Integrand size = 20, antiderivative size = 20 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^7} \, dx=\frac {b^3 \pi }{144 x^3}-\frac {13 b^3 \pi \cos \left (b^2 \pi x^2\right )}{720 x^3}-\frac {7 b^6 \pi ^3 \operatorname {FresnelC}\left (\sqrt {2} b x\right )}{144 \sqrt {2}}-\frac {1}{45} \sqrt {2} b^6 \pi ^3 \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{6 x^6}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{48 x^2}+\frac {b^2 \pi \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{24 x^4}-\frac {b \sin \left (b^2 \pi x^2\right )}{60 x^5}+\frac {67 b^5 \pi ^2 \sin \left (b^2 \pi x^2\right )}{1440 x}+\frac {1}{48} b^6 \pi ^3 \text {Int}\left (\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x},x\right ) \] Output:
1/144*b^3*Pi/x^3-13/720*b^3*Pi*cos(b^2*Pi*x^2)/x^3-67/1440*b^6*Pi^3*Fresne lC(2^(1/2)*b*x)*2^(1/2)-1/6*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/x^6+1/48*b^4 *Pi^2*cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/x^2+1/24*b^2*Pi*FresnelS(b*x)*sin( 1/2*b^2*Pi*x^2)/x^4-1/60*b*sin(b^2*Pi*x^2)/x^5+67/1440*b^5*Pi^2*sin(b^2*Pi *x^2)/x+1/48*b^6*Pi^3*Defer(Int)(FresnelS(b*x)*sin(1/2*b^2*Pi*x^2)/x,x)
Not integrable
Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^7} \, dx=\int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^7} \, dx \] Input:
Integrate[(Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/x^7,x]
Output:
Integrate[(Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/x^7, x]
Not integrable
Time = 1.48 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 13, 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.
\(\displaystyle \int \frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{x^7} \, dx\) |
\(\Big \downarrow \) 7018 |
\(\displaystyle -\frac {1}{6} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5}dx+\frac {1}{12} b \int \frac {\sin \left (b^2 \pi x^2\right )}{x^6}dx-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}\) |
\(\Big \downarrow \) 3868 |
\(\displaystyle -\frac {1}{6} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5}dx+\frac {1}{12} b \left (\frac {2}{5} \pi b^2 \int \frac {\cos \left (b^2 \pi x^2\right )}{x^4}dx-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}\) |
\(\Big \downarrow \) 3869 |
\(\displaystyle -\frac {1}{6} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5}dx+\frac {1}{12} b \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2}dx-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}\) |
\(\Big \downarrow \) 3868 |
\(\displaystyle -\frac {1}{6} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5}dx+\frac {1}{12} b \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (2 \pi b^2 \int \cos \left (b^2 \pi x^2\right )dx-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}\) |
\(\Big \downarrow \) 3833 |
\(\displaystyle -\frac {1}{6} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5}dx+\frac {1}{12} b \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}\) |
\(\Big \downarrow \) 7010 |
\(\displaystyle -\frac {1}{6} \pi b^2 \left (\frac {1}{4} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^3}dx-\frac {1}{8} b \int \frac {\cos \left (b^2 \pi x^2\right )}{x^4}dx-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}-\frac {b}{24 x^3}\right )+\frac {1}{12} b \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}\) |
\(\Big \downarrow \) 3869 |
\(\displaystyle -\frac {1}{6} \pi b^2 \left (\frac {1}{4} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^3}dx-\frac {1}{8} b \left (-\frac {2}{3} \pi b^2 \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2}dx-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}-\frac {b}{24 x^3}\right )+\frac {1}{12} b \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}\) |
\(\Big \downarrow \) 3868 |
\(\displaystyle -\frac {1}{6} \pi b^2 \left (\frac {1}{4} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^3}dx-\frac {1}{8} b \left (-\frac {2}{3} \pi b^2 \left (2 \pi b^2 \int \cos \left (b^2 \pi x^2\right )dx-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}-\frac {b}{24 x^3}\right )+\frac {1}{12} b \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}\) |
\(\Big \downarrow \) 3833 |
\(\displaystyle -\frac {1}{6} \pi b^2 \left (\frac {1}{4} \pi b^2 \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^3}dx-\frac {1}{8} b \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}-\frac {b}{24 x^3}\right )+\frac {1}{12} b \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}\) |
\(\Big \downarrow \) 7018 |
\(\displaystyle -\frac {1}{6} \pi b^2 \left (\frac {1}{4} \pi b^2 \left (-\frac {1}{2} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x}dx+\frac {1}{4} b \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2}dx-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{2 x^2}\right )-\frac {1}{8} b \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}-\frac {b}{24 x^3}\right )+\frac {1}{12} b \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}\) |
\(\Big \downarrow \) 3868 |
\(\displaystyle -\frac {1}{6} \pi b^2 \left (\frac {1}{4} \pi b^2 \left (-\frac {1}{2} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x}dx+\frac {1}{4} b \left (2 \pi b^2 \int \cos \left (b^2 \pi x^2\right )dx-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{2 x^2}\right )-\frac {1}{8} b \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}-\frac {b}{24 x^3}\right )+\frac {1}{12} b \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}\) |
\(\Big \downarrow \) 3833 |
\(\displaystyle -\frac {1}{6} \pi b^2 \left (\frac {1}{4} \pi b^2 \left (-\frac {1}{2} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x}dx+\frac {1}{4} b \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{2 x^2}\right )-\frac {1}{8} b \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}-\frac {b}{24 x^3}\right )+\frac {1}{12} b \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}\) |
\(\Big \downarrow \) 7012 |
\(\displaystyle -\frac {1}{6} \pi b^2 \left (\frac {1}{4} \pi b^2 \left (-\frac {1}{2} \pi b^2 \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x}dx+\frac {1}{4} b \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{2 x^2}\right )-\frac {1}{8} b \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{4 x^4}-\frac {b}{24 x^3}\right )+\frac {1}{12} b \left (\frac {2}{5} \pi b^2 \left (-\frac {2}{3} \pi b^2 \left (\sqrt {2} \pi b \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\sin \left (\pi b^2 x^2\right )}{x}\right )-\frac {\cos \left (\pi b^2 x^2\right )}{3 x^3}\right )-\frac {\sin \left (\pi b^2 x^2\right )}{5 x^5}\right )-\frac {\operatorname {FresnelS}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{6 x^6}\) |
Input:
Int[(Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/x^7,x]
Output:
$Aborted
Not integrable
Time = 0.24 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90
\[\int \frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \operatorname {FresnelS}\left (b x \right )}{x^{7}}d x\]
Input:
int(cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/x^7,x)
Output:
int(cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/x^7,x)
Not integrable
Time = 0.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^7} \, dx=\int { \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right )}{x^{7}} \,d x } \] Input:
integrate(cos(1/2*b^2*pi*x^2)*fresnel_sin(b*x)/x^7,x, algorithm="fricas")
Output:
integral(cos(1/2*pi*b^2*x^2)*fresnel_sin(b*x)/x^7, x)
Not integrable
Time = 11.61 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^7} \, dx=\int \frac {\cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )}{x^{7}}\, dx \] Input:
integrate(cos(1/2*b**2*pi*x**2)*fresnels(b*x)/x**7,x)
Output:
Integral(cos(pi*b**2*x**2/2)*fresnels(b*x)/x**7, x)
Not integrable
Time = 0.13 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^7} \, dx=\int { \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right )}{x^{7}} \,d x } \] Input:
integrate(cos(1/2*b^2*pi*x^2)*fresnel_sin(b*x)/x^7,x, algorithm="maxima")
Output:
integrate(cos(1/2*pi*b^2*x^2)*fresnel_sin(b*x)/x^7, x)
Not integrable
Time = 0.12 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^7} \, dx=\int { \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right )}{x^{7}} \,d x } \] Input:
integrate(cos(1/2*b^2*pi*x^2)*fresnel_sin(b*x)/x^7,x, algorithm="giac")
Output:
integrate(cos(1/2*pi*b^2*x^2)*fresnel_sin(b*x)/x^7, x)
Not integrable
Time = 3.73 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^7} \, dx=\int \frac {\mathrm {FresnelS}\left (b\,x\right )\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right )}{x^7} \,d x \] Input:
int((FresnelS(b*x)*cos((Pi*b^2*x^2)/2))/x^7,x)
Output:
int((FresnelS(b*x)*cos((Pi*b^2*x^2)/2))/x^7, x)
Not integrable
Time = 0.20 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^7} \, dx=\int \frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \mathrm {FresnelS}\left (b x \right )}{x^{7}}d x \] Input:
int(cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/x^7,x)
Output:
int(cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/x^7,x)