Integrand size = 20, antiderivative size = 307 \[ \int x^8 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=\frac {35 x^4}{8 b^5 \pi ^3}-\frac {x^8}{16 b \pi }-\frac {40 \cos \left (b^2 \pi x^2\right )}{b^9 \pi ^5}+\frac {5 x^4 \cos \left (b^2 \pi x^2\right )}{2 b^5 \pi ^3}-\frac {105 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b^8 \pi ^4}+\frac {7 x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b^4 \pi ^2}+\frac {105 \operatorname {FresnelC}(b x) \operatorname {FresnelS}(b x)}{2 b^9 \pi ^4}-\frac {105 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 b^7 \pi ^4}+\frac {105 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 b^7 \pi ^4}-\frac {35 x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac {x^7 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {55 x^2 \sin \left (b^2 \pi x^2\right )}{4 b^7 \pi ^4}+\frac {x^6 \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2} \]
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Time = 0.30 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6597, 3460, 3390, 30, 3377, 2718, 6589, 6581} \[ \int x^8 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=-\frac {105 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi ^4 b^7}+\frac {105 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 \pi ^4 b^7}+\frac {105 \operatorname {FresnelC}(b x) \operatorname {FresnelS}(b x)}{2 \pi ^4 b^9}+\frac {35 x^4}{8 \pi ^3 b^5}+\frac {x^7 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {40 \cos \left (\pi b^2 x^2\right )}{\pi ^5 b^9}-\frac {105 x \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^4 b^8}-\frac {55 x^2 \sin \left (\pi b^2 x^2\right )}{4 \pi ^4 b^7}-\frac {35 x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^3 b^6}+\frac {5 x^4 \cos \left (\pi b^2 x^2\right )}{2 \pi ^3 b^5}+\frac {7 x^5 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}+\frac {x^6 \sin \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3}-\frac {x^8}{16 \pi b} \]
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Rule 30
Rule 2718
Rule 3377
Rule 3390
Rule 3460
Rule 6581
Rule 6589
Rule 6597
Rubi steps \begin{align*} \text {integral}& = \frac {x^7 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {7 \int x^6 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b^2 \pi }-\frac {\int x^7 \sin ^2\left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b \pi } \\ & = \frac {7 x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b^4 \pi ^2}+\frac {x^7 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {35 \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx}{b^4 \pi ^2}-\frac {7 \int x^5 \sin \left (b^2 \pi x^2\right ) \, dx}{2 b^3 \pi ^2}-\frac {\text {Subst}\left (\int x^3 \sin ^2\left (\frac {1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b \pi } \\ & = \frac {7 x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b^4 \pi ^2}-\frac {35 x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac {x^7 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }+\frac {105 \int x^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b^6 \pi ^3}+\frac {35 \int x^3 \sin ^2\left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b^5 \pi ^3}-\frac {7 \text {Subst}\left (\int x^2 \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b^3 \pi ^2}-\frac {\text {Subst}\left (\int x^3 \, dx,x,x^2\right )}{4 b \pi }+\frac {\text {Subst}\left (\int x^3 \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b \pi } \\ & = -\frac {x^8}{16 b \pi }+\frac {7 x^4 \cos \left (b^2 \pi x^2\right )}{4 b^5 \pi ^3}-\frac {105 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b^8 \pi ^4}+\frac {7 x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b^4 \pi ^2}-\frac {35 x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac {x^7 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }+\frac {x^6 \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {105 \int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx}{b^8 \pi ^4}+\frac {105 \int x \sin \left (b^2 \pi x^2\right ) \, dx}{2 b^7 \pi ^4}-\frac {7 \text {Subst}\left (\int x \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b^5 \pi ^3}+\frac {35 \text {Subst}\left (\int x \sin ^2\left (\frac {1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b^5 \pi ^3}-\frac {3 \text {Subst}\left (\int x^2 \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b^3 \pi ^2} \\ & = -\frac {x^8}{16 b \pi }+\frac {5 x^4 \cos \left (b^2 \pi x^2\right )}{2 b^5 \pi ^3}-\frac {105 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b^8 \pi ^4}+\frac {7 x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b^4 \pi ^2}+\frac {105 \operatorname {FresnelC}(b x) \operatorname {FresnelS}(b x)}{2 b^9 \pi ^4}-\frac {105 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 b^7 \pi ^4}+\frac {105 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 b^7 \pi ^4}-\frac {35 x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac {x^7 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {7 x^2 \sin \left (b^2 \pi x^2\right )}{2 b^7 \pi ^4}+\frac {x^6 \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {7 \text {Subst}\left (\int \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b^7 \pi ^4}+\frac {105 \text {Subst}\left (\int \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b^7 \pi ^4}-\frac {3 \text {Subst}\left (\int x \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b^5 \pi ^3}+\frac {35 \text {Subst}\left (\int x \, dx,x,x^2\right )}{4 b^5 \pi ^3}-\frac {35 \text {Subst}\left (\int x \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b^5 \pi ^3} \\ & = \frac {35 x^4}{8 b^5 \pi ^3}-\frac {x^8}{16 b \pi }-\frac {119 \cos \left (b^2 \pi x^2\right )}{4 b^9 \pi ^5}+\frac {5 x^4 \cos \left (b^2 \pi x^2\right )}{2 b^5 \pi ^3}-\frac {105 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b^8 \pi ^4}+\frac {7 x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b^4 \pi ^2}+\frac {105 \operatorname {FresnelC}(b x) \operatorname {FresnelS}(b x)}{2 b^9 \pi ^4}-\frac {105 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 b^7 \pi ^4}+\frac {105 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 b^7 \pi ^4}-\frac {35 x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac {x^7 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {55 x^2 \sin \left (b^2 \pi x^2\right )}{4 b^7 \pi ^4}+\frac {x^6 \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {3 \text {Subst}\left (\int \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b^7 \pi ^4}+\frac {35 \text {Subst}\left (\int \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b^7 \pi ^4} \\ & = \frac {35 x^4}{8 b^5 \pi ^3}-\frac {x^8}{16 b \pi }-\frac {40 \cos \left (b^2 \pi x^2\right )}{b^9 \pi ^5}+\frac {5 x^4 \cos \left (b^2 \pi x^2\right )}{2 b^5 \pi ^3}-\frac {105 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b^8 \pi ^4}+\frac {7 x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{b^4 \pi ^2}+\frac {105 \operatorname {FresnelC}(b x) \operatorname {FresnelS}(b x)}{2 b^9 \pi ^4}-\frac {105 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i b^2 \pi x^2\right )}{8 b^7 \pi ^4}+\frac {105 i x^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i b^2 \pi x^2\right )}{8 b^7 \pi ^4}-\frac {35 x^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac {x^7 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {55 x^2 \sin \left (b^2 \pi x^2\right )}{4 b^7 \pi ^4}+\frac {x^6 \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2} \\ \end{align*}
\[ \int x^8 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=\int x^8 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx \]
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\[\int x^{8} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \operatorname {FresnelS}\left (b x \right )d x\]
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\[ \int x^8 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=\int { x^{8} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right ) \,d x } \]
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\[ \int x^8 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=\int x^{8} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )\, dx \]
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\[ \int x^8 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=\int { x^{8} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right ) \,d x } \]
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\[ \int x^8 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=\int { x^{8} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right ) \,d x } \]
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Timed out. \[ \int x^8 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x) \, dx=\int x^8\,\mathrm {FresnelS}\left (b\,x\right )\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right ) \,d x \]
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