Integrand size = 20, antiderivative size = 215 \[ \int x^7 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx=\frac {24 x}{b^7 \pi ^4}-\frac {3 x^5}{5 b^3 \pi ^2}-\frac {147 x \cos \left (b^2 \pi x^2\right )}{16 b^7 \pi ^4}+\frac {x^5 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}-\frac {48 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^8 \pi ^4}+\frac {6 x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^4 \pi ^2}+\frac {531 \operatorname {FresnelC}\left (\sqrt {2} b x\right )}{16 \sqrt {2} b^8 \pi ^4}-\frac {24 x^2 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac {x^6 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {17 x^3 \sin \left (b^2 \pi x^2\right )}{8 b^5 \pi ^3} \]
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Time = 0.19 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {6590, 6598, 6596, 3439, 3433, 3466, 3473, 30, 3467} \[ \int x^7 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx=\frac {531 \operatorname {FresnelC}\left (\sqrt {2} b x\right )}{16 \sqrt {2} \pi ^4 b^8}+\frac {24 x}{\pi ^4 b^7}-\frac {3 x^5}{5 \pi ^2 b^3}+\frac {x^6 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {48 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^4 b^8}-\frac {147 x \cos \left (\pi b^2 x^2\right )}{16 \pi ^4 b^7}-\frac {24 x^2 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^3 b^6}-\frac {17 x^3 \sin \left (\pi b^2 x^2\right )}{8 \pi ^3 b^5}+\frac {6 x^4 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}+\frac {x^5 \cos \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3} \]
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Rule 30
Rule 3433
Rule 3439
Rule 3466
Rule 3467
Rule 3473
Rule 6590
Rule 6596
Rule 6598
Rubi steps \begin{align*} \text {integral}& = \frac {x^6 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {6 \int x^5 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b^2 \pi }-\frac {\int x^6 \sin \left (b^2 \pi x^2\right ) \, dx}{2 b \pi } \\ & = \frac {x^5 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {6 x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^4 \pi ^2}+\frac {x^6 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {24 \int x^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx}{b^4 \pi ^2}-\frac {5 \int x^4 \cos \left (b^2 \pi x^2\right ) \, dx}{4 b^3 \pi ^2}-\frac {6 \int x^4 \cos ^2\left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b^3 \pi ^2} \\ & = \frac {x^5 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {6 x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^4 \pi ^2}-\frac {24 x^2 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac {x^6 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {5 x^3 \sin \left (b^2 \pi x^2\right )}{8 b^5 \pi ^3}+\frac {48 \int x \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b^6 \pi ^3}+\frac {15 \int x^2 \sin \left (b^2 \pi x^2\right ) \, dx}{8 b^5 \pi ^3}+\frac {12 \int x^2 \sin \left (b^2 \pi x^2\right ) \, dx}{b^5 \pi ^3}-\frac {3 \int x^4 \, dx}{b^3 \pi ^2}-\frac {3 \int x^4 \cos \left (b^2 \pi x^2\right ) \, dx}{b^3 \pi ^2} \\ & = -\frac {3 x^5}{5 b^3 \pi ^2}-\frac {111 x \cos \left (b^2 \pi x^2\right )}{16 b^7 \pi ^4}+\frac {x^5 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}-\frac {48 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^8 \pi ^4}+\frac {6 x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^4 \pi ^2}-\frac {24 x^2 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac {x^6 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {17 x^3 \sin \left (b^2 \pi x^2\right )}{8 b^5 \pi ^3}+\frac {15 \int \cos \left (b^2 \pi x^2\right ) \, dx}{16 b^7 \pi ^4}+\frac {6 \int \cos \left (b^2 \pi x^2\right ) \, dx}{b^7 \pi ^4}+\frac {48 \int \cos ^2\left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b^7 \pi ^4}+\frac {9 \int x^2 \sin \left (b^2 \pi x^2\right ) \, dx}{2 b^5 \pi ^3} \\ & = -\frac {3 x^5}{5 b^3 \pi ^2}-\frac {147 x \cos \left (b^2 \pi x^2\right )}{16 b^7 \pi ^4}+\frac {x^5 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}-\frac {48 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^8 \pi ^4}+\frac {6 x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^4 \pi ^2}+\frac {15 \operatorname {FresnelC}\left (\sqrt {2} b x\right )}{16 \sqrt {2} b^8 \pi ^4}+\frac {3 \sqrt {2} \operatorname {FresnelC}\left (\sqrt {2} b x\right )}{b^8 \pi ^4}-\frac {24 x^2 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac {x^6 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {17 x^3 \sin \left (b^2 \pi x^2\right )}{8 b^5 \pi ^3}+\frac {9 \int \cos \left (b^2 \pi x^2\right ) \, dx}{4 b^7 \pi ^4}+\frac {48 \int \left (\frac {1}{2}+\frac {1}{2} \cos \left (b^2 \pi x^2\right )\right ) \, dx}{b^7 \pi ^4} \\ & = \frac {24 x}{b^7 \pi ^4}-\frac {3 x^5}{5 b^3 \pi ^2}-\frac {147 x \cos \left (b^2 \pi x^2\right )}{16 b^7 \pi ^4}+\frac {x^5 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}-\frac {48 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^8 \pi ^4}+\frac {6 x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^4 \pi ^2}+\frac {51 \operatorname {FresnelC}\left (\sqrt {2} b x\right )}{16 \sqrt {2} b^8 \pi ^4}+\frac {3 \sqrt {2} \operatorname {FresnelC}\left (\sqrt {2} b x\right )}{b^8 \pi ^4}-\frac {24 x^2 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac {x^6 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {17 x^3 \sin \left (b^2 \pi x^2\right )}{8 b^5 \pi ^3}+\frac {24 \int \cos \left (b^2 \pi x^2\right ) \, dx}{b^7 \pi ^4} \\ & = \frac {24 x}{b^7 \pi ^4}-\frac {3 x^5}{5 b^3 \pi ^2}-\frac {147 x \cos \left (b^2 \pi x^2\right )}{16 b^7 \pi ^4}+\frac {x^5 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}-\frac {48 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^8 \pi ^4}+\frac {6 x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^4 \pi ^2}+\frac {51 \operatorname {FresnelC}\left (\sqrt {2} b x\right )}{16 \sqrt {2} b^8 \pi ^4}+\frac {15 \sqrt {2} \operatorname {FresnelC}\left (\sqrt {2} b x\right )}{b^8 \pi ^4}-\frac {24 x^2 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac {x^6 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {17 x^3 \sin \left (b^2 \pi x^2\right )}{8 b^5 \pi ^3} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.72 \[ \int x^7 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx=\frac {2655 \sqrt {2} \operatorname {FresnelC}\left (\sqrt {2} b x\right )+160 \operatorname {FresnelC}(b x) \left (6 \left (-8+b^4 \pi ^2 x^4\right ) \cos \left (\frac {1}{2} b^2 \pi x^2\right )+b^2 \pi x^2 \left (-24+b^4 \pi ^2 x^4\right ) \sin \left (\frac {1}{2} b^2 \pi x^2\right )\right )+2 b x \left (5 \left (-147+4 b^4 \pi ^2 x^4\right ) \cos \left (b^2 \pi x^2\right )-2 \left (-960+24 b^4 \pi ^2 x^4+85 b^2 \pi x^2 \sin \left (b^2 \pi x^2\right )\right )\right )}{160 b^8 \pi ^4} \]
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Time = 7.28 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.47
method | result | size |
default | \(\frac {\frac {\operatorname {FresnelC}\left (b x \right ) \left (\frac {b^{6} x^{6} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }-\frac {6 \left (-\frac {b^{4} x^{4} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }+\frac {\frac {4 b^{2} x^{2} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }+\frac {8 \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi ^{2}}}{\pi }\right )}{\pi }\right )}{b^{7}}-\frac {\frac {\frac {3}{5} b^{5} x^{5} \pi ^{2}-24 b x}{\pi ^{4}}+\frac {\frac {3 \pi \,b^{3} x^{3} \sin \left (b^{2} \pi \,x^{2}\right )}{2}-\frac {9 \pi \left (-\frac {b x \cos \left (b^{2} \pi \,x^{2}\right )}{2 \pi }+\frac {\sqrt {2}\, \operatorname {FresnelC}\left (b x \sqrt {2}\right )}{4 \pi }\right )}{2}-12 \sqrt {2}\, \operatorname {FresnelC}\left (b x \sqrt {2}\right )}{\pi ^{4}}+\frac {-\frac {\pi \,b^{5} x^{5} \cos \left (b^{2} \pi \,x^{2}\right )}{2}+\frac {5 \pi \left (\frac {b^{3} x^{3} \sin \left (b^{2} \pi \,x^{2}\right )}{2 \pi }-\frac {3 \left (-\frac {b x \cos \left (b^{2} \pi \,x^{2}\right )}{2 \pi }+\frac {\sqrt {2}\, \operatorname {FresnelC}\left (b x \sqrt {2}\right )}{4 \pi }\right )}{2 \pi }\right )}{2}+\frac {12 b x \cos \left (b^{2} \pi \,x^{2}\right )}{\pi }-\frac {6 \sqrt {2}\, \operatorname {FresnelC}\left (b x \sqrt {2}\right )}{\pi }}{2 \pi ^{3}}}{b^{7}}}{b}\) | \(317\) |
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Time = 0.27 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.78 \[ \int x^7 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx=-\frac {136 \, \pi ^{2} b^{6} x^{5} - 5310 \, b^{2} x - 20 \, {\left (4 \, \pi ^{2} b^{6} x^{5} - 147 \, b^{2} x\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} - 960 \, {\left (\pi ^{2} b^{5} x^{4} - 8 \, b\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right ) - 2655 \, \sqrt {2} \sqrt {b^{2}} \operatorname {C}\left (\sqrt {2} \sqrt {b^{2}} x\right ) + 40 \, {\left (17 \, \pi b^{4} x^{3} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - 4 \, {\left (\pi ^{3} b^{7} x^{6} - 24 \, \pi b^{3} x^{2}\right )} \operatorname {C}\left (b x\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{160 \, \pi ^{4} b^{9}} \]
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\[ \int x^7 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx=\int x^{7} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )\, dx \]
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\[ \int x^7 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx=\int { x^{7} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right ) \,d x } \]
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\[ \int x^7 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx=\int { x^{7} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right ) \,d x } \]
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Timed out. \[ \int x^7 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x) \, dx=\int x^7\,\mathrm {FresnelC}\left (b\,x\right )\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right ) \,d x \]
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