(b x) \, dx=-\frac {5 x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{6 b^3 \pi ^2}+\frac {1}{6} x^6 \operatorname {FresnelC}(b x)-\frac {5 \operatorname {FresnelS}(b x)}{2 b^6 \pi ^3}+\frac {5 x \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{2 b^5 \pi ^3}-\frac {x^5 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{6 b \pi } \]
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Time = 0.04 (sec) , antiderivative size = 99, 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 = {6562, 3467, 3466, 3432} \[ \int x^5 \operatorname {FresnelC}(b x) \, dx=-\frac {5 \operatorname {FresnelS}(b x)}{2 \pi ^3 b^6}-\frac {x^5 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{6 \pi b}+\frac {5 x \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{2 \pi ^3 b^5}-\frac {5 x^3 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 \pi ^2 b^3}+\frac {1}{6} x^6 \operatorname {FresnelC}(b x) \]
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Rule 3432
Rule 3466
Rule 3467
Rule 6562
Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} x^6 \operatorname {FresnelC}(b x)-\frac {1}{6} b \int x^6 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx \\ & = \frac {1}{6} x^6 \operatorname {FresnelC}(b x)-\frac {x^5 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{6 b \pi }+\frac {5 \int x^4 \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{6 b \pi } \\ & = -\frac {5 x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{6 b^3 \pi ^2}+\frac {1}{6} x^6 \operatorname {FresnelC}(b x)-\frac {x^5 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{6 b \pi }+\frac {5 \int x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{2 b^3 \pi ^2} \\ & = -\frac {5 x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{6 b^3 \pi ^2}+\frac {1}{6} x^6 \operatorname {FresnelC}(b x)+\frac {5 x \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{2 b^5 \pi ^3}-\frac {x^5 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{6 b \pi }-\frac {5 \int \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{2 b^5 \pi ^3} \\ & = -\frac {5 x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{6 b^3 \pi ^2}+\frac {1}{6} x^6 \operatorname {FresnelC}(b x)-\frac {5 \operatorname {FresnelS}(b x)}{2 b^6 \pi ^3}+\frac {5 x \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{2 b^5 \pi ^3}-\frac {x^5 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{6 b \pi } \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.81 \[ \int x^5 \operatorname {FresnelC}(b x) \, dx=\frac {-5 b^3 \pi x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right )+b^6 \pi ^3 x^6 \operatorname {FresnelC}(b x)-15 \operatorname {FresnelS}(b x)+b x \left (15-b^4 \pi ^2 x^4\right ) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{6 b^6 \pi ^3} \]
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Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 0.39 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.26
method | result | size |
meijerg | \(\frac {b \,x^{7} \operatorname {hypergeom}\left (\left [\frac {1}{4}, \frac {7}{4}\right ], \left [\frac {1}{2}, \frac {5}{4}, \frac {11}{4}\right ], -\frac {x^{4} \pi ^{2} b^{4}}{16}\right )}{7}\) | \(26\) |
derivativedivides | \(\frac {\frac {\operatorname {FresnelC}\left (b x \right ) b^{6} x^{6}}{6}-\frac {b^{5} x^{5} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{6 \pi }+\frac {-\frac {5 b^{3} x^{3} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{6 \pi }+\frac {5 \left (\frac {3 b x \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }-\frac {3 \,\operatorname {FresnelS}\left (b x \right )}{\pi }\right )}{6 \pi }}{\pi }}{b^{6}}\) | \(97\) |
default | \(\frac {\frac {\operatorname {FresnelC}\left (b x \right ) b^{6} x^{6}}{6}-\frac {b^{5} x^{5} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{6 \pi }+\frac {-\frac {5 b^{3} x^{3} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{6 \pi }+\frac {5 \left (\frac {3 b x \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }-\frac {3 \,\operatorname {FresnelS}\left (b x \right )}{\pi }\right )}{6 \pi }}{\pi }}{b^{6}}\) | \(97\) |
parts | \(\frac {x^{6} \operatorname {FresnelC}\left (b x \right )}{6}-\frac {b \left (\frac {x^{5} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b^{2} \pi }-\frac {5 \left (-\frac {x^{3} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b^{2} \pi }+\frac {\frac {3 x \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b^{2} \pi }-\frac {3 \,\operatorname {FresnelS}\left (\frac {\sqrt {\pi }\, b^{2} x}{\sqrt {b^{2} \pi }}\right )}{b^{2} \sqrt {\pi }\, \sqrt {b^{2} \pi }}}{b^{2} \pi }\right )}{b^{2} \pi }\right )}{6}\) | \(123\) |
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Time = 0.25 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.87 \[ \int x^5 \operatorname {FresnelC}(b x) \, dx=\frac {\pi ^{3} b^{7} x^{6} \operatorname {C}\left (b x\right ) - 5 \, \pi b^{4} x^{3} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - {\left (\pi ^{2} b^{6} x^{5} - 15 \, b^{2} x\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - 15 \, \sqrt {b^{2}} \operatorname {S}\left (\sqrt {b^{2}} x\right )}{6 \, \pi ^{3} b^{7}} \]
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Time = 0.49 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.49 \[ \int x^5 \operatorname {FresnelC}(b x) \, dx=\frac {b x^{7} \Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{3}\left (\begin {matrix} \frac {1}{4}, \frac {7}{4} \\ \frac {1}{2}, \frac {5}{4}, \frac {11}{4} \end {matrix}\middle | {- \frac {\pi ^{2} b^{4} x^{4}}{16}} \right )}}{16 \Gamma \left (\frac {5}{4}\right ) \Gamma \left (\frac {11}{4}\right )} \]
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Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.11 \[ \int x^5 \operatorname {FresnelC}(b x) \, dx=\frac {1}{6} \, x^{6} \operatorname {C}\left (b x\right ) - \frac {\sqrt {\frac {1}{2}} {\left (20 \, \sqrt {\frac {1}{2}} \pi ^{2} b^{3} x^{3} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + \left (15 i + 15\right ) \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \pi \operatorname {erf}\left (\sqrt {\frac {1}{2} i \, \pi } b x\right ) - \left (15 i - 15\right ) \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \pi \operatorname {erf}\left (\sqrt {-\frac {1}{2} i \, \pi } b x\right ) + 4 \, {\left (\sqrt {\frac {1}{2}} \pi ^{3} b^{5} x^{5} - 15 \, \sqrt {\frac {1}{2}} \pi b x\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )\right )}}{12 \, \pi ^{4} b^{6}} \]
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\[ \int x^5 \operatorname {FresnelC}(b x) \, dx=\int { x^{5} \operatorname {C}\left (b x\right ) \,d x } \]
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Timed out. \[ \int x^5 \operatorname {FresnelC}(b x) \, dx=\int x^5\,\mathrm {FresnelC}\left (b\,x\right ) \,d x \]
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