Integrand size = 10, antiderivative size = 177 \[ \int x^4 \operatorname {FresnelS}(b x)^2 \, dx=\frac {4 x^3}{15 b^2 \pi ^2}+\frac {x^3 \cos \left (b^2 \pi x^2\right )}{10 b^2 \pi ^2}-\frac {16 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{5 b^5 \pi ^3}+\frac {2 x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{5 b \pi }+\frac {1}{5} x^5 \operatorname {FresnelS}(b x)^2+\frac {43 \operatorname {FresnelS}\left (\sqrt {2} b x\right )}{20 \sqrt {2} b^5 \pi ^3}-\frac {8 x^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{5 b^3 \pi ^2}-\frac {11 x \sin \left (b^2 \pi x^2\right )}{20 b^4 \pi ^3} \]
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Time = 0.13 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {6565, 6589, 6597, 3472, 30, 3467, 3432, 6587, 3466} \[ \int x^4 \operatorname {FresnelS}(b x)^2 \, dx=\frac {43 \operatorname {FresnelS}\left (\sqrt {2} b x\right )}{20 \sqrt {2} \pi ^3 b^5}+\frac {2 x^4 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 \pi b}+\frac {4 x^3}{15 \pi ^2 b^2}+\frac {x^3 \cos \left (\pi b^2 x^2\right )}{10 \pi ^2 b^2}-\frac {16 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 \pi ^3 b^5}-\frac {11 x \sin \left (\pi b^2 x^2\right )}{20 \pi ^3 b^4}-\frac {8 x^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{5 \pi ^2 b^3}+\frac {1}{5} x^5 \operatorname {FresnelS}(b x)^2 \]
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Rule 30
Rule 3432
Rule 3466
Rule 3467
Rule 3472
Rule 6565
Rule 6587
Rule 6589
Rule 6597
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} x^5 \operatorname {FresnelS}(b x)^2-\frac {1}{5} (2 b) \int x^5 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx \\ & = \frac {2 x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{5 b \pi }+\frac {1}{5} x^5 \operatorname {FresnelS}(b x)^2-\frac {\int x^4 \sin \left (b^2 \pi x^2\right ) \, dx}{5 \pi }-\frac {8 \int x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx}{5 b \pi } \\ & = \frac {x^3 \cos \left (b^2 \pi x^2\right )}{10 b^2 \pi ^2}+\frac {2 x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{5 b \pi }+\frac {1}{5} x^5 \operatorname {FresnelS}(b x)^2-\frac {8 x^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{5 b^3 \pi ^2}+\frac {16 \int x \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{5 b^3 \pi ^2}-\frac {3 \int x^2 \cos \left (b^2 \pi x^2\right ) \, dx}{10 b^2 \pi ^2}+\frac {8 \int x^2 \sin ^2\left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{5 b^2 \pi ^2} \\ & = \frac {x^3 \cos \left (b^2 \pi x^2\right )}{10 b^2 \pi ^2}-\frac {16 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{5 b^5 \pi ^3}+\frac {2 x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{5 b \pi }+\frac {1}{5} x^5 \operatorname {FresnelS}(b x)^2-\frac {8 x^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{5 b^3 \pi ^2}-\frac {3 x \sin \left (b^2 \pi x^2\right )}{20 b^4 \pi ^3}+\frac {3 \int \sin \left (b^2 \pi x^2\right ) \, dx}{20 b^4 \pi ^3}+\frac {8 \int \sin \left (b^2 \pi x^2\right ) \, dx}{5 b^4 \pi ^3}+\frac {4 \int x^2 \, dx}{5 b^2 \pi ^2}-\frac {4 \int x^2 \cos \left (b^2 \pi x^2\right ) \, dx}{5 b^2 \pi ^2} \\ & = \frac {4 x^3}{15 b^2 \pi ^2}+\frac {x^3 \cos \left (b^2 \pi x^2\right )}{10 b^2 \pi ^2}-\frac {16 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{5 b^5 \pi ^3}+\frac {2 x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{5 b \pi }+\frac {1}{5} x^5 \operatorname {FresnelS}(b x)^2+\frac {3 \operatorname {FresnelS}\left (\sqrt {2} b x\right )}{20 \sqrt {2} b^5 \pi ^3}+\frac {4 \sqrt {2} \operatorname {FresnelS}\left (\sqrt {2} b x\right )}{5 b^5 \pi ^3}-\frac {8 x^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{5 b^3 \pi ^2}-\frac {11 x \sin \left (b^2 \pi x^2\right )}{20 b^4 \pi ^3}+\frac {2 \int \sin \left (b^2 \pi x^2\right ) \, dx}{5 b^4 \pi ^3} \\ & = \frac {4 x^3}{15 b^2 \pi ^2}+\frac {x^3 \cos \left (b^2 \pi x^2\right )}{10 b^2 \pi ^2}-\frac {16 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{5 b^5 \pi ^3}+\frac {2 x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{5 b \pi }+\frac {1}{5} x^5 \operatorname {FresnelS}(b x)^2+\frac {3 \operatorname {FresnelS}\left (\sqrt {2} b x\right )}{20 \sqrt {2} b^5 \pi ^3}+\frac {\sqrt {2} \operatorname {FresnelS}\left (\sqrt {2} b x\right )}{b^5 \pi ^3}-\frac {8 x^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{5 b^3 \pi ^2}-\frac {11 x \sin \left (b^2 \pi x^2\right )}{20 b^4 \pi ^3} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.77 \[ \int x^4 \operatorname {FresnelS}(b x)^2 \, dx=\frac {32 b^3 \pi x^3+12 b^3 \pi x^3 \cos \left (b^2 \pi x^2\right )+24 b^5 \pi ^3 x^5 \operatorname {FresnelS}(b x)^2+129 \sqrt {2} \operatorname {FresnelS}\left (\sqrt {2} b x\right )+48 \operatorname {FresnelS}(b x) \left (\left (-8+b^4 \pi ^2 x^4\right ) \cos \left (\frac {1}{2} b^2 \pi x^2\right )-4 b^2 \pi x^2 \sin \left (\frac {1}{2} b^2 \pi x^2\right )\right )-66 b x \sin \left (b^2 \pi x^2\right )}{120 b^5 \pi ^3} \]
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Time = 0.54 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.18
method | result | size |
derivativedivides | \(\frac {\frac {\operatorname {FresnelS}\left (b x \right )^{2} b^{5} x^{5}}{5}-2 \,\operatorname {FresnelS}\left (b x \right ) \left (-\frac {b^{4} x^{4} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{5 \pi }+\frac {\frac {4 b^{2} x^{2} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{5 \pi }+\frac {8 \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{5 \pi ^{2}}}{\pi }\right )+\frac {4 b^{3} x^{3}}{15 \pi ^{2}}-\frac {4 \left (\frac {b x \sin \left (b^{2} \pi \,x^{2}\right )}{2 \pi }-\frac {\sqrt {2}\, \operatorname {FresnelS}\left (b x \sqrt {2}\right )}{4 \pi }\right )}{5 \pi ^{2}}-\frac {-\frac {\pi \,b^{3} x^{3} \cos \left (b^{2} \pi \,x^{2}\right )}{2}+\frac {3 \pi \left (\frac {b x \sin \left (b^{2} \pi \,x^{2}\right )}{2 \pi }-\frac {\sqrt {2}\, \operatorname {FresnelS}\left (b x \sqrt {2}\right )}{4 \pi }\right )}{2}-4 \sqrt {2}\, \operatorname {FresnelS}\left (b x \sqrt {2}\right )}{5 \pi ^{3}}}{b^{5}}\) | \(208\) |
default | \(\frac {\frac {\operatorname {FresnelS}\left (b x \right )^{2} b^{5} x^{5}}{5}-2 \,\operatorname {FresnelS}\left (b x \right ) \left (-\frac {b^{4} x^{4} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{5 \pi }+\frac {\frac {4 b^{2} x^{2} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{5 \pi }+\frac {8 \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{5 \pi ^{2}}}{\pi }\right )+\frac {4 b^{3} x^{3}}{15 \pi ^{2}}-\frac {4 \left (\frac {b x \sin \left (b^{2} \pi \,x^{2}\right )}{2 \pi }-\frac {\sqrt {2}\, \operatorname {FresnelS}\left (b x \sqrt {2}\right )}{4 \pi }\right )}{5 \pi ^{2}}-\frac {-\frac {\pi \,b^{3} x^{3} \cos \left (b^{2} \pi \,x^{2}\right )}{2}+\frac {3 \pi \left (\frac {b x \sin \left (b^{2} \pi \,x^{2}\right )}{2 \pi }-\frac {\sqrt {2}\, \operatorname {FresnelS}\left (b x \sqrt {2}\right )}{4 \pi }\right )}{2}-4 \sqrt {2}\, \operatorname {FresnelS}\left (b x \sqrt {2}\right )}{5 \pi ^{3}}}{b^{5}}\) | \(208\) |
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Time = 0.26 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.84 \[ \int x^4 \operatorname {FresnelS}(b x)^2 \, dx=\frac {24 \, \pi ^{3} b^{6} x^{5} \operatorname {S}\left (b x\right )^{2} + 24 \, \pi b^{4} x^{3} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} + 20 \, \pi b^{4} x^{3} + 48 \, {\left (\pi ^{2} b^{5} x^{4} - 8 \, b\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right ) + 129 \, \sqrt {2} \sqrt {b^{2}} \operatorname {S}\left (\sqrt {2} \sqrt {b^{2}} x\right ) - 12 \, {\left (16 \, \pi b^{3} x^{2} \operatorname {S}\left (b x\right ) + 11 \, b^{2} x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{120 \, \pi ^{3} b^{6}} \]
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\[ \int x^4 \operatorname {FresnelS}(b x)^2 \, dx=\int x^{4} S^{2}\left (b x\right )\, dx \]
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\[ \int x^4 \operatorname {FresnelS}(b x)^2 \, dx=\int { x^{4} \operatorname {S}\left (b x\right )^{2} \,d x } \]
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\[ \int x^4 \operatorname {FresnelS}(b x)^2 \, dx=\int { x^{4} \operatorname {S}\left (b x\right )^{2} \,d x } \]
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Timed out. \[ \int x^4 \operatorname {FresnelS}(b x)^2 \, dx=\int x^4\,{\mathrm {FresnelS}\left (b\,x\right )}^2 \,d x \]
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