\(\int \frac {\cos (\frac {1}{2} b^2 \pi x^2) \operatorname {FresnelS}(b x)}{x^{10}} \, dx\) [109]

Optimal result

Integrand size = 20, antiderivative size = 278 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^{10}} \, dx=\frac {b^3 \pi }{756 x^6}-\frac {b^7 \pi ^3}{3780 x^2}-\frac {11 b^3 \pi \cos \left (b^2 \pi x^2\right )}{3024 x^6}+\frac {5 b^7 \pi ^3 \cos \left (b^2 \pi x^2\right )}{2016 x^2}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{9 x^9}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{315 x^5}-\frac {b^8 \pi ^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{945 x}-\frac {b^9 \pi ^5 \operatorname {FresnelS}(b x)^2}{1890}+\frac {b^2 \pi \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{63 x^7}-\frac {b^6 \pi ^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{945 x^3}-\frac {b \sin \left (b^2 \pi x^2\right )}{144 x^8}+\frac {67 b^5 \pi ^2 \sin \left (b^2 \pi x^2\right )}{30240 x^4}+\frac {83 b^9 \pi ^4 \text {Si}\left (b^2 \pi x^2\right )}{30240} \] Output:

1/756*b^3*Pi/x^6-1/3780*b^7*Pi^3/x^2-11/3024*b^3*Pi*cos(b^2*Pi*x^2)/x^6+5/ 
2016*b^7*Pi^3*cos(b^2*Pi*x^2)/x^2-1/9*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/x^ 
9+1/315*b^4*Pi^2*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/x^5-1/945*b^8*Pi^4*cos( 
1/2*b^2*Pi*x^2)*FresnelS(b*x)/x-1/1890*b^9*Pi^5*FresnelS(b*x)^2+1/63*b^2*P 
i*FresnelS(b*x)*sin(1/2*b^2*Pi*x^2)/x^7-1/945*b^6*Pi^3*FresnelS(b*x)*sin(1 
/2*b^2*Pi*x^2)/x^3-1/144*b*sin(b^2*Pi*x^2)/x^8+67/30240*b^5*Pi^2*sin(b^2*P 
i*x^2)/x^4+83/30240*b^9*Pi^4*Si(b^2*Pi*x^2)
 

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^{10}} \, dx=\frac {b^3 \pi }{756 x^6}-\frac {b^7 \pi ^3}{3780 x^2}-\frac {11 b^3 \pi \cos \left (b^2 \pi x^2\right )}{3024 x^6}+\frac {5 b^7 \pi ^3 \cos \left (b^2 \pi x^2\right )}{2016 x^2}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{9 x^9}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{315 x^5}-\frac {b^8 \pi ^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{945 x}-\frac {b^9 \pi ^5 \operatorname {FresnelS}(b x)^2}{1890}+\frac {b^2 \pi \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{63 x^7}-\frac {b^6 \pi ^3 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{945 x^3}-\frac {b \sin \left (b^2 \pi x^2\right )}{144 x^8}+\frac {67 b^5 \pi ^2 \sin \left (b^2 \pi x^2\right )}{30240 x^4}+\frac {83 b^9 \pi ^4 \text {Si}\left (b^2 \pi x^2\right )}{30240} \] Input:

Integrate[(Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/x^10,x]
 

Output:

(b^3*Pi)/(756*x^6) - (b^7*Pi^3)/(3780*x^2) - (11*b^3*Pi*Cos[b^2*Pi*x^2])/( 
3024*x^6) + (5*b^7*Pi^3*Cos[b^2*Pi*x^2])/(2016*x^2) - (Cos[(b^2*Pi*x^2)/2] 
*FresnelS[b*x])/(9*x^9) + (b^4*Pi^2*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(31 
5*x^5) - (b^8*Pi^4*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(945*x) - (b^9*Pi^5* 
FresnelS[b*x]^2)/1890 + (b^2*Pi*FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/(63*x^7 
) - (b^6*Pi^3*FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/(945*x^3) - (b*Sin[b^2*Pi 
*x^2])/(144*x^8) + (67*b^5*Pi^2*Sin[b^2*Pi*x^2])/(30240*x^4) + (83*b^9*Pi^ 
4*SinIntegral[b^2*Pi*x^2])/30240
 

Rubi [F]

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.

\(\displaystyle \int \frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{x^{10}} \, dx\)

\(\Big \downarrow \) 7018

\(\displaystyle -\frac {1}{9} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8}dx+\frac {1}{18} b \int \frac {\sin \left (b^2 \pi x^2\right )}{x^9}dx-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}\)

\(\Big \downarrow \) 3860

\(\displaystyle -\frac {1}{9} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8}dx+\frac {1}{36} b \int \frac {\sin \left (b^2 \pi x^2\right )}{x^{10}}dx^2-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {1}{9} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8}dx+\frac {1}{36} b \int \frac {\sin \left (b^2 \pi x^2\right )}{x^{10}}dx^2-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}\)

\(\Big \downarrow \) 3778

\(\displaystyle -\frac {1}{9} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8}dx+\frac {1}{36} b \left (\frac {1}{4} \pi b^2 \int \frac {\cos \left (b^2 \pi x^2\right )}{x^8}dx^2-\frac {\sin \left (\pi b^2 x^2\right )}{4 x^8}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {1}{9} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8}dx+\frac {1}{36} b \left (\frac {1}{4} \pi b^2 \int \frac {\sin \left (b^2 \pi x^2+\frac {\pi }{2}\right )}{x^8}dx^2-\frac {\sin \left (\pi b^2 x^2\right )}{4 x^8}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}\)

\(\Big \downarrow \) 3778

\(\displaystyle -\frac {1}{9} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8}dx+\frac {1}{36} b \left (\frac {1}{4} \pi b^2 \left (\frac {1}{3} \pi b^2 \int -\frac {\sin \left (b^2 \pi x^2\right )}{x^6}dx^2-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{4 x^8}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {1}{9} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8}dx+\frac {1}{36} b \left (\frac {1}{4} \pi b^2 \left (-\frac {1}{3} \pi b^2 \int \frac {\sin \left (b^2 \pi x^2\right )}{x^6}dx^2-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{4 x^8}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {1}{9} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8}dx+\frac {1}{36} b \left (\frac {1}{4} \pi b^2 \left (-\frac {1}{3} \pi b^2 \int \frac {\sin \left (b^2 \pi x^2\right )}{x^6}dx^2-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{4 x^8}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}\)

\(\Big \downarrow \) 3778

\(\displaystyle -\frac {1}{9} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8}dx+\frac {1}{36} b \left (\frac {1}{4} \pi b^2 \left (-\frac {1}{3} \pi b^2 \left (\frac {1}{2} \pi b^2 \int \frac {\cos \left (b^2 \pi x^2\right )}{x^4}dx^2-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{4 x^8}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {1}{9} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8}dx+\frac {1}{36} b \left (\frac {1}{4} \pi b^2 \left (-\frac {1}{3} \pi b^2 \left (\frac {1}{2} \pi b^2 \int \frac {\sin \left (b^2 \pi x^2+\frac {\pi }{2}\right )}{x^4}dx^2-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{4 x^8}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}\)

\(\Big \downarrow \) 3778

\(\displaystyle -\frac {1}{9} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8}dx+\frac {1}{36} b \left (\frac {1}{4} \pi b^2 \left (-\frac {1}{3} \pi b^2 \left (\frac {1}{2} \pi b^2 \left (\pi b^2 \int -\frac {\sin \left (b^2 \pi x^2\right )}{x^2}dx^2-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{4 x^8}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {1}{9} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8}dx+\frac {1}{36} b \left (\frac {1}{4} \pi b^2 \left (-\frac {1}{3} \pi b^2 \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2}dx^2-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{4 x^8}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {1}{9} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8}dx+\frac {1}{36} b \left (\frac {1}{4} \pi b^2 \left (-\frac {1}{3} \pi b^2 \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2}dx^2-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{4 x^8}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}\)

\(\Big \downarrow \) 3780

\(\displaystyle -\frac {1}{9} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8}dx-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}+\frac {1}{36} b \left (\frac {1}{4} \pi b^2 \left (-\frac {1}{3} \pi b^2 \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{4 x^8}\right )\)

\(\Big \downarrow \) 7010

\(\displaystyle -\frac {1}{9} \pi b^2 \left (\frac {1}{7} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^6}dx-\frac {1}{14} b \int \frac {\cos \left (b^2 \pi x^2\right )}{x^7}dx-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}-\frac {b}{84 x^6}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}+\frac {1}{36} b \left (\frac {1}{4} \pi b^2 \left (-\frac {1}{3} \pi b^2 \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{4 x^8}\right )\)

\(\Big \downarrow \) 3861

\(\displaystyle -\frac {1}{9} \pi b^2 \left (\frac {1}{7} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^6}dx-\frac {1}{28} b \int \frac {\cos \left (b^2 \pi x^2\right )}{x^8}dx^2-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}-\frac {b}{84 x^6}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}+\frac {1}{36} b \left (\frac {1}{4} \pi b^2 \left (-\frac {1}{3} \pi b^2 \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{4 x^8}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {1}{9} \pi b^2 \left (\frac {1}{7} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^6}dx-\frac {1}{28} b \int \frac {\sin \left (b^2 \pi x^2+\frac {\pi }{2}\right )}{x^8}dx^2-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}-\frac {b}{84 x^6}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}+\frac {1}{36} b \left (\frac {1}{4} \pi b^2 \left (-\frac {1}{3} \pi b^2 \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{4 x^8}\right )\)

\(\Big \downarrow \) 3778

\(\displaystyle -\frac {1}{9} \pi b^2 \left (\frac {1}{7} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^6}dx-\frac {1}{28} b \left (\frac {1}{3} \pi b^2 \int -\frac {\sin \left (b^2 \pi x^2\right )}{x^6}dx^2-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}-\frac {b}{84 x^6}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}+\frac {1}{36} b \left (\frac {1}{4} \pi b^2 \left (-\frac {1}{3} \pi b^2 \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{4 x^8}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {1}{9} \pi b^2 \left (\frac {1}{7} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^6}dx-\frac {1}{28} b \left (-\frac {1}{3} \pi b^2 \int \frac {\sin \left (b^2 \pi x^2\right )}{x^6}dx^2-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}-\frac {b}{84 x^6}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}+\frac {1}{36} b \left (\frac {1}{4} \pi b^2 \left (-\frac {1}{3} \pi b^2 \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{4 x^8}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {1}{9} \pi b^2 \left (\frac {1}{7} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^6}dx-\frac {1}{28} b \left (-\frac {1}{3} \pi b^2 \int \frac {\sin \left (b^2 \pi x^2\right )}{x^6}dx^2-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}-\frac {b}{84 x^6}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}+\frac {1}{36} b \left (\frac {1}{4} \pi b^2 \left (-\frac {1}{3} \pi b^2 \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{4 x^8}\right )\)

\(\Big \downarrow \) 3778

\(\displaystyle -\frac {1}{9} \pi b^2 \left (\frac {1}{7} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^6}dx-\frac {1}{28} b \left (-\frac {1}{3} \pi b^2 \left (\frac {1}{2} \pi b^2 \int \frac {\cos \left (b^2 \pi x^2\right )}{x^4}dx^2-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}-\frac {b}{84 x^6}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}+\frac {1}{36} b \left (\frac {1}{4} \pi b^2 \left (-\frac {1}{3} \pi b^2 \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{4 x^8}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {1}{9} \pi b^2 \left (\frac {1}{7} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^6}dx-\frac {1}{28} b \left (-\frac {1}{3} \pi b^2 \left (\frac {1}{2} \pi b^2 \int \frac {\sin \left (b^2 \pi x^2+\frac {\pi }{2}\right )}{x^4}dx^2-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}-\frac {b}{84 x^6}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}+\frac {1}{36} b \left (\frac {1}{4} \pi b^2 \left (-\frac {1}{3} \pi b^2 \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{4 x^8}\right )\)

\(\Big \downarrow \) 3778

\(\displaystyle -\frac {1}{9} \pi b^2 \left (\frac {1}{7} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^6}dx-\frac {1}{28} b \left (-\frac {1}{3} \pi b^2 \left (\frac {1}{2} \pi b^2 \left (\pi b^2 \int -\frac {\sin \left (b^2 \pi x^2\right )}{x^2}dx^2-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}-\frac {b}{84 x^6}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}+\frac {1}{36} b \left (\frac {1}{4} \pi b^2 \left (-\frac {1}{3} \pi b^2 \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{4 x^8}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {1}{9} \pi b^2 \left (\frac {1}{7} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^6}dx-\frac {1}{28} b \left (-\frac {1}{3} \pi b^2 \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2}dx^2-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}-\frac {b}{84 x^6}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}+\frac {1}{36} b \left (\frac {1}{4} \pi b^2 \left (-\frac {1}{3} \pi b^2 \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{4 x^8}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {1}{9} \pi b^2 \left (\frac {1}{7} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^6}dx-\frac {1}{28} b \left (-\frac {1}{3} \pi b^2 \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2}dx^2-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}-\frac {b}{84 x^6}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}+\frac {1}{36} b \left (\frac {1}{4} \pi b^2 \left (-\frac {1}{3} \pi b^2 \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{4 x^8}\right )\)

\(\Big \downarrow \) 3780

\(\displaystyle -\frac {1}{9} \pi b^2 \left (\frac {1}{7} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^6}dx-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}-\frac {1}{28} b \left (-\frac {1}{3} \pi b^2 \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {b}{84 x^6}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}+\frac {1}{36} b \left (\frac {1}{4} \pi b^2 \left (-\frac {1}{3} \pi b^2 \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{4 x^8}\right )\)

\(\Big \downarrow \) 7018

\(\displaystyle -\frac {1}{9} \pi b^2 \left (\frac {1}{7} \pi b^2 \left (-\frac {1}{5} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4}dx+\frac {1}{10} b \int \frac {\sin \left (b^2 \pi x^2\right )}{x^5}dx-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}-\frac {1}{28} b \left (-\frac {1}{3} \pi b^2 \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {b}{84 x^6}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}+\frac {1}{36} b \left (\frac {1}{4} \pi b^2 \left (-\frac {1}{3} \pi b^2 \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{4 x^8}\right )\)

\(\Big \downarrow \) 3860

\(\displaystyle -\frac {1}{9} \pi b^2 \left (\frac {1}{7} \pi b^2 \left (-\frac {1}{5} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4}dx+\frac {1}{20} b \int \frac {\sin \left (b^2 \pi x^2\right )}{x^6}dx^2-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}-\frac {1}{28} b \left (-\frac {1}{3} \pi b^2 \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {b}{84 x^6}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}+\frac {1}{36} b \left (\frac {1}{4} \pi b^2 \left (-\frac {1}{3} \pi b^2 \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{4 x^8}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {1}{9} \pi b^2 \left (\frac {1}{7} \pi b^2 \left (-\frac {1}{5} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4}dx+\frac {1}{20} b \int \frac {\sin \left (b^2 \pi x^2\right )}{x^6}dx^2-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}-\frac {1}{28} b \left (-\frac {1}{3} \pi b^2 \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {b}{84 x^6}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}+\frac {1}{36} b \left (\frac {1}{4} \pi b^2 \left (-\frac {1}{3} \pi b^2 \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{4 x^8}\right )\)

\(\Big \downarrow \) 3778

\(\displaystyle -\frac {1}{9} \pi b^2 \left (\frac {1}{7} \pi b^2 \left (-\frac {1}{5} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4}dx+\frac {1}{20} b \left (\frac {1}{2} \pi b^2 \int \frac {\cos \left (b^2 \pi x^2\right )}{x^4}dx^2-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}-\frac {1}{28} b \left (-\frac {1}{3} \pi b^2 \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {b}{84 x^6}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}+\frac {1}{36} b \left (\frac {1}{4} \pi b^2 \left (-\frac {1}{3} \pi b^2 \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{4 x^8}\right )\)

Input:

Int[(Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/x^10,x]
 

Output:

$Aborted
 

Maple [F]

\[\int \frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \operatorname {FresnelS}\left (b x \right )}{x^{10}}d x\]

Input:

int(cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/x^10,x)
 

Output:

int(cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/x^10,x)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.73 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^{10}} \, dx=-\frac {16 \, \pi ^{5} b^{9} x^{9} \operatorname {S}\left (b x\right )^{2} - 83 \, \pi ^{4} b^{9} x^{9} \operatorname {Si}\left (\pi b^{2} x^{2}\right ) + 83 \, \pi ^{3} b^{7} x^{7} - 150 \, \pi b^{3} x^{3} - 10 \, {\left (15 \, \pi ^{3} b^{7} x^{7} - 22 \, \pi b^{3} x^{3}\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} + 32 \, {\left (\pi ^{4} b^{8} x^{8} - 3 \, \pi ^{2} b^{4} x^{4} + 105\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right ) - 2 \, {\left ({\left (67 \, \pi ^{2} b^{5} x^{5} - 210 \, b x\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - 16 \, {\left (\pi ^{3} b^{6} x^{6} - 15 \, \pi b^{2} x^{2}\right )} \operatorname {S}\left (b x\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{30240 \, x^{9}} \] Input:

integrate(cos(1/2*b^2*pi*x^2)*fresnel_sin(b*x)/x^10,x, algorithm="fricas")
 

Output:

-1/30240*(16*pi^5*b^9*x^9*fresnel_sin(b*x)^2 - 83*pi^4*b^9*x^9*sin_integra 
l(pi*b^2*x^2) + 83*pi^3*b^7*x^7 - 150*pi*b^3*x^3 - 10*(15*pi^3*b^7*x^7 - 2 
2*pi*b^3*x^3)*cos(1/2*pi*b^2*x^2)^2 + 32*(pi^4*b^8*x^8 - 3*pi^2*b^4*x^4 + 
105)*cos(1/2*pi*b^2*x^2)*fresnel_sin(b*x) - 2*((67*pi^2*b^5*x^5 - 210*b*x) 
*cos(1/2*pi*b^2*x^2) - 16*(pi^3*b^6*x^6 - 15*pi*b^2*x^2)*fresnel_sin(b*x)) 
*sin(1/2*pi*b^2*x^2))/x^9
 

Sympy [F]

\[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^{10}} \, dx=\int \frac {\cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )}{x^{10}}\, dx \] Input:

integrate(cos(1/2*b**2*pi*x**2)*fresnels(b*x)/x**10,x)
 

Output:

Integral(cos(pi*b**2*x**2/2)*fresnels(b*x)/x**10, x)
 

Maxima [F]

\[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^{10}} \, dx=\int { \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right )}{x^{10}} \,d x } \] Input:

integrate(cos(1/2*b^2*pi*x^2)*fresnel_sin(b*x)/x^10,x, algorithm="maxima")
 

Output:

integrate(cos(1/2*pi*b^2*x^2)*fresnel_sin(b*x)/x^10, x)
 

Giac [F]

\[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^{10}} \, dx=\int { \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right )}{x^{10}} \,d x } \] Input:

integrate(cos(1/2*b^2*pi*x^2)*fresnel_sin(b*x)/x^10,x, algorithm="giac")
 

Output:

integrate(cos(1/2*pi*b^2*x^2)*fresnel_sin(b*x)/x^10, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^{10}} \, dx=\int \frac {\mathrm {FresnelS}\left (b\,x\right )\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right )}{x^{10}} \,d x \] Input:

int((FresnelS(b*x)*cos((Pi*b^2*x^2)/2))/x^10,x)
 

Output:

int((FresnelS(b*x)*cos((Pi*b^2*x^2)/2))/x^10, x)
 

Reduce [F]

\[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^{10}} \, dx=\int \frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \mathrm {FresnelS}\left (b x \right )}{x^{10}}d x \] Input:

int(cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/x^10,x)
 

Output:

int(cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/x^10,x)