Integrand size = 20, antiderivative size = 163 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^6} \, dx=\frac {b^3 \pi }{60 x^2}-\frac {b^3 \pi \cos \left (b^2 \pi x^2\right )}{24 x^2}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{5 x^5}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{15 x}+\frac {1}{30} b^5 \pi ^3 \operatorname {FresnelS}(b x)^2+\frac {b^2 \pi \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{15 x^3}-\frac {b \sin \left (b^2 \pi x^2\right )}{40 x^4}-\frac {7}{120} b^5 \pi ^2 \text {Si}\left (b^2 \pi x^2\right ) \]
1/60*b^3*Pi/x^2-1/24*b^3*Pi*cos(b^2*Pi*x^2)/x^2-1/5*cos(1/2*b^2*Pi*x^2)*Fr esnelS(b*x)/x^5+1/15*b^4*Pi^2*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/x+1/30*b^5 *Pi^3*FresnelS(b*x)^2-7/120*b^5*Pi^2*Si(b^2*Pi*x^2)+1/15*b^2*Pi*FresnelS(b *x)*sin(1/2*b^2*Pi*x^2)/x^3-1/40*b*sin(b^2*Pi*x^2)/x^4
Time = 0.01 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^6} \, dx=\frac {b^3 \pi }{60 x^2}-\frac {b^3 \pi \cos \left (b^2 \pi x^2\right )}{24 x^2}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{5 x^5}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{15 x}+\frac {1}{30} b^5 \pi ^3 \operatorname {FresnelS}(b x)^2+\frac {b^2 \pi \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{15 x^3}-\frac {b \sin \left (b^2 \pi x^2\right )}{40 x^4}-\frac {7}{120} b^5 \pi ^2 \text {Si}\left (b^2 \pi x^2\right ) \]
(b^3*Pi)/(60*x^2) - (b^3*Pi*Cos[b^2*Pi*x^2])/(24*x^2) - (Cos[(b^2*Pi*x^2)/ 2]*FresnelS[b*x])/(5*x^5) + (b^4*Pi^2*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/( 15*x) + (b^5*Pi^3*FresnelS[b*x]^2)/30 + (b^2*Pi*FresnelS[b*x]*Sin[(b^2*Pi* x^2)/2])/(15*x^3) - (b*Sin[b^2*Pi*x^2])/(40*x^4) - (7*b^5*Pi^2*SinIntegral [b^2*Pi*x^2])/120
Time = 1.46 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.32, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {7018, 3860, 3042, 3778, 3042, 3778, 25, 3042, 3780, 7010, 3861, 3042, 3778, 25, 3042, 3780, 7018, 3856, 6994, 15}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{x^6} \, dx\) |
| \(\Big \downarrow \) 7018 |
| \(\displaystyle -\frac {1}{5} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4}dx+\frac {1}{10} b \int \frac {\sin \left (b^2 \pi x^2\right )}{x^5}dx-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}\) |
| \(\Big \downarrow \) 3860 |
| \(\displaystyle -\frac {1}{5} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4}dx+\frac {1}{20} b \int \frac {\sin \left (b^2 \pi x^2\right )}{x^6}dx^2-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {1}{5} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4}dx+\frac {1}{20} b \int \frac {\sin \left (b^2 \pi x^2\right )}{x^6}dx^2-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}\) |
| \(\Big \downarrow \) 3778 |
| \(\displaystyle -\frac {1}{5} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4}dx+\frac {1}{20} b \left (\frac {1}{2} \pi b^2 \int \frac {\cos \left (b^2 \pi x^2\right )}{x^4}dx^2-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {1}{5} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4}dx+\frac {1}{20} b \left (\frac {1}{2} \pi b^2 \int \frac {\sin \left (b^2 \pi x^2+\frac {\pi }{2}\right )}{x^4}dx^2-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}\) |
| \(\Big \downarrow \) 3778 |
| \(\displaystyle -\frac {1}{5} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4}dx+\frac {1}{20} b \left (\frac {1}{2} \pi b^2 \left (\pi b^2 \int -\frac {\sin \left (b^2 \pi x^2\right )}{x^2}dx^2-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{5} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4}dx+\frac {1}{20} b \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2}dx^2-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {1}{5} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4}dx+\frac {1}{20} b \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2}dx^2-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}\) |
| \(\Big \downarrow \) 3780 |
| \(\displaystyle -\frac {1}{5} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4}dx-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}+\frac {1}{20} b \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )\) |
| \(\Big \downarrow \) 7010 |
| \(\displaystyle -\frac {1}{5} \pi b^2 \left (\frac {1}{3} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^2}dx-\frac {1}{6} b \int \frac {\cos \left (b^2 \pi x^2\right )}{x^3}dx-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}-\frac {b}{12 x^2}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}+\frac {1}{20} b \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )\) |
| \(\Big \downarrow \) 3861 |
| \(\displaystyle -\frac {1}{5} \pi b^2 \left (\frac {1}{3} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^2}dx-\frac {1}{12} b \int \frac {\cos \left (b^2 \pi x^2\right )}{x^4}dx^2-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}-\frac {b}{12 x^2}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}+\frac {1}{20} b \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {1}{5} \pi b^2 \left (\frac {1}{3} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^2}dx-\frac {1}{12} b \int \frac {\sin \left (b^2 \pi x^2+\frac {\pi }{2}\right )}{x^4}dx^2-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}-\frac {b}{12 x^2}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}+\frac {1}{20} b \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )\) |
| \(\Big \downarrow \) 3778 |
| \(\displaystyle -\frac {1}{5} \pi b^2 \left (\frac {1}{3} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^2}dx-\frac {1}{12} b \left (\pi b^2 \int -\frac {\sin \left (b^2 \pi x^2\right )}{x^2}dx^2-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}-\frac {b}{12 x^2}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}+\frac {1}{20} b \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{5} \pi b^2 \left (\frac {1}{3} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^2}dx-\frac {1}{12} b \left (-\pi b^2 \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2}dx^2-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}-\frac {b}{12 x^2}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}+\frac {1}{20} b \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {1}{5} \pi b^2 \left (\frac {1}{3} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^2}dx-\frac {1}{12} b \left (-\pi b^2 \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2}dx^2-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}-\frac {b}{12 x^2}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}+\frac {1}{20} b \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )\) |
| \(\Big \downarrow \) 3780 |
| \(\displaystyle -\frac {1}{5} \pi b^2 \left (\frac {1}{3} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^2}dx-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}-\frac {1}{12} b \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {b}{12 x^2}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}+\frac {1}{20} b \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )\) |
| \(\Big \downarrow \) 7018 |
| \(\displaystyle -\frac {1}{5} \pi b^2 \left (\frac {1}{3} \pi b^2 \left (-\pi b^2 \int \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx+\frac {1}{2} b \int \frac {\sin \left (b^2 \pi x^2\right )}{x}dx-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{x}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}-\frac {1}{12} b \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {b}{12 x^2}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}+\frac {1}{20} b \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )\) |
| \(\Big \downarrow \) 3856 |
| \(\displaystyle -\frac {1}{5} \pi b^2 \left (\frac {1}{3} \pi b^2 \left (-\pi b^2 \int \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{x}+\frac {1}{4} b \text {Si}\left (b^2 \pi x^2\right )\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}-\frac {1}{12} b \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {b}{12 x^2}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}+\frac {1}{20} b \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )\) |
| \(\Big \downarrow \) 6994 |
| \(\displaystyle -\frac {1}{5} \pi b^2 \left (\frac {1}{3} \pi b^2 \left (-\pi b \int \operatorname {FresnelS}(b x)d\operatorname {FresnelS}(b x)-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{x}+\frac {1}{4} b \text {Si}\left (b^2 \pi x^2\right )\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}-\frac {1}{12} b \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {b}{12 x^2}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}+\frac {1}{20} b \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -\frac {1}{5} \pi b^2 \left (\frac {1}{3} \pi b^2 \left (-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{x}+\frac {1}{4} b \text {Si}\left (b^2 \pi x^2\right )-\frac {1}{2} \pi b \operatorname {FresnelS}(b x)^2\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}-\frac {1}{12} b \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {b}{12 x^2}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}+\frac {1}{20} b \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )\) |
-1/5*(Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/x^5 - (b^2*Pi*(-1/12*b/x^2 - (Fre snelS[b*x]*Sin[(b^2*Pi*x^2)/2])/(3*x^3) + (b^2*Pi*(-((Cos[(b^2*Pi*x^2)/2]* FresnelS[b*x])/x) - (b*Pi*FresnelS[b*x]^2)/2 + (b*SinIntegral[b^2*Pi*x^2]) /4))/3 - (b*(-(Cos[b^2*Pi*x^2]/x^2) - b^2*Pi*SinIntegral[b^2*Pi*x^2]))/12) )/5 + (b*(-1/2*Sin[b^2*Pi*x^2]/x^4 + (b^2*Pi*(-(Cos[b^2*Pi*x^2]/x^2) - b^2 *Pi*SinIntegral[b^2*Pi*x^2]))/2))/20
3.2.5.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(c + d*x)^(m + 1)*(Sin[e + f*x]/(d*(m + 1))), x] - Simp[f/(d*(m + 1)) Int[( c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[m, - 1]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinInte gral[e + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*e - c*f, 0]
Int[Sin[(d_.)*(x_)^(n_)]/(x_), x_Symbol] :> Simp[SinIntegral[d*x^n]/n, x] / ; FreeQ[{d, n}, x]
Int[(x_)^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol ] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Sin[c + d*x])^ p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Simplify[ (m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[ (m + 1)/n], 0]))
Int[((a_.) + Cos[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol ] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Cos[c + d*x])^ p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Simplify[ (m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[ (m + 1)/n], 0]))
Int[FresnelS[(b_.)*(x_)]^(n_.)*Sin[(d_.)*(x_)^2], x_Symbol] :> Simp[Pi*(b/( 2*d)) Subst[Int[x^n, x], x, FresnelS[b*x]], x] /; FreeQ[{b, d, n}, x] && EqQ[d^2, (Pi^2/4)*b^4]
Int[FresnelS[(b_.)*(x_)]*(x_)^(m_)*Sin[(d_.)*(x_)^2], x_Symbol] :> Simp[x^( m + 1)*Sin[d*x^2]*(FresnelS[b*x]/(m + 1)), x] + (-Simp[d*(x^(m + 2)/(Pi*b*( m + 1)*(m + 2))), x] - Simp[2*(d/(m + 1)) Int[x^(m + 2)*Cos[d*x^2]*Fresne lS[b*x], x], x] + Simp[d/(Pi*b*(m + 1)) Int[x^(m + 1)*Cos[2*d*x^2], x], x ]) /; FreeQ[{b, d}, x] && EqQ[d^2, (Pi^2/4)*b^4] && ILtQ[m, -2]
Int[Cos[(d_.)*(x_)^2]*FresnelS[(b_.)*(x_)]*(x_)^(m_), x_Symbol] :> Simp[x^( m + 1)*Cos[d*x^2]*(FresnelS[b*x]/(m + 1)), x] + (Simp[2*(d/(m + 1)) Int[x ^(m + 2)*Sin[d*x^2]*FresnelS[b*x], x], x] - Simp[d/(Pi*b*(m + 1)) Int[x^( m + 1)*Sin[2*d*x^2], x], x]) /; FreeQ[{b, d}, x] && EqQ[d^2, (Pi^2/4)*b^4] && ILtQ[m, -1]
\[\int \frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \operatorname {FresnelS}\left (b x \right )}{x^{6}}d x\]
Time = 0.27 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.87 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^6} \, dx=\frac {4 \, \pi ^{3} b^{5} x^{5} \operatorname {S}\left (b x\right )^{2} - 7 \, \pi ^{2} b^{5} x^{5} \operatorname {Si}\left (\pi b^{2} x^{2}\right ) - 10 \, \pi b^{3} x^{3} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} + 7 \, \pi b^{3} x^{3} + 8 \, {\left (\pi ^{2} b^{4} x^{4} - 3\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right ) + 2 \, {\left (4 \, \pi b^{2} x^{2} \operatorname {S}\left (b x\right ) - 3 \, b x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{120 \, x^{5}} \]
1/120*(4*pi^3*b^5*x^5*fresnel_sin(b*x)^2 - 7*pi^2*b^5*x^5*sin_integral(pi* b^2*x^2) - 10*pi*b^3*x^3*cos(1/2*pi*b^2*x^2)^2 + 7*pi*b^3*x^3 + 8*(pi^2*b^ 4*x^4 - 3)*cos(1/2*pi*b^2*x^2)*fresnel_sin(b*x) + 2*(4*pi*b^2*x^2*fresnel_ sin(b*x) - 3*b*x*cos(1/2*pi*b^2*x^2))*sin(1/2*pi*b^2*x^2))/x^5
\[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^6} \, dx=\int \frac {\cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )}{x^{6}}\, dx \]
\[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^6} \, dx=\int { \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right )}{x^{6}} \,d x } \]
\[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^6} \, dx=\int { \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right )}{x^{6}} \,d x } \]
Timed out. \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^6} \, dx=\int \frac {\mathrm {FresnelS}\left (b\,x\right )\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right )}{x^6} \,d x \]