\(\int \frac {\operatorname {FresnelS}(b x)^2}{x^9} \, dx\) [47]

Optimal result

Integrand size = 10, antiderivative size = 242 \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^9} \, dx=-\frac {b^2}{336 x^6}+\frac {b^6 \pi ^2}{1680 x^2}+\frac {b^2 \cos \left (b^2 \pi x^2\right )}{336 x^6}-\frac {b^6 \pi ^2 \cos \left (b^2 \pi x^2\right )}{336 x^2}-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{140 x^5}+\frac {b^7 \pi ^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{420 x}+\frac {1}{840} b^8 \pi ^4 \operatorname {FresnelS}(b x)^2-\frac {\operatorname {FresnelS}(b x)^2}{8 x^8}-\frac {b \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{28 x^7}+\frac {b^5 \pi ^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{420 x^3}-\frac {b^4 \pi \sin \left (b^2 \pi x^2\right )}{420 x^4}-\frac {1}{280} b^8 \pi ^3 \text {Si}\left (b^2 \pi x^2\right ) \] Output:

-1/336*b^2/x^6+1/1680*b^6*Pi^2/x^2+1/336*b^2*cos(b^2*Pi*x^2)/x^6-1/336*b^6 
*Pi^2*cos(b^2*Pi*x^2)/x^2-1/140*b^3*Pi*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/x 
^5+1/420*b^7*Pi^3*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/x+1/840*b^8*Pi^4*Fresn 
elS(b*x)^2-1/8*FresnelS(b*x)^2/x^8-1/28*b*FresnelS(b*x)*sin(1/2*b^2*Pi*x^2 
)/x^7+1/420*b^5*Pi^2*FresnelS(b*x)*sin(1/2*b^2*Pi*x^2)/x^3-1/420*b^4*Pi*si 
n(b^2*Pi*x^2)/x^4-1/280*b^8*Pi^3*Si(b^2*Pi*x^2)
 

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^9} \, dx=-\frac {b^2}{336 x^6}+\frac {b^6 \pi ^2}{1680 x^2}+\frac {b^2 \cos \left (b^2 \pi x^2\right )}{336 x^6}-\frac {b^6 \pi ^2 \cos \left (b^2 \pi x^2\right )}{336 x^2}-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{140 x^5}+\frac {b^7 \pi ^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{420 x}+\frac {1}{840} b^8 \pi ^4 \operatorname {FresnelS}(b x)^2-\frac {\operatorname {FresnelS}(b x)^2}{8 x^8}-\frac {b \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{28 x^7}+\frac {b^5 \pi ^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{420 x^3}-\frac {b^4 \pi \sin \left (b^2 \pi x^2\right )}{420 x^4}-\frac {1}{280} b^8 \pi ^3 \text {Si}\left (b^2 \pi x^2\right ) \] Input:

Integrate[FresnelS[b*x]^2/x^9,x]
 

Output:

-1/336*b^2/x^6 + (b^6*Pi^2)/(1680*x^2) + (b^2*Cos[b^2*Pi*x^2])/(336*x^6) - 
 (b^6*Pi^2*Cos[b^2*Pi*x^2])/(336*x^2) - (b^3*Pi*Cos[(b^2*Pi*x^2)/2]*Fresne 
lS[b*x])/(140*x^5) + (b^7*Pi^3*Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(420*x) 
+ (b^8*Pi^4*FresnelS[b*x]^2)/840 - FresnelS[b*x]^2/(8*x^8) - (b*FresnelS[b 
*x]*Sin[(b^2*Pi*x^2)/2])/(28*x^7) + (b^5*Pi^2*FresnelS[b*x]*Sin[(b^2*Pi*x^ 
2)/2])/(420*x^3) - (b^4*Pi*Sin[b^2*Pi*x^2])/(420*x^4) - (b^8*Pi^3*SinInteg 
ral[b^2*Pi*x^2])/280
 

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.

\(\displaystyle \int \frac {\operatorname {FresnelS}(b x)^2}{x^9} \, dx\)

\(\Big \downarrow \) 6984

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

\(\Big \downarrow \) 7010

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

\(\Big \downarrow \) 3861

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3778

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3778

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3778

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3780

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

\(\Big \downarrow \) 7018

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

\(\Big \downarrow \) 3860

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3778

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3778

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3780

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

\(\Big \downarrow \) 7010

\(\displaystyle \frac {1}{4} b \left (\frac {1}{7} \pi b^2 \left (-\frac {1}{5} \pi b^2 \left (\frac {1}{3} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^2}dx-\frac {1}{6} b \int \frac {\cos \left (b^2 \pi x^2\right )}{x^3}dx-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}-\frac {b}{12 x^2}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}+\frac {1}{20} b \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}-\frac {1}{28} b \left (-\frac {1}{3} \pi b^2 \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {b}{84 x^6}\right )-\frac {\operatorname {FresnelS}(b x)^2}{8 x^8}\)

\(\Big \downarrow \) 3861

\(\displaystyle \frac {1}{4} b \left (\frac {1}{7} \pi b^2 \left (-\frac {1}{5} \pi b^2 \left (\frac {1}{3} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^2}dx-\frac {1}{12} b \int \frac {\cos \left (b^2 \pi x^2\right )}{x^4}dx^2-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}-\frac {b}{12 x^2}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}+\frac {1}{20} b \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}-\frac {1}{28} b \left (-\frac {1}{3} \pi b^2 \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {b}{84 x^6}\right )-\frac {\operatorname {FresnelS}(b x)^2}{8 x^8}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{4} b \left (\frac {1}{7} \pi b^2 \left (-\frac {1}{5} \pi b^2 \left (\frac {1}{3} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^2}dx-\frac {1}{12} b \int \frac {\sin \left (b^2 \pi x^2+\frac {\pi }{2}\right )}{x^4}dx^2-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}-\frac {b}{12 x^2}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}+\frac {1}{20} b \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}-\frac {1}{28} b \left (-\frac {1}{3} \pi b^2 \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {b}{84 x^6}\right )-\frac {\operatorname {FresnelS}(b x)^2}{8 x^8}\)

\(\Big \downarrow \) 3778

\(\displaystyle \frac {1}{4} b \left (\frac {1}{7} \pi b^2 \left (-\frac {1}{5} \pi b^2 \left (\frac {1}{3} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^2}dx-\frac {1}{12} b \left (\pi b^2 \int -\frac {\sin \left (b^2 \pi x^2\right )}{x^2}dx^2-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}-\frac {b}{12 x^2}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}+\frac {1}{20} b \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}-\frac {1}{28} b \left (-\frac {1}{3} \pi b^2 \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {b}{84 x^6}\right )-\frac {\operatorname {FresnelS}(b x)^2}{8 x^8}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{4} b \left (\frac {1}{7} \pi b^2 \left (-\frac {1}{5} \pi b^2 \left (\frac {1}{3} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^2}dx-\frac {1}{12} b \left (-\pi b^2 \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2}dx^2-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}-\frac {b}{12 x^2}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}+\frac {1}{20} b \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}-\frac {1}{28} b \left (-\frac {1}{3} \pi b^2 \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {b}{84 x^6}\right )-\frac {\operatorname {FresnelS}(b x)^2}{8 x^8}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{4} b \left (\frac {1}{7} \pi b^2 \left (-\frac {1}{5} \pi b^2 \left (\frac {1}{3} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^2}dx-\frac {1}{12} b \left (-\pi b^2 \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2}dx^2-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}-\frac {b}{12 x^2}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}+\frac {1}{20} b \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}-\frac {1}{28} b \left (-\frac {1}{3} \pi b^2 \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {b}{84 x^6}\right )-\frac {\operatorname {FresnelS}(b x)^2}{8 x^8}\)

\(\Big \downarrow \) 3780

\(\displaystyle \frac {1}{4} b \left (\frac {1}{7} \pi b^2 \left (-\frac {1}{5} \pi b^2 \left (\frac {1}{3} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^2}dx-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}-\frac {1}{12} b \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {b}{12 x^2}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}+\frac {1}{20} b \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}-\frac {1}{28} b \left (-\frac {1}{3} \pi b^2 \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {b}{84 x^6}\right )-\frac {\operatorname {FresnelS}(b x)^2}{8 x^8}\)

\(\Big \downarrow \) 7018

\(\displaystyle \frac {1}{4} b \left (\frac {1}{7} \pi b^2 \left (-\frac {1}{5} \pi b^2 \left (\frac {1}{3} \pi b^2 \left (-\pi b^2 \int \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx+\frac {1}{2} b \int \frac {\sin \left (b^2 \pi x^2\right )}{x}dx-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{x}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}-\frac {1}{12} b \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {b}{12 x^2}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}+\frac {1}{20} b \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}-\frac {1}{28} b \left (-\frac {1}{3} \pi b^2 \left (\frac {1}{2} \pi b^2 \left (-\pi b^2 \text {Si}\left (b^2 \pi x^2\right )-\frac {\cos \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {b}{84 x^6}\right )-\frac {\operatorname {FresnelS}(b x)^2}{8 x^8}\)

Input:

Int[FresnelS[b*x]^2/x^9,x]
 

Output:

$Aborted
 

Maple [F]

\[\int \frac {\operatorname {FresnelS}\left (b x \right )^{2}}{x^{9}}d x\]

Input:

int(FresnelS(b*x)^2/x^9,x)
 

Output:

int(FresnelS(b*x)^2/x^9,x)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.77 \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^9} \, dx=-\frac {3 \, \pi ^{3} b^{8} x^{8} \operatorname {Si}\left (\pi b^{2} x^{2}\right ) - 3 \, \pi ^{2} b^{6} x^{6} + 5 \, b^{2} x^{2} + 5 \, {\left (\pi ^{2} b^{6} x^{6} - b^{2} x^{2}\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} - 2 \, {\left (\pi ^{3} b^{7} x^{7} - 3 \, \pi b^{3} x^{3}\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right ) - {\left (\pi ^{4} b^{8} x^{8} - 105\right )} \operatorname {S}\left (b x\right )^{2} + 2 \, {\left (2 \, \pi b^{4} x^{4} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - {\left (\pi ^{2} b^{5} x^{5} - 15 \, b x\right )} \operatorname {S}\left (b x\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{840 \, x^{8}} \] Input:

integrate(fresnel_sin(b*x)^2/x^9,x, algorithm="fricas")
 

Output:

-1/840*(3*pi^3*b^8*x^8*sin_integral(pi*b^2*x^2) - 3*pi^2*b^6*x^6 + 5*b^2*x 
^2 + 5*(pi^2*b^6*x^6 - b^2*x^2)*cos(1/2*pi*b^2*x^2)^2 - 2*(pi^3*b^7*x^7 - 
3*pi*b^3*x^3)*cos(1/2*pi*b^2*x^2)*fresnel_sin(b*x) - (pi^4*b^8*x^8 - 105)* 
fresnel_sin(b*x)^2 + 2*(2*pi*b^4*x^4*cos(1/2*pi*b^2*x^2) - (pi^2*b^5*x^5 - 
 15*b*x)*fresnel_sin(b*x))*sin(1/2*pi*b^2*x^2))/x^8
 

Sympy [F]

\[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^9} \, dx=\int \frac {S^{2}\left (b x\right )}{x^{9}}\, dx \] Input:

integrate(fresnels(b*x)**2/x**9,x)
 

Output:

Integral(fresnels(b*x)**2/x**9, x)
 

Maxima [F]

\[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^9} \, dx=\int { \frac {\operatorname {S}\left (b x\right )^{2}}{x^{9}} \,d x } \] Input:

integrate(fresnel_sin(b*x)^2/x^9,x, algorithm="maxima")
 

Output:

integrate(fresnel_sin(b*x)^2/x^9, x)
 

Giac [F]

\[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^9} \, dx=\int { \frac {\operatorname {S}\left (b x\right )^{2}}{x^{9}} \,d x } \] Input:

integrate(fresnel_sin(b*x)^2/x^9,x, algorithm="giac")
 

Output:

integrate(fresnel_sin(b*x)^2/x^9, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^9} \, dx=\int \frac {{\mathrm {FresnelS}\left (b\,x\right )}^2}{x^9} \,d x \] Input:

int(FresnelS(b*x)^2/x^9,x)
 

Output:

int(FresnelS(b*x)^2/x^9, x)
 

Reduce [F]

\[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^9} \, dx=\int \frac {\mathrm {FresnelS}\left (b x \right )^{2}}{x^{9}}d x \] Input:

int(FresnelS(b*x)^2/x^9,x)
 

Output:

int(FresnelS(b*x)^2/x^9,x)