Integrand size = 20, antiderivative size = 224 \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8} \, dx=-\frac {b}{84 x^6}+\frac {b^5 \pi ^2}{420 x^2}+\frac {b \cos \left (b^2 \pi x^2\right )}{84 x^6}-\frac {b^5 \pi ^2 \cos \left (b^2 \pi x^2\right )}{84 x^2}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{35 x^5}+\frac {b^6 \pi ^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{105 x}+\frac {1}{210} b^7 \pi ^4 \operatorname {FresnelS}(b x)^2-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 x^7}+\frac {b^4 \pi ^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{105 x^3}-\frac {b^3 \pi \sin \left (b^2 \pi x^2\right )}{105 x^4}-\frac {1}{70} b^7 \pi ^3 \text {Si}\left (b^2 \pi x^2\right ) \]
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Time = 0.27 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {6591, 6599, 6575, 30, 3456, 3461, 3378, 3380, 3460} \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8} \, dx=\frac {1}{210} \pi ^4 b^7 \operatorname {FresnelS}(b x)^2+\frac {\pi ^2 b^5}{420 x^2}-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}-\frac {\pi b^2 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{35 x^5}+\frac {b \cos \left (\pi b^2 x^2\right )}{84 x^6}-\frac {1}{70} \pi ^3 b^7 \text {Si}\left (b^2 \pi x^2\right )+\frac {\pi ^3 b^6 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{105 x}-\frac {\pi ^2 b^5 \cos \left (\pi b^2 x^2\right )}{84 x^2}+\frac {\pi ^2 b^4 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{105 x^3}-\frac {\pi b^3 \sin \left (\pi b^2 x^2\right )}{105 x^4}-\frac {b}{84 x^6} \]
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Rule 30
Rule 3378
Rule 3380
Rule 3456
Rule 3460
Rule 3461
Rule 6575
Rule 6591
Rule 6599
Rubi steps \begin{align*} \text {integral}& = -\frac {b}{84 x^6}-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 x^7}-\frac {1}{14} b \int \frac {\cos \left (b^2 \pi x^2\right )}{x^7} \, dx+\frac {1}{7} \left (b^2 \pi \right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^6} \, dx \\ & = -\frac {b}{84 x^6}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{35 x^5}-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 x^7}-\frac {1}{28} b \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x^4} \, dx,x,x^2\right )+\frac {1}{70} \left (b^3 \pi \right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^5} \, dx-\frac {1}{35} \left (b^4 \pi ^2\right ) \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4} \, dx \\ & = -\frac {b}{84 x^6}+\frac {b^5 \pi ^2}{420 x^2}+\frac {b \cos \left (b^2 \pi x^2\right )}{84 x^6}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{35 x^5}-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 x^7}+\frac {b^4 \pi ^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{105 x^3}+\frac {1}{140} \left (b^3 \pi \right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )+\frac {1}{84} \left (b^3 \pi \right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )+\frac {1}{210} \left (b^5 \pi ^2\right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^3} \, dx-\frac {1}{105} \left (b^6 \pi ^3\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^2} \, dx \\ & = -\frac {b}{84 x^6}+\frac {b^5 \pi ^2}{420 x^2}+\frac {b \cos \left (b^2 \pi x^2\right )}{84 x^6}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{35 x^5}+\frac {b^6 \pi ^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{105 x}-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 x^7}+\frac {b^4 \pi ^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{105 x^3}-\frac {b^3 \pi \sin \left (b^2 \pi x^2\right )}{105 x^4}+\frac {1}{420} \left (b^5 \pi ^2\right ) \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )+\frac {1}{280} \left (b^5 \pi ^2\right ) \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )+\frac {1}{168} \left (b^5 \pi ^2\right ) \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )-\frac {1}{210} \left (b^7 \pi ^3\right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x} \, dx+\frac {1}{105} \left (b^8 \pi ^4\right ) \int \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx \\ & = -\frac {b}{84 x^6}+\frac {b^5 \pi ^2}{420 x^2}+\frac {b \cos \left (b^2 \pi x^2\right )}{84 x^6}-\frac {b^5 \pi ^2 \cos \left (b^2 \pi x^2\right )}{84 x^2}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{35 x^5}+\frac {b^6 \pi ^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{105 x}-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 x^7}+\frac {b^4 \pi ^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{105 x^3}-\frac {b^3 \pi \sin \left (b^2 \pi x^2\right )}{105 x^4}-\frac {1}{420} b^7 \pi ^3 \text {Si}\left (b^2 \pi x^2\right )-\frac {1}{420} \left (b^7 \pi ^3\right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )-\frac {1}{280} \left (b^7 \pi ^3\right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )-\frac {1}{168} \left (b^7 \pi ^3\right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )+\frac {1}{105} \left (b^7 \pi ^4\right ) \text {Subst}(\int x \, dx,x,\operatorname {FresnelS}(b x)) \\ & = -\frac {b}{84 x^6}+\frac {b^5 \pi ^2}{420 x^2}+\frac {b \cos \left (b^2 \pi x^2\right )}{84 x^6}-\frac {b^5 \pi ^2 \cos \left (b^2 \pi x^2\right )}{84 x^2}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{35 x^5}+\frac {b^6 \pi ^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{105 x}+\frac {1}{210} b^7 \pi ^4 \operatorname {FresnelS}(b x)^2-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 x^7}+\frac {b^4 \pi ^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{105 x^3}-\frac {b^3 \pi \sin \left (b^2 \pi x^2\right )}{105 x^4}-\frac {1}{70} b^7 \pi ^3 \text {Si}\left (b^2 \pi x^2\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8} \, dx=-\frac {b}{84 x^6}+\frac {b^5 \pi ^2}{420 x^2}+\frac {b \cos \left (b^2 \pi x^2\right )}{84 x^6}-\frac {b^5 \pi ^2 \cos \left (b^2 \pi x^2\right )}{84 x^2}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{35 x^5}+\frac {b^6 \pi ^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{105 x}+\frac {1}{210} b^7 \pi ^4 \operatorname {FresnelS}(b x)^2-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 x^7}+\frac {b^4 \pi ^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{105 x^3}-\frac {b^3 \pi \sin \left (b^2 \pi x^2\right )}{105 x^4}-\frac {1}{70} b^7 \pi ^3 \text {Si}\left (b^2 \pi x^2\right ) \]
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\[\int \frac {\operatorname {FresnelS}\left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{x^{8}}d x\]
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Time = 0.26 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.77 \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8} \, dx=\frac {\pi ^{4} b^{7} x^{7} \operatorname {S}\left (b x\right )^{2} - 3 \, \pi ^{3} b^{7} x^{7} \operatorname {Si}\left (\pi b^{2} x^{2}\right ) + 3 \, \pi ^{2} b^{5} x^{5} - 5 \, {\left (\pi ^{2} b^{5} x^{5} - b x\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} + 2 \, {\left (\pi ^{3} b^{6} x^{6} - 3 \, \pi b^{2} x^{2}\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right ) - 5 \, b x - 2 \, {\left (2 \, \pi b^{3} x^{3} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - {\left (\pi ^{2} b^{4} x^{4} - 15\right )} \operatorname {S}\left (b x\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{210 \, x^{7}} \]
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\[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8} \, dx=\int \frac {\sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )}{x^{8}}\, dx \]
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\[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8} \, dx=\int { \frac {\operatorname {S}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{x^{8}} \,d x } \]
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\[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8} \, dx=\int { \frac {\operatorname {S}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{x^{8}} \,d x } \]
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Timed out. \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8} \, dx=\int \frac {\mathrm {FresnelS}\left (b\,x\right )\,\sin \left (\frac {\Pi \,b^2\,x^2}{2}\right )}{x^8} \,d x \]
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