\(\int \frac {\cos (\frac {1}{2} b^2 \pi x^2) \operatorname {FresnelS}(b x)}{x^7} \, dx\) [106]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^7} \, dx=\frac {b^3 \pi }{144 x^3}-\frac {13 b^3 \pi \cos \left (b^2 \pi x^2\right )}{720 x^3}-\frac {7 b^6 \pi ^3 \operatorname {FresnelC}\left (\sqrt {2} b x\right )}{144 \sqrt {2}}-\frac {1}{45} \sqrt {2} b^6 \pi ^3 \operatorname {FresnelC}\left (\sqrt {2} b x\right )-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{6 x^6}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{48 x^2}+\frac {b^2 \pi \operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{24 x^4}-\frac {b \sin \left (b^2 \pi x^2\right )}{60 x^5}+\frac {67 b^5 \pi ^2 \sin \left (b^2 \pi x^2\right )}{1440 x}+\frac {1}{48} b^6 \pi ^3 \text {Int}\left (\frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x},x\right ) \]

[Out]

1/144*b^3*Pi/x^3-13/720*b^3*Pi*cos(b^2*Pi*x^2)/x^3-1/6*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-67/1440*b^6*Pi^3*FresnelC(b*x*2^(1/2))*2^(1/2)+1/48*b^6*Pi^3*Unintegrable
(FresnelS(b*x)*sin(1/2*b^2*Pi*x^2)/x,x)

Rubi [N/A]

Not integrable

Time = 0.14 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^7} \, dx=\int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^7} \, dx \]

[In]

Int[(Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/x^7,x]

[Out]

(b^3*Pi)/(144*x^3) - (13*b^3*Pi*Cos[b^2*Pi*x^2])/(720*x^3) - (7*b^6*Pi^3*FresnelC[Sqrt[2]*b*x])/(144*Sqrt[2])
- (Sqrt[2]*b^6*Pi^3*FresnelC[Sqrt[2]*b*x])/45 - (Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/(6*x^6) + (b^4*Pi^2*Cos[(b
^2*Pi*x^2)/2]*FresnelS[b*x])/(48*x^2) + (b^2*Pi*FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2])/(24*x^4) - (b*Sin[b^2*Pi*x^
2])/(60*x^5) + (67*b^5*Pi^2*Sin[b^2*Pi*x^2])/(1440*x) + (b^6*Pi^3*Defer[Int][(FresnelS[b*x]*Sin[(b^2*Pi*x^2)/2
])/x, x])/48

Rubi steps \begin{align*} \text {integral}& = -\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{6 x^6}+\frac {1}{12} b \int \frac {\sin \left (b^2 \pi x^2\right )}{x^6} \, dx-\frac {1}{6} \left (b^2 \pi \right ) \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5} \, dx \\ & = \frac {b^3 \pi }{144 x^3}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{6 x^6}+\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 {1}{48} \left (b^3 \pi \right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^4} \, dx+\frac {1}{30} \left (b^3 \pi \right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^4} \, dx-\frac {1}{24} \left (b^4 \pi ^2\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^3} \, 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 {\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 {1}{96} \left (b^5 \pi ^2\right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2} \, dx-\frac {1}{72} \left (b^5 \pi ^2\right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2} \, dx-\frac {1}{45} \left (b^5 \pi ^2\right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2} \, dx+\frac {1}{48} \left (b^6 \pi ^3\right ) \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x} \, 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 {\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} \left (b^6 \pi ^3\right ) \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x} \, dx-\frac {1}{48} \left (b^7 \pi ^3\right ) \int \cos \left (b^2 \pi x^2\right ) \, dx-\frac {1}{36} \left (b^7 \pi ^3\right ) \int \cos \left (b^2 \pi x^2\right ) \, dx-\frac {1}{45} \left (2 b^7 \pi ^3\right ) \int \cos \left (b^2 \pi x^2\right ) \, 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} \left (b^6 \pi ^3\right ) \int \frac {\operatorname {FresnelS}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^7} \, dx=\int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^7} \, dx \]

[In]

Integrate[(Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/x^7,x]

[Out]

Integrate[(Cos[(b^2*Pi*x^2)/2]*FresnelS[b*x])/x^7, x]

Maple [N/A] (verified)

Not integrable

Time = 0.14 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90

\[\int \frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \operatorname {FresnelS}\left (b x \right )}{x^{7}}d x\]

[In]

int(cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/x^7,x)

[Out]

int(cos(1/2*b^2*Pi*x^2)*FresnelS(b*x)/x^7,x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^7} \, dx=\int { \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right )}{x^{7}} \,d x } \]

[In]

integrate(cos(1/2*b^2*pi*x^2)*fresnel_sin(b*x)/x^7,x, algorithm="fricas")

[Out]

integral(cos(1/2*pi*b^2*x^2)*fresnel_sin(b*x)/x^7, x)

Sympy [N/A]

Not integrable

Time = 11.23 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^7} \, dx=\int \frac {\cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )}{x^{7}}\, dx \]

[In]

integrate(cos(1/2*b**2*pi*x**2)*fresnels(b*x)/x**7,x)

[Out]

Integral(cos(pi*b**2*x**2/2)*fresnels(b*x)/x**7, x)

Maxima [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^7} \, dx=\int { \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right )}{x^{7}} \,d x } \]

[In]

integrate(cos(1/2*b^2*pi*x^2)*fresnel_sin(b*x)/x^7,x, algorithm="maxima")

[Out]

integrate(cos(1/2*pi*b^2*x^2)*fresnel_sin(b*x)/x^7, x)

Giac [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^7} \, dx=\int { \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {S}\left (b x\right )}{x^{7}} \,d x } \]

[In]

integrate(cos(1/2*b^2*pi*x^2)*fresnel_sin(b*x)/x^7,x, algorithm="giac")

[Out]

integrate(cos(1/2*pi*b^2*x^2)*fresnel_sin(b*x)/x^7, x)

Mupad [N/A]

Not integrable

Time = 4.72 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelS}(b x)}{x^7} \, dx=\int \frac {\mathrm {FresnelS}\left (b\,x\right )\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right )}{x^7} \,d x \]

[In]

int((FresnelS(b*x)*cos((Pi*b^2*x^2)/2))/x^7,x)

[Out]

int((FresnelS(b*x)*cos((Pi*b^2*x^2)/2))/x^7, x)