3.1.88 \(\int \frac {\operatorname {FresnelS}(b x) \sin (\frac {1}{2} b^2 \pi x^2)}{x^9} \, dx\) [88]

3.1.88.1 Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^9} \, dx=-\frac {b}{112 x^7}+\frac {b^5 \pi ^2}{1152 x^3}+\frac {b \cos \left (b^2 \pi x^2\right )}{112 x^7}-\frac {187 b^5 \pi ^2 \cos \left (b^2 \pi x^2\right )}{40320 x^3}-\frac {853 b^8 \pi ^4 \operatorname {FresnelC}\left (\sqrt {2} b x\right )}{40320 \sqrt {2}}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{48 x^6}+\frac {b^6 \pi ^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{384 x^2}-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{8 x^8}+\frac {b^4 \pi ^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{192 x^4}-\frac {19 b^3 \pi \sin \left (b^2 \pi x^2\right )}{3360 x^5}+\frac {853 b^7 \pi ^3 \sin \left (b^2 \pi x^2\right )}{80640 x}+\frac {1}{384} b^8 \pi ^4 \text {Int}\left (\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x},x\right ) \]

output

-1/112*b/x^7+1/1152*b^5*Pi^2/x^3+1/112*b*cos(b^2*Pi*x^2)/x^7-187/40320*b^5 
*Pi^2*cos(b^2*Pi*x^2)/x^3-1/48*b^2*Pi*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/x^ 
6+1/384*b^6*Pi^3*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/x^2-1/8*FresnelS(b*x)*s 
in(1/2*b^2*Pi*x^2)/x^8+1/192*b^4*Pi^2*FresnelS(b*x)*sin(1/2*b^2*Pi*x^2)/x^ 
4-19/3360*b^3*Pi*sin(b^2*Pi*x^2)/x^5+853/80640*b^7*Pi^3*sin(b^2*Pi*x^2)/x- 
853/80640*b^8*Pi^4*FresnelC(b*x*2^(1/2))*2^(1/2)+1/384*b^8*Pi^4*Unintegrab 
le(FresnelS(b*x)*sin(1/2*b^2*Pi*x^2)/x,x)
 

3.1.88.2 Mathematica [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^9} \, dx=\int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^9} \, dx \]

input

Integrate[(FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/x^9,x]
 

output

Integrate[(FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/x^9, x]
 

3.1.88.3 Rubi [N/A]

Not integrable

Time = 2.42 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {7010, 3869, 3868, 3869, 3868, 3833, 7018, 3868, 3869, 3868, 3833, 7010, 3869, 3868, 3833, 7018, 3868, 3833, 7012}

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.

\(\displaystyle \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{x^9} \, dx\)

\(\Big \downarrow \) 7010

\(\displaystyle \frac {1}{8} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^7}dx-\frac {1}{16} b \int \frac {\cos \left (b^2 \pi x^2\right )}{x^8}dx-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{8 x^8}-\frac {b}{112 x^7}\)

\(\Big \downarrow \) 3869

\(\displaystyle \frac {1}{8} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^7}dx-\frac {1}{16} b \left (-\frac {2}{7} \pi b^2 \int \frac {\sin \left (b^2 \pi x^2\right )}{x^6}dx-\frac {\cos \left (\pi b^2 x^2\right )}{7 x^7}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{8 x^8}-\frac {b}{112 x^7}\)

\(\Big \downarrow \) 3868

\(\displaystyle \frac {1}{8} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^7}dx-\frac {1}{16} b \left (-\frac {2}{7} \pi b^2 \left (\frac {2}{5} \pi b^2 \int \frac {\cos \left (b^2 \pi x^2\right )}{x^4}dx-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{7 x^7}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{8 x^8}-\frac {b}{112 x^7}\)

\(\Big \downarrow \) 3869

\(\displaystyle \frac {1}{8} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^7}dx-\frac {1}{16} b \left (-\frac {2}{7} \pi b^2 \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2}dx-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{7 x^7}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{8 x^8}-\frac {b}{112 x^7}\)

\(\Big \downarrow \) 3868

\(\displaystyle \frac {1}{8} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^7}dx-\frac {1}{16} b \left (-\frac {2}{7} \pi b^2 \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (2 \pi b^2 \int \cos \left (b^2 \pi x^2\right )dx-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{7 x^7}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{8 x^8}-\frac {b}{112 x^7}\)

\(\Big \downarrow \) 3833

\(\displaystyle \frac {1}{8} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^7}dx-\frac {1}{16} b \left (-\frac {2}{7} \pi b^2 \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{7 x^7}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{8 x^8}-\frac {b}{112 x^7}\)

\(\Big \downarrow \) 7018

\(\displaystyle \frac {1}{8} \pi b^2 \left (-\frac {1}{6} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5}dx+\frac {1}{12} b \int \frac {\sin \left (b^2 \pi x^2\right )}{x^6}dx-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}\right )-\frac {1}{16} b \left (-\frac {2}{7} \pi b^2 \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{7 x^7}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{8 x^8}-\frac {b}{112 x^7}\)

\(\Big \downarrow \) 3868

\(\displaystyle \frac {1}{8} \pi b^2 \left (-\frac {1}{6} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5}dx+\frac {1}{12} b \left (\frac {2}{5} \pi b^2 \int \frac {\cos \left (b^2 \pi x^2\right )}{x^4}dx-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}\right )-\frac {1}{16} b \left (-\frac {2}{7} \pi b^2 \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{7 x^7}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{8 x^8}-\frac {b}{112 x^7}\)

\(\Big \downarrow \) 3869

\(\displaystyle \frac {1}{8} \pi b^2 \left (-\frac {1}{6} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5}dx+\frac {1}{12} b \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2}dx-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}\right )-\frac {1}{16} b \left (-\frac {2}{7} \pi b^2 \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{7 x^7}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{8 x^8}-\frac {b}{112 x^7}\)

\(\Big \downarrow \) 3868

\(\displaystyle \frac {1}{8} \pi b^2 \left (-\frac {1}{6} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5}dx+\frac {1}{12} b \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (2 \pi b^2 \int \cos \left (b^2 \pi x^2\right )dx-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}\right )-\frac {1}{16} b \left (-\frac {2}{7} \pi b^2 \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{7 x^7}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{8 x^8}-\frac {b}{112 x^7}\)

\(\Big \downarrow \) 3833

\(\displaystyle \frac {1}{8} \pi b^2 \left (-\frac {1}{6} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5}dx+\frac {1}{12} b \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}\right )-\frac {1}{16} b \left (-\frac {2}{7} \pi b^2 \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{7 x^7}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{8 x^8}-\frac {b}{112 x^7}\)

\(\Big \downarrow \) 7010

\(\displaystyle \frac {1}{8} \pi b^2 \left (-\frac {1}{6} \pi b^2 \left (\frac {1}{4} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^3}dx-\frac {1}{8} b \int \frac {\cos \left (b^2 \pi x^2\right )}{x^4}dx-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}-\frac {b}{24 x^3}\right )+\frac {1}{12} b \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}\right )-\frac {1}{16} b \left (-\frac {2}{7} \pi b^2 \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{7 x^7}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{8 x^8}-\frac {b}{112 x^7}\)

\(\Big \downarrow \) 3869

\(\displaystyle \frac {1}{8} \pi b^2 \left (-\frac {1}{6} \pi b^2 \left (\frac {1}{4} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^3}dx-\frac {1}{8} b \left (-\frac {2}{3} \pi b^2 \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2}dx-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}-\frac {b}{24 x^3}\right )+\frac {1}{12} b \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}\right )-\frac {1}{16} b \left (-\frac {2}{7} \pi b^2 \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{7 x^7}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{8 x^8}-\frac {b}{112 x^7}\)

\(\Big \downarrow \) 3868

\(\displaystyle \frac {1}{8} \pi b^2 \left (-\frac {1}{6} \pi b^2 \left (\frac {1}{4} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^3}dx-\frac {1}{8} b \left (-\frac {2}{3} \pi b^2 \left (2 \pi b^2 \int \cos \left (b^2 \pi x^2\right )dx-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}-\frac {b}{24 x^3}\right )+\frac {1}{12} b \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}\right )-\frac {1}{16} b \left (-\frac {2}{7} \pi b^2 \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{7 x^7}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{8 x^8}-\frac {b}{112 x^7}\)

\(\Big \downarrow \) 3833

\(\displaystyle \frac {1}{8} \pi b^2 \left (-\frac {1}{6} \pi b^2 \left (\frac {1}{4} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^3}dx-\frac {1}{8} b \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}-\frac {b}{24 x^3}\right )+\frac {1}{12} b \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}\right )-\frac {1}{16} b \left (-\frac {2}{7} \pi b^2 \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{7 x^7}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{8 x^8}-\frac {b}{112 x^7}\)

\(\Big \downarrow \) 7018

\(\displaystyle \frac {1}{8} \pi b^2 \left (-\frac {1}{6} \pi b^2 \left (\frac {1}{4} \pi b^2 \left (-\frac {1}{2} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x}dx+\frac {1}{4} b \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2}dx-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{2 x^2}\right )-\frac {1}{8} b \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}-\frac {b}{24 x^3}\right )+\frac {1}{12} b \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}\right )-\frac {1}{16} b \left (-\frac {2}{7} \pi b^2 \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{7 x^7}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{8 x^8}-\frac {b}{112 x^7}\)

\(\Big \downarrow \) 3868

\(\displaystyle \frac {1}{8} \pi b^2 \left (-\frac {1}{6} \pi b^2 \left (\frac {1}{4} \pi b^2 \left (-\frac {1}{2} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x}dx+\frac {1}{4} b \left (2 \pi b^2 \int \cos \left (b^2 \pi x^2\right )dx-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{2 x^2}\right )-\frac {1}{8} b \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}-\frac {b}{24 x^3}\right )+\frac {1}{12} b \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}\right )-\frac {1}{16} b \left (-\frac {2}{7} \pi b^2 \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{7 x^7}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{8 x^8}-\frac {b}{112 x^7}\)

\(\Big \downarrow \) 3833

\(\displaystyle \frac {1}{8} \pi b^2 \left (-\frac {1}{6} \pi b^2 \left (\frac {1}{4} \pi b^2 \left (-\frac {1}{2} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x}dx+\frac {1}{4} b \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{2 x^2}\right )-\frac {1}{8} b \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}-\frac {b}{24 x^3}\right )+\frac {1}{12} b \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}\right )-\frac {1}{16} b \left (-\frac {2}{7} \pi b^2 \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{7 x^7}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{8 x^8}-\frac {b}{112 x^7}\)

\(\Big \downarrow \) 7012

\(\displaystyle \frac {1}{8} \pi b^2 \left (-\frac {1}{6} \pi b^2 \left (\frac {1}{4} \pi b^2 \left (-\frac {1}{2} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x}dx+\frac {1}{4} b \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{2 x^2}\right )-\frac {1}{8} b \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}-\frac {b}{24 x^3}\right )+\frac {1}{12} b \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}\right )-\frac {1}{16} b \left (-\frac {2}{7} \pi b^2 \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{7 x^7}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{8 x^8}-\frac {b}{112 x^7}\)

input

Int[(FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/x^9,x]
 

output

$Aborted
 

3.1.88.3.1 Defintions of rubi rules used

rule 3833

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ 
d, 2]))*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
 

rule 3868

Int[((e_.)*(x_))^(m_)*Sin[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> Simp[(e*x) 
^(m + 1)*(Sin[c + d*x^n]/(e*(m + 1))), x] - Simp[d*(n/(e^n*(m + 1)))   Int[ 
(e*x)^(m + n)*Cos[c + d*x^n], x], x] /; FreeQ[{c, d, e}, x] && IGtQ[n, 0] & 
& LtQ[m, -1]
 

rule 3869

Int[Cos[(c_.) + (d_.)*(x_)^(n_)]*((e_.)*(x_))^(m_), x_Symbol] :> Simp[(e*x) 
^(m + 1)*(Cos[c + d*x^n]/(e*(m + 1))), x] + Simp[d*(n/(e^n*(m + 1)))   Int[ 
(e*x)^(m + n)*Sin[c + d*x^n], x], x] /; FreeQ[{c, d, e}, x] && IGtQ[n, 0] & 
& LtQ[m, -1]
 

rule 7010

Int[FresnelS[(b_.)*(x_)]*(x_)^(m_)*Sin[(d_.)*(x_)^2], x_Symbol] :> Simp[x^( 
m + 1)*Sin[d*x^2]*(FresnelS[b*x]/(m + 1)), x] + (-Simp[d*(x^(m + 2)/(Pi*b*( 
m + 1)*(m + 2))), x] - Simp[2*(d/(m + 1))   Int[x^(m + 2)*Cos[d*x^2]*Fresne 
lS[b*x], x], x] + Simp[d/(Pi*b*(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 7012

Int[FresnelS[(a_.) + (b_.)*(x_)]^(n_.)*((e_.)*(x_))^(m_.)*Sin[(c_.) + (d_.) 
*(x_)^2], x_Symbol] :> Unintegrable[(e*x)^m*FresnelS[a + b*x]^n*Sin[c + d*x 
^2], x] /; FreeQ[{a, b, c, d, e, m, n}, x]
 

rule 7018

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

3.1.88.4 Maple [N/A] (verified)

Not integrable

Time = 0.12 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90

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

input

int(FresnelS(b*x)*sin(1/2*b^2*Pi*x^2)/x^9,x)
 

output

int(FresnelS(b*x)*sin(1/2*b^2*Pi*x^2)/x^9,x)
 

3.1.88.5 Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^9} \, dx=\int { \frac {\operatorname {S}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{x^{9}} \,d x } \]

input

integrate(fresnel_sin(b*x)*sin(1/2*b^2*pi*x^2)/x^9,x, algorithm="fricas")
 

output

integral(fresnel_sin(b*x)*sin(1/2*pi*b^2*x^2)/x^9, x)
 

3.1.88.6 Sympy [N/A]

Not integrable

Time = 36.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^9} \, dx=\int \frac {\sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )}{x^{9}}\, dx \]

input

integrate(fresnels(b*x)*sin(1/2*b**2*pi*x**2)/x**9,x)
 

output

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

3.1.88.7 Maxima [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^9} \, dx=\int { \frac {\operatorname {S}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{x^{9}} \,d x } \]

input

integrate(fresnel_sin(b*x)*sin(1/2*b^2*pi*x^2)/x^9,x, algorithm="maxima")
 

output

integrate(fresnel_sin(b*x)*sin(1/2*pi*b^2*x^2)/x^9, x)
 

3.1.88.8 Giac [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^9} \, dx=\int { \frac {\operatorname {S}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{x^{9}} \,d x } \]

input

integrate(fresnel_sin(b*x)*sin(1/2*b^2*pi*x^2)/x^9,x, algorithm="giac")
 

output

integrate(fresnel_sin(b*x)*sin(1/2*pi*b^2*x^2)/x^9, x)
 

3.1.88.9 Mupad [N/A]

Not integrable

Time = 4.98 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^9} \, dx=\int \frac {\mathrm {FresnelS}\left (b\,x\right )\,\sin \left (\frac {\Pi \,b^2\,x^2}{2}\right )}{x^9} \,d x \]

input

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

output

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