\(\int \frac {\operatorname {FresnelC}(b x)^2}{x^8} \, dx\) [155]

Optimal result

Integrand size = 10, antiderivative size = 10 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^8} \, dx=-\frac {b^2}{210 x^5}+\frac {b^6 \pi ^2}{336 x}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{210 x^5}+\frac {67 b^6 \pi ^2 \cos \left (b^2 \pi x^2\right )}{5040 x}-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{21 x^6}+\frac {b^5 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{168 x^2}-\frac {\operatorname {FresnelC}(b x)^2}{7 x^7}+\frac {b^7 \pi ^3 \operatorname {FresnelS}\left (\sqrt {2} b x\right )}{72 \sqrt {2}}+\frac {2}{315} \sqrt {2} b^7 \pi ^3 \operatorname {FresnelS}\left (\sqrt {2} b x\right )+\frac {b^3 \pi \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{84 x^4}+\frac {13 b^4 \pi \sin \left (b^2 \pi x^2\right )}{2520 x^3}+\frac {1}{168} b^7 \pi ^3 \text {Int}\left (\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x},x\right ) \] Output:

-1/210*b^2/x^5+1/336*b^6*Pi^2/x-1/210*b^2*cos(b^2*Pi*x^2)/x^5+67/5040*b^6* 
Pi^2*cos(b^2*Pi*x^2)/x-1/21*b*cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/x^6+1/168* 
b^5*Pi^2*cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/x^2-1/7*FresnelC(b*x)^2/x^7+67/ 
5040*b^7*Pi^3*FresnelS(2^(1/2)*b*x)*2^(1/2)+1/84*b^3*Pi*FresnelC(b*x)*sin( 
1/2*b^2*Pi*x^2)/x^4+13/2520*b^4*Pi*sin(b^2*Pi*x^2)/x^3+1/168*b^7*Pi^3*Defe 
r(Int)(FresnelC(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 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^8} \, dx=\int \frac {\operatorname {FresnelC}(b x)^2}{x^8} \, dx \] Input:

Integrate[FresnelC[b*x]^2/x^8,x]
 

Output:

Integrate[FresnelC[b*x]^2/x^8, x]
 

Rubi [N/A]

Not integrable

Time = 1.64 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 14, 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.

\(\displaystyle \int \frac {\operatorname {FresnelC}(b x)^2}{x^8} \, dx\)

\(\Big \downarrow \) 6985

\(\displaystyle \frac {2}{7} b \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^7}dx-\frac {\operatorname {FresnelC}(b x)^2}{7 x^7}\)

\(\Big \downarrow \) 7011

\(\displaystyle \frac {2}{7} b \left (-\frac {1}{6} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5}dx+\frac {1}{12} b \int \frac {\cos \left (b^2 \pi x^2\right )}{x^6}dx-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}-\frac {b}{60 x^5}\right )-\frac {\operatorname {FresnelC}(b x)^2}{7 x^7}\)

\(\Big \downarrow \) 3869

\(\displaystyle \frac {2}{7} b \left (-\frac {1}{6} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5}dx+\frac {1}{12} b \left (-\frac {2}{5} \pi b^2 \int \frac {\sin \left (b^2 \pi x^2\right )}{x^4}dx-\frac {\cos \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}-\frac {b}{60 x^5}\right )-\frac {\operatorname {FresnelC}(b x)^2}{7 x^7}\)

\(\Big \downarrow \) 3868

\(\displaystyle \frac {2}{7} b \left (-\frac {1}{6} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5}dx+\frac {1}{12} b \left (-\frac {2}{5} \pi b^2 \left (\frac {2}{3} \pi b^2 \int \frac {\cos \left (b^2 \pi x^2\right )}{x^2}dx-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}-\frac {b}{60 x^5}\right )-\frac {\operatorname {FresnelC}(b x)^2}{7 x^7}\)

\(\Big \downarrow \) 3869

\(\displaystyle \frac {2}{7} b \left (-\frac {1}{6} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5}dx+\frac {1}{12} b \left (-\frac {2}{5} \pi b^2 \left (\frac {2}{3} \pi b^2 \left (-2 \pi b^2 \int \sin \left (b^2 \pi x^2\right )dx-\frac {\cos \left (\pi b^2 x^2\right )}{x}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}-\frac {b}{60 x^5}\right )-\frac {\operatorname {FresnelC}(b x)^2}{7 x^7}\)

\(\Big \downarrow \) 3832

\(\displaystyle \frac {2}{7} b \left (-\frac {1}{6} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5}dx-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}+\frac {1}{12} b \left (-\frac {2}{5} \pi b^2 \left (\frac {2}{3} \pi b^2 \left (-\frac {\cos \left (\pi b^2 x^2\right )}{x}-\sqrt {2} \pi b \operatorname {FresnelS}\left (\sqrt {2} b x\right )\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {b}{60 x^5}\right )-\frac {\operatorname {FresnelC}(b x)^2}{7 x^7}\)

\(\Big \downarrow \) 7019

\(\displaystyle \frac {2}{7} b \left (-\frac {1}{6} \pi b^2 \left (\frac {1}{4} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^3}dx+\frac {1}{8} b \int \frac {\sin \left (b^2 \pi x^2\right )}{x^4}dx-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}+\frac {1}{12} b \left (-\frac {2}{5} \pi b^2 \left (\frac {2}{3} \pi b^2 \left (-\frac {\cos \left (\pi b^2 x^2\right )}{x}-\sqrt {2} \pi b \operatorname {FresnelS}\left (\sqrt {2} b x\right )\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {b}{60 x^5}\right )-\frac {\operatorname {FresnelC}(b x)^2}{7 x^7}\)

\(\Big \downarrow \) 3868

\(\displaystyle \frac {2}{7} b \left (-\frac {1}{6} \pi b^2 \left (\frac {1}{4} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^3}dx+\frac {1}{8} b \left (\frac {2}{3} \pi b^2 \int \frac {\cos \left (b^2 \pi x^2\right )}{x^2}dx-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}+\frac {1}{12} b \left (-\frac {2}{5} \pi b^2 \left (\frac {2}{3} \pi b^2 \left (-\frac {\cos \left (\pi b^2 x^2\right )}{x}-\sqrt {2} \pi b \operatorname {FresnelS}\left (\sqrt {2} b x\right )\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {b}{60 x^5}\right )-\frac {\operatorname {FresnelC}(b x)^2}{7 x^7}\)

\(\Big \downarrow \) 3869

\(\displaystyle \frac {2}{7} b \left (-\frac {1}{6} \pi b^2 \left (\frac {1}{4} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^3}dx+\frac {1}{8} b \left (\frac {2}{3} \pi b^2 \left (-2 \pi b^2 \int \sin \left (b^2 \pi x^2\right )dx-\frac {\cos \left (\pi b^2 x^2\right )}{x}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}+\frac {1}{12} b \left (-\frac {2}{5} \pi b^2 \left (\frac {2}{3} \pi b^2 \left (-\frac {\cos \left (\pi b^2 x^2\right )}{x}-\sqrt {2} \pi b \operatorname {FresnelS}\left (\sqrt {2} b x\right )\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {b}{60 x^5}\right )-\frac {\operatorname {FresnelC}(b x)^2}{7 x^7}\)

\(\Big \downarrow \) 3832

\(\displaystyle \frac {2}{7} b \left (-\frac {1}{6} \pi b^2 \left (\frac {1}{4} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^3}dx-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}+\frac {1}{8} b \left (\frac {2}{3} \pi b^2 \left (-\frac {\cos \left (\pi b^2 x^2\right )}{x}-\sqrt {2} \pi b \operatorname {FresnelS}\left (\sqrt {2} b x\right )\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}+\frac {1}{12} b \left (-\frac {2}{5} \pi b^2 \left (\frac {2}{3} \pi b^2 \left (-\frac {\cos \left (\pi b^2 x^2\right )}{x}-\sqrt {2} \pi b \operatorname {FresnelS}\left (\sqrt {2} b x\right )\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {b}{60 x^5}\right )-\frac {\operatorname {FresnelC}(b x)^2}{7 x^7}\)

\(\Big \downarrow \) 7011

\(\displaystyle \frac {2}{7} b \left (-\frac {1}{6} \pi b^2 \left (\frac {1}{4} \pi b^2 \left (-\frac {1}{2} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x}dx+\frac {1}{4} b \int \frac {\cos \left (b^2 \pi x^2\right )}{x^2}dx-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{2 x^2}-\frac {b}{4 x}\right )-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}+\frac {1}{8} b \left (\frac {2}{3} \pi b^2 \left (-\frac {\cos \left (\pi b^2 x^2\right )}{x}-\sqrt {2} \pi b \operatorname {FresnelS}\left (\sqrt {2} b x\right )\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}+\frac {1}{12} b \left (-\frac {2}{5} \pi b^2 \left (\frac {2}{3} \pi b^2 \left (-\frac {\cos \left (\pi b^2 x^2\right )}{x}-\sqrt {2} \pi b \operatorname {FresnelS}\left (\sqrt {2} b x\right )\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {b}{60 x^5}\right )-\frac {\operatorname {FresnelC}(b x)^2}{7 x^7}\)

\(\Big \downarrow \) 3869

\(\displaystyle \frac {2}{7} b \left (-\frac {1}{6} \pi b^2 \left (\frac {1}{4} \pi b^2 \left (-\frac {1}{2} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x}dx+\frac {1}{4} b \left (-2 \pi b^2 \int \sin \left (b^2 \pi x^2\right )dx-\frac {\cos \left (\pi b^2 x^2\right )}{x}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{2 x^2}-\frac {b}{4 x}\right )-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}+\frac {1}{8} b \left (\frac {2}{3} \pi b^2 \left (-\frac {\cos \left (\pi b^2 x^2\right )}{x}-\sqrt {2} \pi b \operatorname {FresnelS}\left (\sqrt {2} b x\right )\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}+\frac {1}{12} b \left (-\frac {2}{5} \pi b^2 \left (\frac {2}{3} \pi b^2 \left (-\frac {\cos \left (\pi b^2 x^2\right )}{x}-\sqrt {2} \pi b \operatorname {FresnelS}\left (\sqrt {2} b x\right )\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {b}{60 x^5}\right )-\frac {\operatorname {FresnelC}(b x)^2}{7 x^7}\)

\(\Big \downarrow \) 3832

\(\displaystyle \frac {2}{7} b \left (-\frac {1}{6} \pi b^2 \left (\frac {1}{4} \pi b^2 \left (-\frac {1}{2} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x}dx-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{2 x^2}+\frac {1}{4} b \left (-\frac {\cos \left (\pi b^2 x^2\right )}{x}-\sqrt {2} \pi b \operatorname {FresnelS}\left (\sqrt {2} b x\right )\right )-\frac {b}{4 x}\right )-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}+\frac {1}{8} b \left (\frac {2}{3} \pi b^2 \left (-\frac {\cos \left (\pi b^2 x^2\right )}{x}-\sqrt {2} \pi b \operatorname {FresnelS}\left (\sqrt {2} b x\right )\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}+\frac {1}{12} b \left (-\frac {2}{5} \pi b^2 \left (\frac {2}{3} \pi b^2 \left (-\frac {\cos \left (\pi b^2 x^2\right )}{x}-\sqrt {2} \pi b \operatorname {FresnelS}\left (\sqrt {2} b x\right )\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {b}{60 x^5}\right )-\frac {\operatorname {FresnelC}(b x)^2}{7 x^7}\)

\(\Big \downarrow \) 7021

\(\displaystyle \frac {2}{7} b \left (-\frac {1}{6} \pi b^2 \left (\frac {1}{4} \pi b^2 \left (-\frac {1}{2} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x}dx-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{2 x^2}+\frac {1}{4} b \left (-\frac {\cos \left (\pi b^2 x^2\right )}{x}-\sqrt {2} \pi b \operatorname {FresnelS}\left (\sqrt {2} b x\right )\right )-\frac {b}{4 x}\right )-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}+\frac {1}{8} b \left (\frac {2}{3} \pi b^2 \left (-\frac {\cos \left (\pi b^2 x^2\right )}{x}-\sqrt {2} \pi b \operatorname {FresnelS}\left (\sqrt {2} b x\right )\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}+\frac {1}{12} b \left (-\frac {2}{5} \pi b^2 \left (\frac {2}{3} \pi b^2 \left (-\frac {\cos \left (\pi b^2 x^2\right )}{x}-\sqrt {2} \pi b \operatorname {FresnelS}\left (\sqrt {2} b x\right )\right )-\frac {\sin \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {b}{60 x^5}\right )-\frac {\operatorname {FresnelC}(b x)^2}{7 x^7}\)

Input:

Int[FresnelC[b*x]^2/x^8,x]
 

Output:

$Aborted
 

Maple [N/A]

Not integrable

Time = 0.16 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00

\[\int \frac {\operatorname {FresnelC}\left (b x \right )^{2}}{x^{8}}d x\]

Input:

int(FresnelC(b*x)^2/x^8,x)
 

Output:

int(FresnelC(b*x)^2/x^8,x)
 

Fricas [N/A]

Not integrable

Time = 0.10 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^8} \, dx=\int { \frac {\operatorname {C}\left (b x\right )^{2}}{x^{8}} \,d x } \] Input:

integrate(fresnel_cos(b*x)^2/x^8,x, algorithm="fricas")
 

Output:

integral(fresnel_cos(b*x)^2/x^8, x)
 

Sympy [N/A]

Not integrable

Time = 1.59 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^8} \, dx=\int \frac {C^{2}\left (b x\right )}{x^{8}}\, dx \] Input:

integrate(fresnelc(b*x)**2/x**8,x)
 

Output:

Integral(fresnelc(b*x)**2/x**8, x)
 

Maxima [N/A]

Not integrable

Time = 0.06 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^8} \, dx=\int { \frac {\operatorname {C}\left (b x\right )^{2}}{x^{8}} \,d x } \] Input:

integrate(fresnel_cos(b*x)^2/x^8,x, algorithm="maxima")
 

Output:

integrate(fresnel_cos(b*x)^2/x^8, x)
 

Giac [N/A]

Not integrable

Time = 0.11 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^8} \, dx=\int { \frac {\operatorname {C}\left (b x\right )^{2}}{x^{8}} \,d x } \] Input:

integrate(fresnel_cos(b*x)^2/x^8,x, algorithm="giac")
 

Output:

integrate(fresnel_cos(b*x)^2/x^8, x)
 

Mupad [N/A]

Not integrable

Time = 4.14 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^8} \, dx=\int \frac {{\mathrm {FresnelC}\left (b\,x\right )}^2}{x^8} \,d x \] Input:

int(FresnelC(b*x)^2/x^8,x)
 

Output:

int(FresnelC(b*x)^2/x^8, x)
 

Reduce [N/A]

Not integrable

Time = 0.21 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^8} \, dx=\int \frac {\mathrm {FresnelC}\left (b x \right )^{2}}{x^{8}}d x \] Input:

int(FresnelC(b*x)^2/x^8,x)
 

Output:

int(FresnelC(b*x)^2/x^8,x)