Integrand size = 20, antiderivative size = 108 \[ \int x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=-\frac {x^3}{6 b \pi }+\frac {2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b^4 \pi ^2}-\frac {5 \operatorname {FresnelS}\left (\sqrt {2} b x\right )}{4 \sqrt {2} b^4 \pi ^2}+\frac {x^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }+\frac {x \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2} \]
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Time = 0.06 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6597, 3472, 30, 3467, 3432, 6587} \[ \int x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=-\frac {5 \operatorname {FresnelS}\left (\sqrt {2} b x\right )}{4 \sqrt {2} \pi ^2 b^4}+\frac {x^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {2 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}+\frac {x \sin \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3}-\frac {x^3}{6 \pi b} \]
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Rule 30
Rule 3432
Rule 3467
Rule 3472
Rule 6587
Rule 6597
Rubi steps \begin{align*} \text {integral}& = \frac {x^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {2 \int x \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b^2 \pi }-\frac {\int x^2 \sin ^2\left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b \pi } \\ & = \frac {2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b^4 \pi ^2}+\frac {x^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {\int \sin \left (b^2 \pi x^2\right ) \, dx}{b^3 \pi ^2}-\frac {\int x^2 \, dx}{2 b \pi }+\frac {\int x^2 \cos \left (b^2 \pi x^2\right ) \, dx}{2 b \pi } \\ & = -\frac {x^3}{6 b \pi }+\frac {2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b^4 \pi ^2}-\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{\sqrt {2} b^4 \pi ^2}+\frac {x^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }+\frac {x \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}-\frac {\int \sin \left (b^2 \pi x^2\right ) \, dx}{4 b^3 \pi ^2} \\ & = -\frac {x^3}{6 b \pi }+\frac {2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b^4 \pi ^2}-\frac {5 \operatorname {FresnelS}\left (\sqrt {2} b x\right )}{4 \sqrt {2} b^4 \pi ^2}+\frac {x^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }+\frac {x \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.83 \[ \int x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=\frac {-4 b^3 \pi x^3-15 \sqrt {2} \operatorname {FresnelS}\left (\sqrt {2} b x\right )+24 \operatorname {FresnelS}(b x) \left (2 \cos \left (\frac {1}{2} b^2 \pi x^2\right )+b^2 \pi x^2 \sin \left (\frac {1}{2} b^2 \pi x^2\right )\right )+6 b x \sin \left (b^2 \pi x^2\right )}{24 b^4 \pi ^2} \]
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Time = 1.13 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.10
method | result | size |
default | \(\frac {\frac {\operatorname {FresnelS}\left (b x \right ) \left (\frac {b^{2} x^{2} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }+\frac {2 \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi ^{2}}\right )}{b^{3}}-\frac {\frac {\sqrt {2}\, \operatorname {FresnelS}\left (b x \sqrt {2}\right )}{2 \pi ^{2}}+\frac {b^{3} x^{3}}{6 \pi }-\frac {\frac {b x \sin \left (b^{2} \pi \,x^{2}\right )}{2 \pi }-\frac {\sqrt {2}\, \operatorname {FresnelS}\left (b x \sqrt {2}\right )}{4 \pi }}{2 \pi }}{b^{3}}}{b}\) | \(119\) |
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Time = 0.26 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.90 \[ \int x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=-\frac {4 \, \pi b^{4} x^{3} - 48 \, b \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right ) + 15 \, \sqrt {2} \sqrt {b^{2}} \operatorname {S}\left (\sqrt {2} \sqrt {b^{2}} x\right ) - 12 \, {\left (2 \, \pi b^{3} x^{2} \operatorname {S}\left (b x\right ) + b^{2} x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{24 \, \pi ^{2} b^{5}} \]
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\[ \int x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=\int x^{3} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )\, dx \]
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\[ \int x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=\int { x^{3} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right ) \,d x } \]
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\[ \int x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=\int { x^{3} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right ) \,d x } \]
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Timed out. \[ \int x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=\int x^3\,\mathrm {FresnelS}\left (b\,x\right )\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right ) \,d x \]
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