\(\int \frac {\cos (\frac {1}{2} b^2 \pi x^2) \operatorname {FresnelC}(b x)}{x^6} \, dx\) [194]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^6} \, dx=-\frac {b}{40 x^4}-\frac {b \cos \left (b^2 \pi x^2\right )}{40 x^4}-\frac {1}{24} b^5 \pi ^2 \operatorname {CosIntegral}\left (b^2 \pi x^2\right )-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{5 x^5}+\frac {b^2 \pi \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{15 x^3}+\frac {b^3 \pi \sin \left (b^2 \pi x^2\right )}{24 x^2}-\frac {1}{15} b^4 \pi ^2 \text {Int}\left (\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^2},x\right ) \] Output:

-1/40*b/x^4-1/40*b*cos(b^2*Pi*x^2)/x^4-1/24*b^5*Pi^2*Ci(b^2*Pi*x^2)-1/5*co 
s(1/2*b^2*Pi*x^2)*FresnelC(b*x)/x^5+1/15*b^2*Pi*FresnelC(b*x)*sin(1/2*b^2* 
Pi*x^2)/x^3+1/24*b^3*Pi*sin(b^2*Pi*x^2)/x^2-1/15*b^4*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 = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^6} \, dx=\int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^6} \, dx \] Input:

Integrate[(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x^6,x]
 

Output:

Integrate[(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x^6, x]
 

Rubi [N/A]

Not integrable

Time = 1.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 17, 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.

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

\(\Big \downarrow \) 7011

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

\(\Big \downarrow \) 3861

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3778

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3778

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3783

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

\(\Big \downarrow \) 7019

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

\(\Big \downarrow \) 3860

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3778

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3783

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

\(\Big \downarrow \) 7013

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

Input:

Int[(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x^6,x]
 

Output:

$Aborted
 

Maple [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90

\[\int \frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \operatorname {FresnelC}\left (b x \right )}{x^{6}}d x\]

Input:

int(cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/x^6,x)
 

Output:

int(cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/x^6,x)
 

Fricas [N/A]

Not integrable

Time = 0.09 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^6} \, dx=\int { \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right )}{x^{6}} \,d x } \] Input:

integrate(cos(1/2*b^2*pi*x^2)*fresnel_cos(b*x)/x^6,x, algorithm="fricas")
 

Output:

integral(cos(1/2*pi*b^2*x^2)*fresnel_cos(b*x)/x^6, x)
 

Sympy [N/A]

Not integrable

Time = 5.71 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^6} \, dx=\int \frac {\cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )}{x^{6}}\, dx \] Input:

integrate(cos(1/2*b**2*pi*x**2)*fresnelc(b*x)/x**6,x)
 

Output:

Integral(cos(pi*b**2*x**2/2)*fresnelc(b*x)/x**6, x)
 

Maxima [N/A]

Not integrable

Time = 0.14 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^6} \, dx=\int { \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right )}{x^{6}} \,d x } \] Input:

integrate(cos(1/2*b^2*pi*x^2)*fresnel_cos(b*x)/x^6,x, algorithm="maxima")
 

Output:

integrate(cos(1/2*pi*b^2*x^2)*fresnel_cos(b*x)/x^6, x)
 

Giac [N/A]

Not integrable

Time = 0.12 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^6} \, dx=\int { \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right )}{x^{6}} \,d x } \] Input:

integrate(cos(1/2*b^2*pi*x^2)*fresnel_cos(b*x)/x^6,x, algorithm="giac")
 

Output:

integrate(cos(1/2*pi*b^2*x^2)*fresnel_cos(b*x)/x^6, x)
 

Mupad [N/A]

Not integrable

Time = 3.66 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^6} \, dx=\int \frac {\mathrm {FresnelC}\left (b\,x\right )\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right )}{x^6} \,d x \] Input:

int((FresnelC(b*x)*cos((Pi*b^2*x^2)/2))/x^6,x)
 

Output:

int((FresnelC(b*x)*cos((Pi*b^2*x^2)/2))/x^6, x)
 

Reduce [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^6} \, dx=\int \frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \mathrm {FresnelC}\left (b x \right )}{x^{6}}d x \] Input:

int(cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/x^6,x)
 

Output:

int(cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/x^6,x)