\(\int \frac {\operatorname {FresnelC}(b x)^2}{x^7} \, dx\) [154]

Optimal result

Integrand size = 10, antiderivative size = 10 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^7} \, dx=-\frac {b^2}{120 x^4}-\frac {b^2 \cos \left (b^2 \pi x^2\right )}{120 x^4}-\frac {1}{72} b^6 \pi ^2 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {b \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{15 x^5}-\frac {\operatorname {FresnelC}(b x)^2}{6 x^6}+\frac {b^3 \pi \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{45 x^3}+\frac {b^4 \pi \sin \left (b^2 \pi x^2\right )}{72 x^2}-\frac {1}{45} b^5 \pi ^2 \text {Int}\left (\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^2},x\right ) \] Output:

-1/120*b^2/x^4-1/120*b^2*cos(b^2*Pi*x^2)/x^4-1/72*b^6*Pi^2*Ci(b^2*Pi*x^2)- 
1/15*b*cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/x^5-1/6*FresnelC(b*x)^2/x^6+1/45* 
b^3*Pi*FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/x^3+1/72*b^4*Pi*sin(b^2*Pi*x^2)/x 
^2-1/45*b^5*Pi^2*Defer(Int)(cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/x^2,x)
 

Mathematica [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^7} \, dx=\int \frac {\operatorname {FresnelC}(b x)^2}{x^7} \, dx \] Input:

Integrate[FresnelC[b*x]^2/x^7,x]
 

Output:

Integrate[FresnelC[b*x]^2/x^7, x]
 

Rubi [N/A]

Not integrable

Time = 1.40 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 18, 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.

\(\displaystyle \int \frac {\operatorname {FresnelC}(b x)^2}{x^7} \, dx\)

\(\Big \downarrow \) 6985

\(\displaystyle \frac {1}{3} b \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^6}dx-\frac {\operatorname {FresnelC}(b x)^2}{6 x^6}\)

\(\Big \downarrow \) 7011

\(\displaystyle \frac {1}{3} b \left (-\frac {1}{5} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4}dx+\frac {1}{10} b \int \frac {\cos \left (b^2 \pi x^2\right )}{x^5}dx-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}-\frac {b}{40 x^4}\right )-\frac {\operatorname {FresnelC}(b x)^2}{6 x^6}\)

\(\Big \downarrow \) 3861

\(\displaystyle \frac {1}{3} b \left (-\frac {1}{5} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4}dx+\frac {1}{20} b \int \frac {\cos \left (b^2 \pi x^2\right )}{x^6}dx^2-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}-\frac {b}{40 x^4}\right )-\frac {\operatorname {FresnelC}(b x)^2}{6 x^6}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{3} b \left (-\frac {1}{5} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4}dx+\frac {1}{20} b \int \frac {\sin \left (b^2 \pi x^2+\frac {\pi }{2}\right )}{x^6}dx^2-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}-\frac {b}{40 x^4}\right )-\frac {\operatorname {FresnelC}(b x)^2}{6 x^6}\)

\(\Big \downarrow \) 3778

\(\displaystyle \frac {1}{3} b \left (-\frac {1}{5} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4}dx+\frac {1}{20} b \left (\frac {1}{2} \pi b^2 \int -\frac {\sin \left (b^2 \pi x^2\right )}{x^4}dx^2-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}-\frac {b}{40 x^4}\right )-\frac {\operatorname {FresnelC}(b x)^2}{6 x^6}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{3} b \left (-\frac {1}{5} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4}dx+\frac {1}{20} b \left (-\frac {1}{2} \pi b^2 \int \frac {\sin \left (b^2 \pi x^2\right )}{x^4}dx^2-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}-\frac {b}{40 x^4}\right )-\frac {\operatorname {FresnelC}(b x)^2}{6 x^6}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{3} b \left (-\frac {1}{5} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4}dx+\frac {1}{20} b \left (-\frac {1}{2} \pi b^2 \int \frac {\sin \left (b^2 \pi x^2\right )}{x^4}dx^2-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}-\frac {b}{40 x^4}\right )-\frac {\operatorname {FresnelC}(b x)^2}{6 x^6}\)

\(\Big \downarrow \) 3778

\(\displaystyle \frac {1}{3} b \left (-\frac {1}{5} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4}dx+\frac {1}{20} b \left (-\frac {1}{2} \pi b^2 \left (\pi b^2 \int \frac {\cos \left (b^2 \pi x^2\right )}{x^2}dx^2-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}-\frac {b}{40 x^4}\right )-\frac {\operatorname {FresnelC}(b x)^2}{6 x^6}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{3} b \left (-\frac {1}{5} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4}dx+\frac {1}{20} b \left (-\frac {1}{2} \pi b^2 \left (\pi b^2 \int \frac {\sin \left (b^2 \pi x^2+\frac {\pi }{2}\right )}{x^2}dx^2-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}-\frac {b}{40 x^4}\right )-\frac {\operatorname {FresnelC}(b x)^2}{6 x^6}\)

\(\Big \downarrow \) 3783

\(\displaystyle \frac {1}{3} b \left (-\frac {1}{5} \pi b^2 \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4}dx+\frac {1}{20} b \left (-\frac {1}{2} \pi b^2 \left (\pi b^2 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}-\frac {b}{40 x^4}\right )-\frac {\operatorname {FresnelC}(b x)^2}{6 x^6}\)

\(\Big \downarrow \) 7019

\(\displaystyle \frac {1}{3} b \left (-\frac {1}{5} \pi b^2 \left (\frac {1}{3} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^2}dx+\frac {1}{6} b \int \frac {\sin \left (b^2 \pi x^2\right )}{x^3}dx-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}\right )+\frac {1}{20} b \left (-\frac {1}{2} \pi b^2 \left (\pi b^2 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}-\frac {b}{40 x^4}\right )-\frac {\operatorname {FresnelC}(b x)^2}{6 x^6}\)

\(\Big \downarrow \) 3860

\(\displaystyle \frac {1}{3} b \left (-\frac {1}{5} \pi b^2 \left (\frac {1}{3} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^2}dx+\frac {1}{12} b \int \frac {\sin \left (b^2 \pi x^2\right )}{x^4}dx^2-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}\right )+\frac {1}{20} b \left (-\frac {1}{2} \pi b^2 \left (\pi b^2 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}-\frac {b}{40 x^4}\right )-\frac {\operatorname {FresnelC}(b x)^2}{6 x^6}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{3} b \left (-\frac {1}{5} \pi b^2 \left (\frac {1}{3} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^2}dx+\frac {1}{12} b \int \frac {\sin \left (b^2 \pi x^2\right )}{x^4}dx^2-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}\right )+\frac {1}{20} b \left (-\frac {1}{2} \pi b^2 \left (\pi b^2 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}-\frac {b}{40 x^4}\right )-\frac {\operatorname {FresnelC}(b x)^2}{6 x^6}\)

\(\Big \downarrow \) 3778

\(\displaystyle \frac {1}{3} b \left (-\frac {1}{5} \pi b^2 \left (\frac {1}{3} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^2}dx+\frac {1}{12} b \left (\pi b^2 \int \frac {\cos \left (b^2 \pi x^2\right )}{x^2}dx^2-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}\right )+\frac {1}{20} b \left (-\frac {1}{2} \pi b^2 \left (\pi b^2 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}-\frac {b}{40 x^4}\right )-\frac {\operatorname {FresnelC}(b x)^2}{6 x^6}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{3} b \left (-\frac {1}{5} \pi b^2 \left (\frac {1}{3} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^2}dx+\frac {1}{12} b \left (\pi b^2 \int \frac {\sin \left (b^2 \pi x^2+\frac {\pi }{2}\right )}{x^2}dx^2-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}\right )+\frac {1}{20} b \left (-\frac {1}{2} \pi b^2 \left (\pi b^2 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}-\frac {b}{40 x^4}\right )-\frac {\operatorname {FresnelC}(b x)^2}{6 x^6}\)

\(\Big \downarrow \) 3783

\(\displaystyle \frac {1}{3} b \left (-\frac {1}{5} \pi b^2 \left (\frac {1}{3} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^2}dx+\frac {1}{12} b \left (\pi b^2 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}\right )+\frac {1}{20} b \left (-\frac {1}{2} \pi b^2 \left (\pi b^2 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}-\frac {b}{40 x^4}\right )-\frac {\operatorname {FresnelC}(b x)^2}{6 x^6}\)

\(\Big \downarrow \) 7013

\(\displaystyle \frac {1}{3} b \left (-\frac {1}{5} \pi b^2 \left (\frac {1}{3} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^2}dx+\frac {1}{12} b \left (\pi b^2 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{3 x^3}\right )+\frac {1}{20} b \left (-\frac {1}{2} \pi b^2 \left (\pi b^2 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x^2}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{2 x^4}\right )-\frac {\operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{5 x^5}-\frac {b}{40 x^4}\right )-\frac {\operatorname {FresnelC}(b x)^2}{6 x^6}\)

Input:

Int[FresnelC[b*x]^2/x^7,x]
 

Output:

$Aborted
 

Maple [N/A]

Not integrable

Time = 0.20 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00

\[\int \frac {\operatorname {FresnelC}\left (b x \right )^{2}}{x^{7}}d x\]

Input:

int(FresnelC(b*x)^2/x^7,x)
 

Output:

int(FresnelC(b*x)^2/x^7,x)
 

Fricas [N/A]

Not integrable

Time = 0.09 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^7} \, dx=\int { \frac {\operatorname {C}\left (b x\right )^{2}}{x^{7}} \,d x } \] Input:

integrate(fresnel_cos(b*x)^2/x^7,x, algorithm="fricas")
 

Output:

integral(fresnel_cos(b*x)^2/x^7, x)
 

Sympy [N/A]

Not integrable

Time = 1.37 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^7} \, dx=\int \frac {C^{2}\left (b x\right )}{x^{7}}\, dx \] Input:

integrate(fresnelc(b*x)**2/x**7,x)
 

Output:

Integral(fresnelc(b*x)**2/x**7, x)
 

Maxima [N/A]

Not integrable

Time = 0.06 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^7} \, dx=\int { \frac {\operatorname {C}\left (b x\right )^{2}}{x^{7}} \,d x } \] Input:

integrate(fresnel_cos(b*x)^2/x^7,x, algorithm="maxima")
 

Output:

integrate(fresnel_cos(b*x)^2/x^7, x)
 

Giac [N/A]

Not integrable

Time = 0.10 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^7} \, dx=\int { \frac {\operatorname {C}\left (b x\right )^{2}}{x^{7}} \,d x } \] Input:

integrate(fresnel_cos(b*x)^2/x^7,x, algorithm="giac")
 

Output:

integrate(fresnel_cos(b*x)^2/x^7, x)
 

Mupad [N/A]

Not integrable

Time = 3.81 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^7} \, dx=\int \frac {{\mathrm {FresnelC}\left (b\,x\right )}^2}{x^7} \,d x \] Input:

int(FresnelC(b*x)^2/x^7,x)
 

Output:

int(FresnelC(b*x)^2/x^7, x)
 

Reduce [N/A]

Not integrable

Time = 0.19 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {\operatorname {FresnelC}(b x)^2}{x^7} \, dx=\int \frac {\mathrm {FresnelC}\left (b x \right )^{2}}{x^{7}}d x \] Input:

int(FresnelC(b*x)^2/x^7,x)
 

Output:

int(FresnelC(b*x)^2/x^7,x)