3.2.57 \(\int \frac {\operatorname {FresnelC}(b x)^2}{x^{10}} \, dx\) [157]

3.2.57.1 Optimal result

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

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^3-1/36*b*cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/x^ 
8+1/864*b^5*Pi^2*cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/x^4-1/9*FresnelC(b*x)^2 
/x^9+1/216*b^3*Pi*FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/x^6-1/1728*b^7*Pi^3*Fr 
esnelC(b*x)*sin(1/2*b^2*Pi*x^2)/x^2+19/15120*b^4*Pi*sin(b^2*Pi*x^2)/x^5-85 
3/362880*b^8*Pi^3*sin(b^2*Pi*x^2)/x+853/362880*b^9*Pi^4*FresnelC(b*x*2^(1/ 
2))*2^(1/2)+1/1728*b^9*Pi^4*Unintegrable(cos(1/2*b^2*Pi*x^2)*FresnelC(b*x) 
/x,x)
 

3.2.57.2 Mathematica [N/A]

Not integrable

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

input

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

output

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

3.2.57.3 Rubi [N/A]

Not integrable

Time = 2.61 (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 = {6985, 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.

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

\(\Big \downarrow \) 6985

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

\(\Big \downarrow \) 7011

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

\(\Big \downarrow \) 3869

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

\(\Big \downarrow \) 3868

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

\(\Big \downarrow \) 3869

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

\(\Big \downarrow \) 3868

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

\(\Big \downarrow \) 3833

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

\(\Big \downarrow \) 7019

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

\(\Big \downarrow \) 3868

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

\(\Big \downarrow \) 3869

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

\(\Big \downarrow \) 3868

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

\(\Big \downarrow \) 3833

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

\(\Big \downarrow \) 7011

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

\(\Big \downarrow \) 3869

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

\(\Big \downarrow \) 3868

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

\(\Big \downarrow \) 3833

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

\(\Big \downarrow \) 7019

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

\(\Big \downarrow \) 3868

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

\(\Big \downarrow \) 3833

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

\(\Big \downarrow \) 7013

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

input

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

output

$Aborted
 

3.2.57.3.1 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 6985

Int[FresnelC[(b_.)*(x_)]^2*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*(Fresnel 
C[b*x]^2/(m + 1)), x] - Simp[2*(b/(m + 1))   Int[x^(m + 1)*Cos[(Pi/2)*b^2*x 
^2]*FresnelC[b*x], x], x] /; FreeQ[b, x] && IntegerQ[m] && NeQ[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]
 

3.2.57.4 Maple [N/A] (verified)

Not integrable

Time = 0.06 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00

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

input

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

output

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

3.2.57.5 Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^{10}} \, dx=\int { \frac {\operatorname {C}\left (b x\right )^{2}}{x^{10}} \,d x } \]

input

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

output

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

3.2.57.6 Sympy [N/A]

Not integrable

Time = 2.18 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^{10}} \, dx=\int \frac {C^{2}\left (b x\right )}{x^{10}}\, dx \]

input

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

output

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

3.2.57.7 Maxima [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^{10}} \, dx=\int { \frac {\operatorname {C}\left (b x\right )^{2}}{x^{10}} \,d x } \]

input

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

output

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

3.2.57.8 Giac [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^{10}} \, dx=\int { \frac {\operatorname {C}\left (b x\right )^{2}}{x^{10}} \,d x } \]

input

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

output

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

3.2.57.9 Mupad [N/A]

Not integrable

Time = 4.72 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^{10}} \, dx=\int \frac {{\mathrm {FresnelC}\left (b\,x\right )}^2}{x^{10}} \,d x \]

input

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

output

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