\(\int \frac {\cos (\frac {1}{2} b^2 \pi x^2) \operatorname {FresnelC}(b x)}{x^{10}} \, dx\) [198]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^{10}} \, dx=-\frac {b}{144 x^8}+\frac {b^5 \pi ^2}{2520 x^4}-\frac {b \cos \left (b^2 \pi x^2\right )}{144 x^8}+\frac {67 b^5 \pi ^2 \cos \left (b^2 \pi x^2\right )}{30240 x^4}+\frac {5 b^9 \pi ^4 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )}{2016}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{9 x^9}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{315 x^5}+\frac {b^2 \pi \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{63 x^7}-\frac {b^6 \pi ^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{945 x^3}+\frac {11 b^3 \pi \sin \left (b^2 \pi x^2\right )}{3024 x^6}-\frac {5 b^7 \pi ^3 \sin \left (b^2 \pi x^2\right )}{2016 x^2}+\frac {1}{945} b^8 \pi ^4 \text {Int}\left (\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^2},x\right ) \] Output:

-1/144*b/x^8+1/2520*b^5*Pi^2/x^4-1/144*b*cos(b^2*Pi*x^2)/x^8+67/30240*b^5* 
Pi^2*cos(b^2*Pi*x^2)/x^4+5/2016*b^9*Pi^4*Ci(b^2*Pi*x^2)-1/9*cos(1/2*b^2*Pi 
*x^2)*FresnelC(b*x)/x^9+1/315*b^4*Pi^2*cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/x 
^5+1/63*b^2*Pi*FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/x^7-1/945*b^6*Pi^3*Fresne 
lC(b*x)*sin(1/2*b^2*Pi*x^2)/x^3+11/3024*b^3*Pi*sin(b^2*Pi*x^2)/x^6-5/2016* 
b^7*Pi^3*sin(b^2*Pi*x^2)/x^2+1/945*b^8*Pi^4*Defer(Int)(cos(1/2*b^2*Pi*x^2) 
*FresnelC(b*x)/x^2,x)
 

Mathematica [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^{10}} \, dx=\int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^{10}} \, dx \] Input:

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

Output:

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

Rubi [N/A]

Not integrable

Time = 2.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 31, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}

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.

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

\(\Big \downarrow \) 7011

\(\displaystyle -\frac {1}{9} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8}dx+\frac {1}{18} b \int \frac {\cos \left (b^2 \pi x^2\right )}{x^9}dx-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}-\frac {b}{144 x^8}\)

\(\Big \downarrow \) 3861

\(\displaystyle -\frac {1}{9} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8}dx+\frac {1}{36} b \int \frac {\cos \left (b^2 \pi x^2\right )}{x^{10}}dx^2-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}-\frac {b}{144 x^8}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {1}{9} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8}dx+\frac {1}{36} b \int \frac {\sin \left (b^2 \pi x^2+\frac {\pi }{2}\right )}{x^{10}}dx^2-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}-\frac {b}{144 x^8}\)

\(\Big \downarrow \) 3778

\(\displaystyle -\frac {1}{9} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8}dx+\frac {1}{36} b \left (\frac {1}{4} \pi b^2 \int -\frac {\sin \left (b^2 \pi x^2\right )}{x^8}dx^2-\frac {\cos \left (\pi b^2 x^2\right )}{4 x^8}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}-\frac {b}{144 x^8}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {1}{9} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8}dx+\frac {1}{36} b \left (-\frac {1}{4} \pi b^2 \int \frac {\sin \left (b^2 \pi x^2\right )}{x^8}dx^2-\frac {\cos \left (\pi b^2 x^2\right )}{4 x^8}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}-\frac {b}{144 x^8}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {1}{9} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8}dx+\frac {1}{36} b \left (-\frac {1}{4} \pi b^2 \int \frac {\sin \left (b^2 \pi x^2\right )}{x^8}dx^2-\frac {\cos \left (\pi b^2 x^2\right )}{4 x^8}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}-\frac {b}{144 x^8}\)

\(\Big \downarrow \) 3778

\(\displaystyle -\frac {1}{9} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8}dx+\frac {1}{36} b \left (-\frac {1}{4} \pi b^2 \left (\frac {1}{3} \pi b^2 \int \frac {\cos \left (b^2 \pi x^2\right )}{x^6}dx^2-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{4 x^8}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}-\frac {b}{144 x^8}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {1}{9} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8}dx+\frac {1}{36} b \left (-\frac {1}{4} \pi b^2 \left (\frac {1}{3} \pi b^2 \int \frac {\sin \left (b^2 \pi x^2+\frac {\pi }{2}\right )}{x^6}dx^2-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{4 x^8}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}-\frac {b}{144 x^8}\)

\(\Big \downarrow \) 3778

\(\displaystyle -\frac {1}{9} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8}dx+\frac {1}{36} b \left (-\frac {1}{4} \pi b^2 \left (\frac {1}{3} \pi b^2 \left (\frac {1}{2} \pi b^2 \int -\frac {\sin \left (b^2 \pi x^2\right )}{x^4}dx^2-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{4 x^8}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}-\frac {b}{144 x^8}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {1}{9} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8}dx+\frac {1}{36} b \left (-\frac {1}{4} \pi b^2 \left (\frac {1}{3} \pi b^2 \left (-\frac {1}{2} \pi b^2 \int \frac {\sin \left (b^2 \pi x^2\right )}{x^4}dx^2-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{4 x^8}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}-\frac {b}{144 x^8}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {1}{9} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8}dx+\frac {1}{36} b \left (-\frac {1}{4} \pi b^2 \left (\frac {1}{3} \pi b^2 \left (-\frac {1}{2} \pi b^2 \int \frac {\sin \left (b^2 \pi x^2\right )}{x^4}dx^2-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{4 x^8}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}-\frac {b}{144 x^8}\)

\(\Big \downarrow \) 3778

\(\displaystyle -\frac {1}{9} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8}dx+\frac {1}{36} b \left (-\frac {1}{4} \pi b^2 \left (\frac {1}{3} \pi b^2 \left (-\frac {1}{2} \pi b^2 \left (\pi b^2 \int \frac {\cos \left (b^2 \pi x^2\right )}{x^2}dx^2-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{4 x^8}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}-\frac {b}{144 x^8}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {1}{9} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8}dx+\frac {1}{36} b \left (-\frac {1}{4} \pi b^2 \left (\frac {1}{3} \pi b^2 \left (-\frac {1}{2} \pi b^2 \left (\pi b^2 \int \frac {\sin \left (b^2 \pi x^2+\frac {\pi }{2}\right )}{x^2}dx^2-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{4 x^8}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}-\frac {b}{144 x^8}\)

\(\Big \downarrow \) 3783

\(\displaystyle -\frac {1}{9} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8}dx+\frac {1}{36} b \left (-\frac {1}{4} \pi b^2 \left (\frac {1}{3} \pi b^2 \left (-\frac {1}{2} \pi b^2 \left (\pi b^2 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{4 x^8}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}-\frac {b}{144 x^8}\)

\(\Big \downarrow \) 7019

\(\displaystyle -\frac {1}{9} \pi b^2 \left (\frac {1}{7} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^6}dx+\frac {1}{14} b \int \frac {\sin \left (b^2 \pi x^2\right )}{x^7}dx-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}\right )+\frac {1}{36} b \left (-\frac {1}{4} \pi b^2 \left (\frac {1}{3} \pi b^2 \left (-\frac {1}{2} \pi b^2 \left (\pi b^2 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{4 x^8}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}-\frac {b}{144 x^8}\)

\(\Big \downarrow \) 3860

\(\displaystyle -\frac {1}{9} \pi b^2 \left (\frac {1}{7} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^6}dx+\frac {1}{28} b \int \frac {\sin \left (b^2 \pi x^2\right )}{x^8}dx^2-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}\right )+\frac {1}{36} b \left (-\frac {1}{4} \pi b^2 \left (\frac {1}{3} \pi b^2 \left (-\frac {1}{2} \pi b^2 \left (\pi b^2 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{4 x^8}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}-\frac {b}{144 x^8}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {1}{9} \pi b^2 \left (\frac {1}{7} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^6}dx+\frac {1}{28} b \int \frac {\sin \left (b^2 \pi x^2\right )}{x^8}dx^2-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}\right )+\frac {1}{36} b \left (-\frac {1}{4} \pi b^2 \left (\frac {1}{3} \pi b^2 \left (-\frac {1}{2} \pi b^2 \left (\pi b^2 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{4 x^8}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}-\frac {b}{144 x^8}\)

\(\Big \downarrow \) 3778

\(\displaystyle -\frac {1}{9} \pi b^2 \left (\frac {1}{7} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^6}dx+\frac {1}{28} b \left (\frac {1}{3} \pi b^2 \int \frac {\cos \left (b^2 \pi x^2\right )}{x^6}dx^2-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}\right )+\frac {1}{36} b \left (-\frac {1}{4} \pi b^2 \left (\frac {1}{3} \pi b^2 \left (-\frac {1}{2} \pi b^2 \left (\pi b^2 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{4 x^8}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}-\frac {b}{144 x^8}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {1}{9} \pi b^2 \left (\frac {1}{7} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^6}dx+\frac {1}{28} b \left (\frac {1}{3} \pi b^2 \int \frac {\sin \left (b^2 \pi x^2+\frac {\pi }{2}\right )}{x^6}dx^2-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}\right )+\frac {1}{36} b \left (-\frac {1}{4} \pi b^2 \left (\frac {1}{3} \pi b^2 \left (-\frac {1}{2} \pi b^2 \left (\pi b^2 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{4 x^8}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}-\frac {b}{144 x^8}\)

\(\Big \downarrow \) 3778

\(\displaystyle -\frac {1}{9} \pi b^2 \left (\frac {1}{7} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^6}dx+\frac {1}{28} b \left (\frac {1}{3} \pi b^2 \left (\frac {1}{2} \pi b^2 \int -\frac {\sin \left (b^2 \pi x^2\right )}{x^4}dx^2-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}\right )+\frac {1}{36} b \left (-\frac {1}{4} \pi b^2 \left (\frac {1}{3} \pi b^2 \left (-\frac {1}{2} \pi b^2 \left (\pi b^2 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{4 x^8}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}-\frac {b}{144 x^8}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {1}{9} \pi b^2 \left (\frac {1}{7} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^6}dx+\frac {1}{28} b \left (\frac {1}{3} \pi b^2 \left (-\frac {1}{2} \pi b^2 \int \frac {\sin \left (b^2 \pi x^2\right )}{x^4}dx^2-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}\right )+\frac {1}{36} b \left (-\frac {1}{4} \pi b^2 \left (\frac {1}{3} \pi b^2 \left (-\frac {1}{2} \pi b^2 \left (\pi b^2 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{4 x^8}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}-\frac {b}{144 x^8}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {1}{9} \pi b^2 \left (\frac {1}{7} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^6}dx+\frac {1}{28} b \left (\frac {1}{3} \pi b^2 \left (-\frac {1}{2} \pi b^2 \int \frac {\sin \left (b^2 \pi x^2\right )}{x^4}dx^2-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}\right )+\frac {1}{36} b \left (-\frac {1}{4} \pi b^2 \left (\frac {1}{3} \pi b^2 \left (-\frac {1}{2} \pi b^2 \left (\pi b^2 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{4 x^8}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}-\frac {b}{144 x^8}\)

\(\Big \downarrow \) 3778

\(\displaystyle -\frac {1}{9} \pi b^2 \left (\frac {1}{7} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^6}dx+\frac {1}{28} b \left (\frac {1}{3} \pi b^2 \left (-\frac {1}{2} \pi b^2 \left (\pi b^2 \int \frac {\cos \left (b^2 \pi x^2\right )}{x^2}dx^2-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}\right )+\frac {1}{36} b \left (-\frac {1}{4} \pi b^2 \left (\frac {1}{3} \pi b^2 \left (-\frac {1}{2} \pi b^2 \left (\pi b^2 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{4 x^8}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}-\frac {b}{144 x^8}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {1}{9} \pi b^2 \left (\frac {1}{7} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^6}dx+\frac {1}{28} b \left (\frac {1}{3} \pi b^2 \left (-\frac {1}{2} \pi b^2 \left (\pi b^2 \int \frac {\sin \left (b^2 \pi x^2+\frac {\pi }{2}\right )}{x^2}dx^2-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}\right )+\frac {1}{36} b \left (-\frac {1}{4} \pi b^2 \left (\frac {1}{3} \pi b^2 \left (-\frac {1}{2} \pi b^2 \left (\pi b^2 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{4 x^8}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}-\frac {b}{144 x^8}\)

\(\Big \downarrow \) 3783

\(\displaystyle -\frac {1}{9} \pi b^2 \left (\frac {1}{7} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^6}dx+\frac {1}{28} b \left (\frac {1}{3} \pi b^2 \left (-\frac {1}{2} \pi b^2 \left (\pi b^2 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}\right )+\frac {1}{36} b \left (-\frac {1}{4} \pi b^2 \left (\frac {1}{3} \pi b^2 \left (-\frac {1}{2} \pi b^2 \left (\pi b^2 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{4 x^8}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}-\frac {b}{144 x^8}\)

\(\Big \downarrow \) 7011

\(\displaystyle -\frac {1}{9} \pi b^2 \left (\frac {1}{7} \pi b^2 \left (-\frac {1}{5} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4}dx+\frac {1}{10} b \int \frac {\cos \left (b^2 \pi x^2\right )}{x^5}dx-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}-\frac {b}{40 x^4}\right )+\frac {1}{28} b \left (\frac {1}{3} \pi b^2 \left (-\frac {1}{2} \pi b^2 \left (\pi b^2 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}\right )+\frac {1}{36} b \left (-\frac {1}{4} \pi b^2 \left (\frac {1}{3} \pi b^2 \left (-\frac {1}{2} \pi b^2 \left (\pi b^2 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{4 x^8}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}-\frac {b}{144 x^8}\)

\(\Big \downarrow \) 3861

\(\displaystyle -\frac {1}{9} \pi b^2 \left (\frac {1}{7} \pi b^2 \left (-\frac {1}{5} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4}dx+\frac {1}{20} b \int \frac {\cos \left (b^2 \pi x^2\right )}{x^6}dx^2-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}-\frac {b}{40 x^4}\right )+\frac {1}{28} b \left (\frac {1}{3} \pi b^2 \left (-\frac {1}{2} \pi b^2 \left (\pi b^2 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}\right )+\frac {1}{36} b \left (-\frac {1}{4} \pi b^2 \left (\frac {1}{3} \pi b^2 \left (-\frac {1}{2} \pi b^2 \left (\pi b^2 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{4 x^8}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}-\frac {b}{144 x^8}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {1}{9} \pi b^2 \left (\frac {1}{7} \pi b^2 \left (-\frac {1}{5} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4}dx+\frac {1}{20} b \int \frac {\sin \left (b^2 \pi x^2+\frac {\pi }{2}\right )}{x^6}dx^2-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}-\frac {b}{40 x^4}\right )+\frac {1}{28} b \left (\frac {1}{3} \pi b^2 \left (-\frac {1}{2} \pi b^2 \left (\pi b^2 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}\right )+\frac {1}{36} b \left (-\frac {1}{4} \pi b^2 \left (\frac {1}{3} \pi b^2 \left (-\frac {1}{2} \pi b^2 \left (\pi b^2 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{4 x^8}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}-\frac {b}{144 x^8}\)

\(\Big \downarrow \) 3778

\(\displaystyle -\frac {1}{9} \pi b^2 \left (\frac {1}{7} \pi b^2 \left (-\frac {1}{5} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4}dx+\frac {1}{20} b \left (\frac {1}{2} \pi b^2 \int -\frac {\sin \left (b^2 \pi x^2\right )}{x^4}dx^2-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}-\frac {b}{40 x^4}\right )+\frac {1}{28} b \left (\frac {1}{3} \pi b^2 \left (-\frac {1}{2} \pi b^2 \left (\pi b^2 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}\right )+\frac {1}{36} b \left (-\frac {1}{4} \pi b^2 \left (\frac {1}{3} \pi b^2 \left (-\frac {1}{2} \pi b^2 \left (\pi b^2 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{4 x^8}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}-\frac {b}{144 x^8}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {1}{9} \pi b^2 \left (\frac {1}{7} \pi b^2 \left (-\frac {1}{5} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4}dx+\frac {1}{20} b \left (-\frac {1}{2} \pi b^2 \int \frac {\sin \left (b^2 \pi x^2\right )}{x^4}dx^2-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}-\frac {b}{40 x^4}\right )+\frac {1}{28} b \left (\frac {1}{3} \pi b^2 \left (-\frac {1}{2} \pi b^2 \left (\pi b^2 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 x^7}\right )+\frac {1}{36} b \left (-\frac {1}{4} \pi b^2 \left (\frac {1}{3} \pi b^2 \left (-\frac {1}{2} \pi b^2 \left (\pi b^2 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^6}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{4 x^8}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{9 x^9}-\frac {b}{144 x^8}\)

Input:

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

Output:

$Aborted
 

Maple [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90

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

Input:

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

Output:

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

Fricas [N/A]

Not integrable

Time = 0.10 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^{10}} \, dx=\int { \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right )}{x^{10}} \,d x } \] Input:

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

Output:

integral(cos(1/2*pi*b^2*x^2)*fresnel_cos(b*x)/x^10, x)
 

Sympy [N/A]

Not integrable

Time = 65.73 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^{10}} \, dx=\int \frac {\cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )}{x^{10}}\, dx \] Input:

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

Output:

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

Maxima [N/A]

Not integrable

Time = 0.12 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^{10}} \, dx=\int { \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right )}{x^{10}} \,d x } \] Input:

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

Output:

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

Giac [N/A]

Not integrable

Time = 0.13 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^{10}} \, dx=\int { \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right )}{x^{10}} \,d x } \] Input:

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

Output:

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

Mupad [N/A]

Not integrable

Time = 3.61 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^{10}} \, dx=\int \frac {\mathrm {FresnelC}\left (b\,x\right )\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right )}{x^{10}} \,d x \] Input:

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

Output:

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

Reduce [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^{10}} \, dx=\int \frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \mathrm {FresnelC}\left (b x \right )}{x^{10}}d x \] Input:

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

Output:

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