Integrand size = 8, antiderivative size = 74 \[ \int x^3 \operatorname {FresnelS}(b x) \, dx=\frac {x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b \pi }+\frac {3 \operatorname {FresnelS}(b x)}{4 b^4 \pi ^2}+\frac {1}{4} x^4 \operatorname {FresnelS}(b x)-\frac {3 x \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b^3 \pi ^2} \]
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Time = 0.03 (sec) , antiderivative size = 74, 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, 3466, 3467, 3432} \[ \int x^3 \operatorname {FresnelS}(b x) \, dx=\frac {3 \operatorname {FresnelS}(b x)}{4 \pi ^2 b^4}+\frac {x^3 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{4 \pi b}-\frac {3 x \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 \pi ^2 b^3}+\frac {1}{4} x^4 \operatorname {FresnelS}(b x) \]
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Rule 3432
Rule 3466
Rule 3467
Rule 6561
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x^4 \operatorname {FresnelS}(b x)-\frac {1}{4} b \int x^4 \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx \\ & = \frac {x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b \pi }+\frac {1}{4} x^4 \operatorname {FresnelS}(b x)-\frac {3 \int x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{4 b \pi } \\ & = \frac {x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b \pi }+\frac {1}{4} x^4 \operatorname {FresnelS}(b x)-\frac {3 x \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {3 \int \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{4 b^3 \pi ^2} \\ & = \frac {x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b \pi }+\frac {3 \operatorname {FresnelS}(b x)}{4 b^4 \pi ^2}+\frac {1}{4} x^4 \operatorname {FresnelS}(b x)-\frac {3 x \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b^3 \pi ^2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00 \[ \int x^3 \operatorname {FresnelS}(b x) \, dx=\frac {x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b \pi }+\frac {3 \operatorname {FresnelS}(b x)}{4 b^4 \pi ^2}+\frac {1}{4} x^4 \operatorname {FresnelS}(b x)-\frac {3 x \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b^3 \pi ^2} \]
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Time = 0.38 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.84
method | result | size |
meijerg | \(\frac {\frac {\pi \,x^{3} b^{3} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{4}-\frac {3 \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) b x}{4}+\frac {\left (21 x^{4} \pi ^{2} b^{4}+63\right ) \operatorname {FresnelS}\left (b x \right )}{84}}{\pi ^{2} b^{4}}\) | \(62\) |
derivativedivides | \(\frac {\frac {\operatorname {FresnelS}\left (b x \right ) b^{4} x^{4}}{4}+\frac {b^{3} x^{3} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{4 \pi }-\frac {3 \left (\frac {b x \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }-\frac {\operatorname {FresnelS}\left (b x \right )}{\pi }\right )}{4 \pi }}{b^{4}}\) | \(70\) |
default | \(\frac {\frac {\operatorname {FresnelS}\left (b x \right ) b^{4} x^{4}}{4}+\frac {b^{3} x^{3} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{4 \pi }-\frac {3 \left (\frac {b x \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }-\frac {\operatorname {FresnelS}\left (b x \right )}{\pi }\right )}{4 \pi }}{b^{4}}\) | \(70\) |
parts | \(\frac {x^{4} \operatorname {FresnelS}\left (b x \right )}{4}-\frac {b \left (-\frac {x^{3} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b^{2} \pi }+\frac {\frac {3 x \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b^{2} \pi }-\frac {3 \,\operatorname {FresnelS}\left (\frac {\sqrt {\pi }\, b^{2} x}{\sqrt {b^{2} \pi }}\right )}{b^{2} \sqrt {\pi }\, \sqrt {b^{2} \pi }}}{b^{2} \pi }\right )}{4}\) | \(94\) |
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none
Time = 0.27 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.78 \[ \int x^3 \operatorname {FresnelS}(b x) \, dx=\frac {\pi b^{3} x^{3} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - 3 \, b x \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + {\left (\pi ^{2} b^{4} x^{4} + 3\right )} \operatorname {S}\left (b x\right )}{4 \, \pi ^{2} b^{4}} \]
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Time = 0.57 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.51 \[ \int x^3 \operatorname {FresnelS}(b x) \, dx=\frac {21 x^{4} S\left (b x\right ) \Gamma \left (\frac {3}{4}\right )}{64 \Gamma \left (\frac {11}{4}\right )} + \frac {21 x^{3} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {3}{4}\right )}{64 \pi b \Gamma \left (\frac {11}{4}\right )} - \frac {63 x \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {3}{4}\right )}{64 \pi ^{2} b^{3} \Gamma \left (\frac {11}{4}\right )} + \frac {63 S\left (b x\right ) \Gamma \left (\frac {3}{4}\right )}{64 \pi ^{2} b^{4} \Gamma \left (\frac {11}{4}\right )} \]
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Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.27 \[ \int x^3 \operatorname {FresnelS}(b x) \, dx=\frac {1}{4} \, x^{4} \operatorname {S}\left (b x\right ) + \frac {\sqrt {\frac {1}{2}} {\left (4 \, \sqrt {\frac {1}{2}} \pi ^{2} b^{3} x^{3} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - 12 \, \sqrt {\frac {1}{2}} \pi b x \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + \left (3 i + 3\right ) \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \pi \operatorname {erf}\left (\sqrt {\frac {1}{2} i \, \pi } b x\right ) - \left (3 i - 3\right ) \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \pi \operatorname {erf}\left (\sqrt {-\frac {1}{2} i \, \pi } b x\right )\right )}}{8 \, \pi ^{3} b^{4}} \]
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\[ \int x^3 \operatorname {FresnelS}(b x) \, dx=\int { x^{3} \operatorname {S}\left (b x\right ) \,d x } \]
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Timed out. \[ \int x^3 \operatorname {FresnelS}(b x) \, dx=\int x^3\,\mathrm {FresnelS}\left (b\,x\right ) \,d x \]
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