\(\int \frac {\operatorname {FresnelS}(b x) \sin (\frac {1}{2} b^2 \pi x^2)}{x^{10}} \, dx\) [89]

Optimal result

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

Mathematica [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^{10}} \, dx=\int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^{10}} \, dx \] Input:

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

Output:

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

Rubi [N/A]

Not integrable

Time = 2.31 (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.

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

\(\Big \downarrow \) 7010

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

\(\Big \downarrow \) 3861

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3778

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3778

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3778

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3778

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3783

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

\(\Big \downarrow \) 7018

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

\(\Big \downarrow \) 3860

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3778

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3778

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3778

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3783

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

\(\Big \downarrow \) 7010

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

\(\Big \downarrow \) 3861

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3778

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

\(\Big \downarrow \) 25

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

Input:

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

Output:

$Aborted
 

Maple [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90

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

Input:

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

Output:

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

Fricas [N/A]

Not integrable

Time = 0.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^{10}} \, dx=\int { \frac {\operatorname {S}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{x^{10}} \,d x } \] Input:

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

Output:

integral(fresnel_sin(b*x)*sin(1/2*pi*b^2*x^2)/x^10, x)
 

Sympy [N/A]

Not integrable

Time = 65.38 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^{10}} \, dx=\int \frac {\sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )}{x^{10}}\, dx \] Input:

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

Output:

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

Maxima [N/A]

Not integrable

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

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

Output:

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

Giac [N/A]

Not integrable

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

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

Output:

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

Mupad [N/A]

Not integrable

Time = 3.39 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^{10}} \, dx=\int \frac {\mathrm {FresnelS}\left (b\,x\right )\,\sin \left (\frac {\Pi \,b^2\,x^2}{2}\right )}{x^{10}} \,d x \] Input:

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

Output:

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

Reduce [N/A]

Not integrable

Time = 0.21 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^{10}} \, dx=\int \frac {\mathrm {FresnelS}\left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{x^{10}}d x \] Input:

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

Output:

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