\(\int \frac {\operatorname {FresnelS}(b x)^2}{x^{10}} \, dx\) [48]

Optimal result

Integrand size = 10, antiderivative size = 10 \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^{10}} \, dx=-\frac {b^2}{504 x^7}+\frac {b^6 \pi ^2}{5184 x^3}+\frac {b^2 \cos \left (b^2 \pi x^2\right )}{504 x^7}-\frac {187 b^6 \pi ^2 \cos \left (b^2 \pi x^2\right )}{181440 x^3}-\frac {853 b^9 \pi ^4 \operatorname {FresnelC}\left (\sqrt {2} b x\right )}{181440 \sqrt {2}}-\frac {b^3 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{216 x^6}+\frac {b^7 \pi ^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{1728 x^2}-\frac {\operatorname {FresnelS}(b x)^2}{9 x^9}-\frac {b \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{36 x^8}+\frac {b^5 \pi ^2 \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{864 x^4}-\frac {19 b^4 \pi \sin \left (b^2 \pi x^2\right )}{15120 x^5}+\frac {853 b^8 \pi ^3 \sin \left (b^2 \pi x^2\right )}{362880 x}+\frac {b^9 \pi ^4 \text {Int}\left (\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x},x\right )}{1728} \] Output:

-1/504*b^2/x^7+1/5184*b^6*Pi^2/x^3+1/504*b^2*cos(b^2*Pi*x^2)/x^7-187/18144 
0*b^6*Pi^2*cos(b^2*Pi*x^2)/x^3-853/362880*b^9*Pi^4*FresnelC(2^(1/2)*b*x)*2 
^(1/2)-1/216*b^3*Pi*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/x^6+1/1728*b^7*Pi^3* 
cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/x^2-1/9*FresnelS(b*x)^2/x^9-1/36*b*Fresn 
elS(b*x)*sin(1/2*b^2*Pi*x^2)/x^8+1/864*b^5*Pi^2*FresnelS(b*x)*sin(1/2*b^2* 
Pi*x^2)/x^4-19/15120*b^4*Pi*sin(b^2*Pi*x^2)/x^5+853/362880*b^8*Pi^3*sin(b^ 
2*Pi*x^2)/x+1/1728*b^9*Pi^4*Defer(Int)(FresnelS(b*x)*sin(1/2*b^2*Pi*x^2)/x 
,x)
 

Mathematica [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^{10}} \, dx=\int \frac {\operatorname {FresnelS}(b x)^2}{x^{10}} \, dx \] Input:

Integrate[FresnelS[b*x]^2/x^10,x]
 

Output:

Integrate[FresnelS[b*x]^2/x^10, x]
 

Rubi [N/A]

Not integrable

Time = 2.78 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 20, 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.

\(\displaystyle \int \frac {\operatorname {FresnelS}(b x)^2}{x^{10}} \, dx\)

\(\Big \downarrow \) 6984

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

\(\Big \downarrow \) 7010

\(\displaystyle \frac {2}{9} b \left (\frac {1}{8} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^7}dx-\frac {1}{16} b \int \frac {\cos \left (b^2 \pi x^2\right )}{x^8}dx-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{8 x^8}-\frac {b}{112 x^7}\right )-\frac {\operatorname {FresnelS}(b x)^2}{9 x^9}\)

\(\Big \downarrow \) 3869

\(\displaystyle \frac {2}{9} b \left (\frac {1}{8} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^7}dx-\frac {1}{16} b \left (-\frac {2}{7} \pi b^2 \int \frac {\sin \left (b^2 \pi x^2\right )}{x^6}dx-\frac {\cos \left (\pi b^2 x^2\right )}{7 x^7}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{8 x^8}-\frac {b}{112 x^7}\right )-\frac {\operatorname {FresnelS}(b x)^2}{9 x^9}\)

\(\Big \downarrow \) 3868

\(\displaystyle \frac {2}{9} b \left (\frac {1}{8} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^7}dx-\frac {1}{16} b \left (-\frac {2}{7} \pi b^2 \left (\frac {2}{5} \pi b^2 \int \frac {\cos \left (b^2 \pi x^2\right )}{x^4}dx-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{7 x^7}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{8 x^8}-\frac {b}{112 x^7}\right )-\frac {\operatorname {FresnelS}(b x)^2}{9 x^9}\)

\(\Big \downarrow \) 3869

\(\displaystyle \frac {2}{9} b \left (\frac {1}{8} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^7}dx-\frac {1}{16} b \left (-\frac {2}{7} \pi b^2 \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2}dx-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{7 x^7}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{8 x^8}-\frac {b}{112 x^7}\right )-\frac {\operatorname {FresnelS}(b x)^2}{9 x^9}\)

\(\Big \downarrow \) 3868

\(\displaystyle \frac {2}{9} b \left (\frac {1}{8} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^7}dx-\frac {1}{16} b \left (-\frac {2}{7} \pi b^2 \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (2 \pi b^2 \int \cos \left (b^2 \pi x^2\right )dx-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{7 x^7}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{8 x^8}-\frac {b}{112 x^7}\right )-\frac {\operatorname {FresnelS}(b x)^2}{9 x^9}\)

\(\Big \downarrow \) 3833

\(\displaystyle \frac {2}{9} b \left (\frac {1}{8} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^7}dx-\frac {1}{16} b \left (-\frac {2}{7} \pi b^2 \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{7 x^7}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{8 x^8}-\frac {b}{112 x^7}\right )-\frac {\operatorname {FresnelS}(b x)^2}{9 x^9}\)

\(\Big \downarrow \) 7018

\(\displaystyle \frac {2}{9} b \left (\frac {1}{8} \pi b^2 \left (-\frac {1}{6} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5}dx+\frac {1}{12} b \int \frac {\sin \left (b^2 \pi x^2\right )}{x^6}dx-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}\right )-\frac {1}{16} b \left (-\frac {2}{7} \pi b^2 \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{7 x^7}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{8 x^8}-\frac {b}{112 x^7}\right )-\frac {\operatorname {FresnelS}(b x)^2}{9 x^9}\)

\(\Big \downarrow \) 3868

\(\displaystyle \frac {2}{9} b \left (\frac {1}{8} \pi b^2 \left (-\frac {1}{6} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5}dx+\frac {1}{12} b \left (\frac {2}{5} \pi b^2 \int \frac {\cos \left (b^2 \pi x^2\right )}{x^4}dx-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}\right )-\frac {1}{16} b \left (-\frac {2}{7} \pi b^2 \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{7 x^7}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{8 x^8}-\frac {b}{112 x^7}\right )-\frac {\operatorname {FresnelS}(b x)^2}{9 x^9}\)

\(\Big \downarrow \) 3869

\(\displaystyle \frac {2}{9} b \left (\frac {1}{8} \pi b^2 \left (-\frac {1}{6} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5}dx+\frac {1}{12} b \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2}dx-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}\right )-\frac {1}{16} b \left (-\frac {2}{7} \pi b^2 \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{7 x^7}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{8 x^8}-\frac {b}{112 x^7}\right )-\frac {\operatorname {FresnelS}(b x)^2}{9 x^9}\)

\(\Big \downarrow \) 3868

\(\displaystyle \frac {2}{9} b \left (\frac {1}{8} \pi b^2 \left (-\frac {1}{6} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5}dx+\frac {1}{12} b \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (2 \pi b^2 \int \cos \left (b^2 \pi x^2\right )dx-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}\right )-\frac {1}{16} b \left (-\frac {2}{7} \pi b^2 \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{7 x^7}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{8 x^8}-\frac {b}{112 x^7}\right )-\frac {\operatorname {FresnelS}(b x)^2}{9 x^9}\)

\(\Big \downarrow \) 3833

\(\displaystyle \frac {2}{9} b \left (\frac {1}{8} \pi b^2 \left (-\frac {1}{6} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5}dx+\frac {1}{12} b \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}\right )-\frac {1}{16} b \left (-\frac {2}{7} \pi b^2 \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{7 x^7}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{8 x^8}-\frac {b}{112 x^7}\right )-\frac {\operatorname {FresnelS}(b x)^2}{9 x^9}\)

\(\Big \downarrow \) 7010

\(\displaystyle \frac {2}{9} b \left (\frac {1}{8} \pi b^2 \left (-\frac {1}{6} \pi b^2 \left (\frac {1}{4} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^3}dx-\frac {1}{8} b \int \frac {\cos \left (b^2 \pi x^2\right )}{x^4}dx-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}-\frac {b}{24 x^3}\right )+\frac {1}{12} b \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}\right )-\frac {1}{16} b \left (-\frac {2}{7} \pi b^2 \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{7 x^7}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{8 x^8}-\frac {b}{112 x^7}\right )-\frac {\operatorname {FresnelS}(b x)^2}{9 x^9}\)

\(\Big \downarrow \) 3869

\(\displaystyle \frac {2}{9} b \left (\frac {1}{8} \pi b^2 \left (-\frac {1}{6} \pi b^2 \left (\frac {1}{4} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^3}dx-\frac {1}{8} b \left (-\frac {2}{3} \pi b^2 \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2}dx-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}-\frac {b}{24 x^3}\right )+\frac {1}{12} b \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}\right )-\frac {1}{16} b \left (-\frac {2}{7} \pi b^2 \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{7 x^7}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{8 x^8}-\frac {b}{112 x^7}\right )-\frac {\operatorname {FresnelS}(b x)^2}{9 x^9}\)

\(\Big \downarrow \) 3868

\(\displaystyle \frac {2}{9} b \left (\frac {1}{8} \pi b^2 \left (-\frac {1}{6} \pi b^2 \left (\frac {1}{4} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^3}dx-\frac {1}{8} b \left (-\frac {2}{3} \pi b^2 \left (2 \pi b^2 \int \cos \left (b^2 \pi x^2\right )dx-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}-\frac {b}{24 x^3}\right )+\frac {1}{12} b \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}\right )-\frac {1}{16} b \left (-\frac {2}{7} \pi b^2 \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{7 x^7}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{8 x^8}-\frac {b}{112 x^7}\right )-\frac {\operatorname {FresnelS}(b x)^2}{9 x^9}\)

\(\Big \downarrow \) 3833

\(\displaystyle \frac {2}{9} b \left (\frac {1}{8} \pi b^2 \left (-\frac {1}{6} \pi b^2 \left (\frac {1}{4} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^3}dx-\frac {1}{8} b \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}-\frac {b}{24 x^3}\right )+\frac {1}{12} b \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}\right )-\frac {1}{16} b \left (-\frac {2}{7} \pi b^2 \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{7 x^7}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{8 x^8}-\frac {b}{112 x^7}\right )-\frac {\operatorname {FresnelS}(b x)^2}{9 x^9}\)

\(\Big \downarrow \) 7018

\(\displaystyle \frac {2}{9} b \left (\frac {1}{8} \pi b^2 \left (-\frac {1}{6} \pi b^2 \left (\frac {1}{4} \pi b^2 \left (-\frac {1}{2} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x}dx+\frac {1}{4} b \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2}dx-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{2 x^2}\right )-\frac {1}{8} b \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}-\frac {b}{24 x^3}\right )+\frac {1}{12} b \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}\right )-\frac {1}{16} b \left (-\frac {2}{7} \pi b^2 \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{7 x^7}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{8 x^8}-\frac {b}{112 x^7}\right )-\frac {\operatorname {FresnelS}(b x)^2}{9 x^9}\)

\(\Big \downarrow \) 3868

\(\displaystyle \frac {2}{9} b \left (\frac {1}{8} \pi b^2 \left (-\frac {1}{6} \pi b^2 \left (\frac {1}{4} \pi b^2 \left (-\frac {1}{2} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x}dx+\frac {1}{4} b \left (2 \pi b^2 \int \cos \left (b^2 \pi x^2\right )dx-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{2 x^2}\right )-\frac {1}{8} b \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}-\frac {b}{24 x^3}\right )+\frac {1}{12} b \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}\right )-\frac {1}{16} b \left (-\frac {2}{7} \pi b^2 \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{7 x^7}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{8 x^8}-\frac {b}{112 x^7}\right )-\frac {\operatorname {FresnelS}(b x)^2}{9 x^9}\)

\(\Big \downarrow \) 3833

\(\displaystyle \frac {2}{9} b \left (\frac {1}{8} \pi b^2 \left (-\frac {1}{6} \pi b^2 \left (\frac {1}{4} \pi b^2 \left (-\frac {1}{2} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x}dx+\frac {1}{4} b \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{2 x^2}\right )-\frac {1}{8} b \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}-\frac {b}{24 x^3}\right )+\frac {1}{12} b \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}\right )-\frac {1}{16} b \left (-\frac {2}{7} \pi b^2 \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{7 x^7}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{8 x^8}-\frac {b}{112 x^7}\right )-\frac {\operatorname {FresnelS}(b x)^2}{9 x^9}\)

\(\Big \downarrow \) 7012

\(\displaystyle \frac {2}{9} b \left (\frac {1}{8} \pi b^2 \left (-\frac {1}{6} \pi b^2 \left (\frac {1}{4} \pi b^2 \left (-\frac {1}{2} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x}dx+\frac {1}{4} b \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{2 x^2}\right )-\frac {1}{8} b \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}-\frac {b}{24 x^3}\right )+\frac {1}{12} b \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}\right )-\frac {1}{16} b \left (-\frac {2}{7} \pi b^2 \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{7 x^7}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{8 x^8}-\frac {b}{112 x^7}\right )-\frac {\operatorname {FresnelS}(b x)^2}{9 x^9}\)

Input:

Int[FresnelS[b*x]^2/x^10,x]
 

Output:

$Aborted
 

Maple [N/A]

Not integrable

Time = 0.19 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00

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

Input:

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

Output:

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

Fricas [N/A]

Not integrable

Time = 0.08 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^{10}} \, dx=\int { \frac {\operatorname {S}\left (b x\right )^{2}}{x^{10}} \,d x } \] Input:

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

Output:

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

Sympy [N/A]

Not integrable

Time = 2.23 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^{10}} \, dx=\int \frac {S^{2}\left (b x\right )}{x^{10}}\, dx \] Input:

integrate(fresnels(b*x)**2/x**10,x)
 

Output:

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

Maxima [N/A]

Not integrable

Time = 0.06 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^{10}} \, dx=\int { \frac {\operatorname {S}\left (b x\right )^{2}}{x^{10}} \,d x } \] Input:

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

Output:

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

Giac [N/A]

Not integrable

Time = 0.11 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^{10}} \, dx=\int { \frac {\operatorname {S}\left (b x\right )^{2}}{x^{10}} \,d x } \] Input:

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

Output:

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

Mupad [N/A]

Not integrable

Time = 3.79 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^{10}} \, dx=\int \frac {{\mathrm {FresnelS}\left (b\,x\right )}^2}{x^{10}} \,d x \] Input:

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

Output:

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

Reduce [N/A]

Not integrable

Time = 0.21 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelS}(b x)^2}{x^{10}} \, dx=\int \frac {\mathrm {FresnelS}\left (b x \right )^{2}}{x^{10}}d x \] Input:

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

Output:

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