Integrand size = 20, antiderivative size = 105 \[ \int x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=-\frac {x}{b^3 \pi ^2}-\frac {x \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {5 \operatorname {FresnelC}\left (\sqrt {2} b x\right )}{4 \sqrt {2} b^4 \pi ^2}-\frac {x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b^2 \pi }+\frac {2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2} \]
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Time = 0.05 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6589, 6595, 3438, 3433, 3466} \[ \int x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\frac {5 \operatorname {FresnelC}\left (\sqrt {2} b x\right )}{4 \sqrt {2} \pi ^2 b^4}-\frac {x}{\pi ^2 b^3}-\frac {x^2 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}-\frac {x \cos \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3} \]
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Rule 3433
Rule 3438
Rule 3466
Rule 6589
Rule 6595
Rubi steps \begin{align*} \text {integral}& = -\frac {x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b^2 \pi }+\frac {2 \int x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx}{b^2 \pi }+\frac {\int x^2 \sin \left (b^2 \pi x^2\right ) \, dx}{2 b \pi } \\ & = -\frac {x \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}-\frac {x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b^2 \pi }+\frac {2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}+\frac {\int \cos \left (b^2 \pi x^2\right ) \, dx}{4 b^3 \pi ^2}-\frac {2 \int \sin ^2\left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b^3 \pi ^2} \\ & = -\frac {x \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {\operatorname {FresnelC}\left (\sqrt {2} b x\right )}{4 \sqrt {2} b^4 \pi ^2}-\frac {x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b^2 \pi }+\frac {2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}-\frac {2 \int \left (\frac {1}{2}-\frac {1}{2} \cos \left (b^2 \pi x^2\right )\right ) \, dx}{b^3 \pi ^2} \\ & = -\frac {x}{b^3 \pi ^2}-\frac {x \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {\operatorname {FresnelC}\left (\sqrt {2} b x\right )}{4 \sqrt {2} b^4 \pi ^2}-\frac {x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b^2 \pi }+\frac {2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}+\frac {\int \cos \left (b^2 \pi x^2\right ) \, dx}{b^3 \pi ^2} \\ & = -\frac {x}{b^3 \pi ^2}-\frac {x \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {5 \operatorname {FresnelC}\left (\sqrt {2} b x\right )}{4 \sqrt {2} b^4 \pi ^2}-\frac {x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b^2 \pi }+\frac {2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.79 \[ \int x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\frac {-2 b x \left (4+\cos \left (b^2 \pi x^2\right )\right )+5 \sqrt {2} \operatorname {FresnelC}\left (\sqrt {2} b x\right )-8 \operatorname {FresnelS}(b x) \left (b^2 \pi x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right )-2 \sin \left (\frac {1}{2} b^2 \pi x^2\right )\right )}{8 b^4 \pi ^2} \]
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Time = 0.82 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.10
method | result | size |
default | \(\frac {\frac {\operatorname {FresnelS}\left (b x \right ) \left (-\frac {b^{2} x^{2} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }+\frac {2 \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi ^{2}}\right )}{b^{3}}-\frac {\frac {b x}{\pi ^{2}}-\frac {\sqrt {2}\, \operatorname {FresnelC}\left (b x \sqrt {2}\right )}{2 \pi ^{2}}-\frac {-\frac {b x \cos \left (b^{2} \pi \,x^{2}\right )}{2 \pi }+\frac {\sqrt {2}\, \operatorname {FresnelC}\left (b x \sqrt {2}\right )}{4 \pi }}{2 \pi }}{b^{3}}}{b}\) | \(115\) |
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Time = 0.27 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=-\frac {8 \, \pi b^{3} x^{2} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right ) + 4 \, b^{2} x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} + 6 \, b^{2} x - 16 \, b \operatorname {S}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - 5 \, \sqrt {2} \sqrt {b^{2}} \operatorname {C}\left (\sqrt {2} \sqrt {b^{2}} x\right )}{8 \, \pi ^{2} b^{5}} \]
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\[ \int x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\int x^{3} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )\, dx \]
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\[ \int x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\int { x^{3} \operatorname {S}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \,d x } \]
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\[ \int x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\int { x^{3} \operatorname {S}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \,d x } \]
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Timed out. \[ \int x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\int x^3\,\mathrm {FresnelS}\left (b\,x\right )\,\sin \left (\frac {\Pi \,b^2\,x^2}{2}\right ) \,d x \]
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