3.2.56 \(\int \frac {\operatorname {FresnelC}(b x)^2}{x^9} \, dx\) [156]

3.2.56.1 Optimal result

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

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^2-1/28*b*cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/x^7+1/4 
20*b^5*Pi^2*cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/x^3+1/840*b^8*Pi^4*FresnelC( 
b*x)^2-1/8*FresnelC(b*x)^2/x^8+1/280*b^8*Pi^3*Si(b^2*Pi*x^2)+1/140*b^3*Pi* 
FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/x^5-1/420*b^7*Pi^3*FresnelC(b*x)*sin(1/2 
*b^2*Pi*x^2)/x+1/420*b^4*Pi*sin(b^2*Pi*x^2)/x^4
 

3.2.56.2 Mathematica [A] (verified)

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

input

Integrate[FresnelC[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*Cos[(b^2*Pi*x^2)/2]*FresnelC[b* 
x])/(28*x^7) + (b^5*Pi^2*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(420*x^3) + (b 
^8*Pi^4*FresnelC[b*x]^2)/840 - FresnelC[b*x]^2/(8*x^8) + (b^3*Pi*FresnelC[ 
b*x]*Sin[(b^2*Pi*x^2)/2])/(140*x^5) - (b^7*Pi^3*FresnelC[b*x]*Sin[(b^2*Pi* 
x^2)/2])/(420*x) + (b^4*Pi*Sin[b^2*Pi*x^2])/(420*x^4) + (b^8*Pi^3*SinInteg 
ral[b^2*Pi*x^2])/280
 

3.2.56.3 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.

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

\(\Big \downarrow \) 6985

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

\(\Big \downarrow \) 7011

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

\(\Big \downarrow \) 3861

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3778

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3778

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3778

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3780

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

\(\Big \downarrow \) 7019

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

\(\Big \downarrow \) 3860

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3778

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3778

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3780

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

\(\Big \downarrow \) 7011

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

\(\Big \downarrow \) 3861

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3778

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3780

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

\(\Big \downarrow \) 7019

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

input

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

output

$Aborted
 

3.2.56.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3778

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]
 

rule 3780

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]
 

rule 3860

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]))
 

rule 3861

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]))
 

rule 6985

Int[FresnelC[(b_.)*(x_)]^2*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*(Fresnel 
C[b*x]^2/(m + 1)), x] - Simp[2*(b/(m + 1))   Int[x^(m + 1)*Cos[(Pi/2)*b^2*x 
^2]*FresnelC[b*x], x], x] /; FreeQ[b, x] && IntegerQ[m] && NeQ[m, -1]
 

rule 7011

Int[Cos[(d_.)*(x_)^2]*FresnelC[(b_.)*(x_)]*(x_)^(m_), x_Symbol] :> Simp[x^( 
m + 1)*Cos[d*x^2]*(FresnelC[b*x]/(m + 1)), x] + (-Simp[b*(x^(m + 2)/(2*(m + 
 1)*(m + 2))), x] + Simp[2*(d/(m + 1))   Int[x^(m + 2)*Sin[d*x^2]*FresnelC[ 
b*x], x], x] - Simp[b/(2*(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]
 

rule 7019

Int[FresnelC[(b_.)*(x_)]*(x_)^(m_)*Sin[(d_.)*(x_)^2], x_Symbol] :> Simp[x^( 
m + 1)*Sin[d*x^2]*(FresnelC[b*x]/(m + 1)), x] + (-Simp[2*(d/(m + 1))   Int[ 
x^(m + 2)*Cos[d*x^2]*FresnelC[b*x], x], x] - Simp[b/(2*(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]
 

3.2.56.4 Maple [F]

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

input

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

output

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

3.2.56.5 Fricas [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.74 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^9} \, dx=\frac {3 \, \pi ^{3} b^{8} x^{8} \operatorname {Si}\left (\pi b^{2} x^{2}\right ) - 2 \, \pi ^{2} b^{6} x^{6} + 5 \, {\left (\pi ^{2} b^{6} x^{6} - b^{2} x^{2}\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} + 2 \, {\left (\pi ^{2} b^{5} x^{5} - 15 \, b x\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right ) + {\left (\pi ^{4} b^{8} x^{8} - 105\right )} \operatorname {C}\left (b x\right )^{2} + 2 \, {\left (2 \, \pi b^{4} x^{4} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - {\left (\pi ^{3} b^{7} x^{7} - 3 \, \pi b^{3} x^{3}\right )} \operatorname {C}\left (b x\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{840 \, x^{8}} \]

input

integrate(fresnel_cos(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) - 2*pi^2*b^6*x^6 + 5*(pi^2* 
b^6*x^6 - b^2*x^2)*cos(1/2*pi*b^2*x^2)^2 + 2*(pi^2*b^5*x^5 - 15*b*x)*cos(1 
/2*pi*b^2*x^2)*fresnel_cos(b*x) + (pi^4*b^8*x^8 - 105)*fresnel_cos(b*x)^2 
+ 2*(2*pi*b^4*x^4*cos(1/2*pi*b^2*x^2) - (pi^3*b^7*x^7 - 3*pi*b^3*x^3)*fres 
nel_cos(b*x))*sin(1/2*pi*b^2*x^2))/x^8
 

3.2.56.6 Sympy [F]

\[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^9} \, dx=\int \frac {C^{2}\left (b x\right )}{x^{9}}\, dx \]

input

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

output

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

3.2.56.7 Maxima [F]

\[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^9} \, dx=\int { \frac {\operatorname {C}\left (b x\right )^{2}}{x^{9}} \,d x } \]

input

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

output

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

3.2.56.8 Giac [F]

\[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^9} \, dx=\int { \frac {\operatorname {C}\left (b x\right )^{2}}{x^{9}} \,d x } \]

input

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

output

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

3.2.56.9 Mupad [F(-1)]

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

input

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

output

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