Integrand size = 20, antiderivative size = 163 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^6} \, dx=\frac {b^3 \pi }{60 x^2}-\frac {b^3 \pi \cos \left (b^2 \pi x^2\right )}{24 x^2}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{5 x^5}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{15 x}+\frac {1}{30} b^5 \pi ^3 \operatorname {FresnelS}(b x)^2+\frac {b^2 \pi \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{15 x^3}-\frac {b \sin \left (b^2 \pi x^2\right )}{40 x^4}-\frac {7}{120} b^5 \pi ^2 \text {Si}\left (b^2 \pi x^2\right ) \]
[Out]
Time = 0.15 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {6599, 6591, 6575, 30, 3456, 3461, 3378, 3380, 3460} \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^6} \, dx=\frac {1}{30} \pi ^3 b^5 \operatorname {FresnelS}(b x)^2+\frac {\pi b^3}{60 x^2}-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}+\frac {\pi b^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{15 x^3}-\frac {b \sin \left (\pi b^2 x^2\right )}{40 x^4}-\frac {7}{120} \pi ^2 b^5 \text {Si}\left (b^2 \pi x^2\right )+\frac {\pi ^2 b^4 \operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{15 x}-\frac {\pi b^3 \cos \left (\pi b^2 x^2\right )}{24 x^2} \]
[In]
[Out]
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 {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{5 x^5}+\frac {1}{10} b \int \frac {\sin \left (b^2 \pi x^2\right )}{x^5} \, dx-\frac {1}{5} \left (b^2 \pi \right ) \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4} \, dx \\ & = \frac {b^3 \pi }{60 x^2}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{5 x^5}+\frac {b^2 \pi \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{15 x^3}+\frac {1}{20} b \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )+\frac {1}{30} \left (b^3 \pi \right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^3} \, dx-\frac {1}{15} \left (b^4 \pi ^2\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^2} \, dx \\ & = \frac {b^3 \pi }{60 x^2}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{5 x^5}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{15 x}+\frac {b^2 \pi \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{15 x^3}-\frac {b \sin \left (b^2 \pi x^2\right )}{40 x^4}+\frac {1}{60} \left (b^3 \pi \right ) \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )+\frac {1}{40} \left (b^3 \pi \right ) \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )-\frac {1}{30} \left (b^5 \pi ^2\right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x} \, dx+\frac {1}{15} \left (b^6 \pi ^3\right ) \int \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx \\ & = \frac {b^3 \pi }{60 x^2}-\frac {b^3 \pi \cos \left (b^2 \pi x^2\right )}{24 x^2}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{5 x^5}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{15 x}+\frac {b^2 \pi \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{15 x^3}-\frac {b \sin \left (b^2 \pi x^2\right )}{40 x^4}-\frac {1}{60} b^5 \pi ^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {1}{60} \left (b^5 \pi ^2\right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )-\frac {1}{40} \left (b^5 \pi ^2\right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )+\frac {1}{15} \left (b^5 \pi ^3\right ) \text {Subst}(\int x \, dx,x,\operatorname {FresnelS}(b x)) \\ & = \frac {b^3 \pi }{60 x^2}-\frac {b^3 \pi \cos \left (b^2 \pi x^2\right )}{24 x^2}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{5 x^5}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{15 x}+\frac {1}{30} b^5 \pi ^3 \operatorname {FresnelS}(b x)^2+\frac {b^2 \pi \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{15 x^3}-\frac {b \sin \left (b^2 \pi x^2\right )}{40 x^4}-\frac {7}{120} b^5 \pi ^2 \text {Si}\left (b^2 \pi x^2\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^6} \, dx=\frac {b^3 \pi }{60 x^2}-\frac {b^3 \pi \cos \left (b^2 \pi x^2\right )}{24 x^2}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{5 x^5}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{15 x}+\frac {1}{30} b^5 \pi ^3 \operatorname {FresnelS}(b x)^2+\frac {b^2 \pi \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{15 x^3}-\frac {b \sin \left (b^2 \pi x^2\right )}{40 x^4}-\frac {7}{120} b^5 \pi ^2 \text {Si}\left (b^2 \pi x^2\right ) \]
[In]
[Out]
\[\int \frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \operatorname {FresnelS}\left (b x \right )}{x^{6}}d x\]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.87 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^6} \, dx=\frac {4 \, \pi ^{3} b^{5} x^{5} \operatorname {S}\left (b x\right )^{2} - 7 \, \pi ^{2} b^{5} x^{5} \operatorname {Si}\left (\pi b^{2} x^{2}\right ) - 10 \, \pi b^{3} x^{3} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} + 7 \, \pi b^{3} x^{3} + 8 \, {\left (\pi ^{2} b^{4} x^{4} - 3\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right ) + 2 \, {\left (4 \, \pi b^{2} x^{2} \operatorname {S}\left (b x\right ) - 3 \, b x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{120 \, x^{5}} \]
[In]
[Out]
\[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^6} \, dx=\int \frac {\cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )}{x^{6}}\, dx \]
[In]
[Out]
\[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^6} \, dx=\int { \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right )}{x^{6}} \,d x } \]
[In]
[Out]
\[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^6} \, dx=\int { \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right )}{x^{6}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^6} \, dx=\int \frac {\mathrm {FresnelS}\left (b\,x\right )\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right )}{x^6} \,d x \]
[In]
[Out]