3.2.97 \(\int \frac {\cos (\frac {1}{2} b^2 \pi x^2) \operatorname {FresnelC}(b x)}{x^9} \, dx\) [197]

3.2.97.1 Optimal result

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

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^3-1/8*cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/x^8+1/192* 
b^4*Pi^2*cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/x^4+1/48*b^2*Pi*FresnelC(b*x)*s 
in(1/2*b^2*Pi*x^2)/x^6-1/384*b^6*Pi^3*FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/x^ 
2+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(cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/x,x)
 

3.2.97.2 Mathematica [N/A]

Not integrable

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

input

Integrate[(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x^9,x]
 

output

Integrate[(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x^9, x]
 

3.2.97.3 Rubi [N/A]

Not integrable

Time = 2.39 (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 = {7011, 3869, 3868, 3869, 3868, 3833, 7019, 3868, 3869, 3868, 3833, 7011, 3869, 3868, 3833, 7019, 3868, 3833, 7013}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 7011

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

\(\Big \downarrow \) 3869

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

\(\Big \downarrow \) 3868

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

\(\Big \downarrow \) 3869

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

\(\Big \downarrow \) 3868

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

\(\Big \downarrow \) 3833

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

\(\Big \downarrow \) 7019

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

\(\Big \downarrow \) 3868

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

\(\Big \downarrow \) 3869

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

\(\Big \downarrow \) 3868

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

\(\Big \downarrow \) 3833

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

\(\Big \downarrow \) 7011

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

\(\Big \downarrow \) 3869

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

\(\Big \downarrow \) 3868

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

\(\Big \downarrow \) 3833

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

\(\Big \downarrow \) 7019

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

\(\Big \downarrow \) 3868

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

\(\Big \downarrow \) 3833

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

\(\Big \downarrow \) 7013

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

input

Int[(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x^9,x]
 

output

$Aborted
 

3.2.97.3.1 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 7011

Int[Cos[(d_.)*(x_)^2]*FresnelC[(b_.)*(x_)]*(x_)^(m_), x_Symbol] :> Simp[x^( 
m + 1)*Cos[d*x^2]*(FresnelC[b*x]/(m + 1)), x] + (-Simp[b*(x^(m + 2)/(2*(m + 
 1)*(m + 2))), x] + Simp[2*(d/(m + 1))   Int[x^(m + 2)*Sin[d*x^2]*FresnelC[ 
b*x], x], x] - Simp[b/(2*(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 7013

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

rule 7019

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

3.2.97.4 Maple [N/A] (verified)

Not integrable

Time = 0.13 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90

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

input

int(cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/x^9,x)
 

output

int(cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/x^9,x)
 

3.2.97.5 Fricas [N/A]

Not integrable

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

input

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

output

integral(cos(1/2*pi*b^2*x^2)*fresnel_cos(b*x)/x^9, x)
 

3.2.97.6 Sympy [N/A]

Not integrable

Time = 36.97 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^9} \, dx=\int \frac {\cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )}{x^{9}}\, dx \]

input

integrate(cos(1/2*b**2*pi*x**2)*fresnelc(b*x)/x**9,x)
 

output

Integral(cos(pi*b**2*x**2/2)*fresnelc(b*x)/x**9, x)
 

3.2.97.7 Maxima [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^9} \, dx=\int { \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right )}{x^{9}} \,d x } \]

input

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

output

integrate(cos(1/2*pi*b^2*x^2)*fresnel_cos(b*x)/x^9, x)
 

3.2.97.8 Giac [N/A]

Not integrable

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

input

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

output

integrate(cos(1/2*pi*b^2*x^2)*fresnel_cos(b*x)/x^9, x)
 

3.2.97.9 Mupad [N/A]

Not integrable

Time = 4.62 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^9} \, dx=\int \frac {\mathrm {FresnelC}\left (b\,x\right )\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right )}{x^9} \,d x \]

input

int((FresnelC(b*x)*cos((Pi*b^2*x^2)/2))/x^9,x)
 

output

int((FresnelC(b*x)*cos((Pi*b^2*x^2)/2))/x^9, x)