Integrand size = 10, antiderivative size = 124 \[ \int x^2 \operatorname {FresnelS}(b x)^2 \, dx=\frac {2 x}{3 b^2 \pi ^2}+\frac {x \cos \left (b^2 \pi x^2\right )}{6 b^2 \pi ^2}-\frac {5 \operatorname {FresnelC}\left (\sqrt {2} b x\right )}{6 \sqrt {2} b^3 \pi ^2}+\frac {2 x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{3 b \pi }+\frac {1}{3} x^3 \operatorname {FresnelS}(b x)^2-\frac {4 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{3 b^3 \pi ^2} \]
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Time = 0.07 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6565, 6589, 6595, 3438, 3433, 3466} \[ \int x^2 \operatorname {FresnelS}(b x)^2 \, dx=-\frac {5 \operatorname {FresnelC}\left (\sqrt {2} b x\right )}{6 \sqrt {2} \pi ^2 b^3}+\frac {2 x^2 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{3 \pi b}+\frac {x \cos \left (\pi b^2 x^2\right )}{6 \pi ^2 b^2}+\frac {2 x}{3 \pi ^2 b^2}-\frac {4 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 \pi ^2 b^3}+\frac {1}{3} x^3 \operatorname {FresnelS}(b x)^2 \]
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Rule 3433
Rule 3438
Rule 3466
Rule 6565
Rule 6589
Rule 6595
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x^3 \operatorname {FresnelS}(b x)^2-\frac {1}{3} (2 b) \int x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx \\ & = \frac {2 x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{3 b \pi }+\frac {1}{3} x^3 \operatorname {FresnelS}(b x)^2-\frac {\int x^2 \sin \left (b^2 \pi x^2\right ) \, dx}{3 \pi }-\frac {4 \int x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx}{3 b \pi } \\ & = \frac {x \cos \left (b^2 \pi x^2\right )}{6 b^2 \pi ^2}+\frac {2 x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{3 b \pi }+\frac {1}{3} x^3 \operatorname {FresnelS}(b x)^2-\frac {4 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{3 b^3 \pi ^2}-\frac {\int \cos \left (b^2 \pi x^2\right ) \, dx}{6 b^2 \pi ^2}+\frac {4 \int \sin ^2\left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{3 b^2 \pi ^2} \\ & = \frac {x \cos \left (b^2 \pi x^2\right )}{6 b^2 \pi ^2}-\frac {\operatorname {FresnelC}\left (\sqrt {2} b x\right )}{6 \sqrt {2} b^3 \pi ^2}+\frac {2 x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{3 b \pi }+\frac {1}{3} x^3 \operatorname {FresnelS}(b x)^2-\frac {4 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{3 b^3 \pi ^2}+\frac {4 \int \left (\frac {1}{2}-\frac {1}{2} \cos \left (b^2 \pi x^2\right )\right ) \, dx}{3 b^2 \pi ^2} \\ & = \frac {2 x}{3 b^2 \pi ^2}+\frac {x \cos \left (b^2 \pi x^2\right )}{6 b^2 \pi ^2}-\frac {\operatorname {FresnelC}\left (\sqrt {2} b x\right )}{6 \sqrt {2} b^3 \pi ^2}+\frac {2 x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{3 b \pi }+\frac {1}{3} x^3 \operatorname {FresnelS}(b x)^2-\frac {4 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{3 b^3 \pi ^2}-\frac {2 \int \cos \left (b^2 \pi x^2\right ) \, dx}{3 b^2 \pi ^2} \\ & = \frac {2 x}{3 b^2 \pi ^2}+\frac {x \cos \left (b^2 \pi x^2\right )}{6 b^2 \pi ^2}-\frac {\operatorname {FresnelC}\left (\sqrt {2} b x\right )}{6 \sqrt {2} b^3 \pi ^2}-\frac {\sqrt {2} \operatorname {FresnelC}\left (\sqrt {2} b x\right )}{3 b^3 \pi ^2}+\frac {2 x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{3 b \pi }+\frac {1}{3} x^3 \operatorname {FresnelS}(b x)^2-\frac {4 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{3 b^3 \pi ^2} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.81 \[ \int x^2 \operatorname {FresnelS}(b x)^2 \, dx=\frac {2 b x \left (4+\cos \left (b^2 \pi x^2\right )\right )-5 \sqrt {2} \operatorname {FresnelC}\left (\sqrt {2} b x\right )+4 b^3 \pi ^2 x^3 \operatorname {FresnelS}(b x)^2+8 \operatorname {FresnelS}(b x) \left (b^2 \pi x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right )-2 \sin \left (\frac {1}{2} b^2 \pi x^2\right )\right )}{12 b^3 \pi ^2} \]
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Time = 0.47 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.98
method | result | size |
derivativedivides | \(\frac {\frac {\operatorname {FresnelS}\left (b x \right )^{2} b^{3} x^{3}}{3}-2 \,\operatorname {FresnelS}\left (b x \right ) \left (-\frac {b^{2} x^{2} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3 \pi }+\frac {2 \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3 \pi ^{2}}\right )+\frac {2 b x}{3 \pi ^{2}}-\frac {\sqrt {2}\, \operatorname {FresnelC}\left (b x \sqrt {2}\right )}{3 \pi ^{2}}-\frac {-\frac {b x \cos \left (b^{2} \pi \,x^{2}\right )}{2 \pi }+\frac {\sqrt {2}\, \operatorname {FresnelC}\left (b x \sqrt {2}\right )}{4 \pi }}{3 \pi }}{b^{3}}\) | \(122\) |
default | \(\frac {\frac {\operatorname {FresnelS}\left (b x \right )^{2} b^{3} x^{3}}{3}-2 \,\operatorname {FresnelS}\left (b x \right ) \left (-\frac {b^{2} x^{2} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3 \pi }+\frac {2 \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3 \pi ^{2}}\right )+\frac {2 b x}{3 \pi ^{2}}-\frac {\sqrt {2}\, \operatorname {FresnelC}\left (b x \sqrt {2}\right )}{3 \pi ^{2}}-\frac {-\frac {b x \cos \left (b^{2} \pi \,x^{2}\right )}{2 \pi }+\frac {\sqrt {2}\, \operatorname {FresnelC}\left (b x \sqrt {2}\right )}{4 \pi }}{3 \pi }}{b^{3}}\) | \(122\) |
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Time = 0.28 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.90 \[ \int x^2 \operatorname {FresnelS}(b x)^2 \, dx=\frac {4 \, \pi ^{2} b^{4} x^{3} \operatorname {S}\left (b x\right )^{2} + 8 \, \pi b^{3} x^{2} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right ) + 4 \, b^{2} x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} + 6 \, b^{2} x - 16 \, b \operatorname {S}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - 5 \, \sqrt {2} \sqrt {b^{2}} \operatorname {C}\left (\sqrt {2} \sqrt {b^{2}} x\right )}{12 \, \pi ^{2} b^{4}} \]
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\[ \int x^2 \operatorname {FresnelS}(b x)^2 \, dx=\int x^{2} S^{2}\left (b x\right )\, dx \]
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\[ \int x^2 \operatorname {FresnelS}(b x)^2 \, dx=\int { x^{2} \operatorname {S}\left (b x\right )^{2} \,d x } \]
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\[ \int x^2 \operatorname {FresnelS}(b x)^2 \, dx=\int { x^{2} \operatorname {S}\left (b x\right )^{2} \,d x } \]
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Timed out. \[ \int x^2 \operatorname {FresnelS}(b x)^2 \, dx=\int x^2\,{\mathrm {FresnelS}\left (b\,x\right )}^2 \,d x \]
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