\(\int \frac {\operatorname {FresnelC}(b x) \sin (\frac {1}{2} b^2 \pi x^2)}{x^9} \, dx\) [217]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^9} \, dx=-\frac {b^3 \pi }{480 x^5}+\frac {b^7 \pi ^3}{768 x}-\frac {19 b^3 \pi \cos \left (b^2 \pi x^2\right )}{3360 x^5}+\frac {853 b^7 \pi ^3 \cos \left (b^2 \pi x^2\right )}{80640 x}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{48 x^6}+\frac {b^6 \pi ^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{384 x^2}+\frac {853 b^8 \pi ^4 \operatorname {FresnelS}\left (\sqrt {2} b x\right )}{40320 \sqrt {2}}-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{8 x^8}+\frac {b^4 \pi ^2 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{192 x^4}-\frac {b \sin \left (b^2 \pi x^2\right )}{112 x^7}+\frac {187 b^5 \pi ^2 \sin \left (b^2 \pi x^2\right )}{40320 x^3}+\frac {1}{384} b^8 \pi ^4 \text {Int}\left (\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x},x\right ) \]

[Out]

-1/480*b^3*Pi/x^5+1/768*b^7*Pi^3/x-19/3360*b^3*Pi*cos(b^2*Pi*x^2)/x^5+853/80640*b^7*Pi^3*cos(b^2*Pi*x^2)/x-1/4
8*b^2*Pi*cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/x^6+1/384*b^6*Pi^3*cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/x^2-1/8*Fresne
lC(b*x)*sin(1/2*b^2*Pi*x^2)/x^8+1/192*b^4*Pi^2*FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/x^4-1/112*b*sin(b^2*Pi*x^2)/x
^7+187/40320*b^5*Pi^2*sin(b^2*Pi*x^2)/x^3+853/80640*b^8*Pi^4*FresnelS(b*x*2^(1/2))*2^(1/2)+1/384*b^8*Pi^4*Unin
tegrable(FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/x,x)

Rubi [N/A]

Not integrable

Time = 0.22 (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 = {} \[ \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^9} \, dx=\int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^9} \, dx \]

[In]

Int[(FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/x^9,x]

[Out]

-1/480*(b^3*Pi)/x^5 + (b^7*Pi^3)/(768*x) - (19*b^3*Pi*Cos[b^2*Pi*x^2])/(3360*x^5) + (853*b^7*Pi^3*Cos[b^2*Pi*x
^2])/(80640*x) - (b^2*Pi*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(48*x^6) + (b^6*Pi^3*Cos[(b^2*Pi*x^2)/2]*FresnelC[
b*x])/(384*x^2) + (853*b^8*Pi^4*FresnelS[Sqrt[2]*b*x])/(40320*Sqrt[2]) - (FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(
8*x^8) + (b^4*Pi^2*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(192*x^4) - (b*Sin[b^2*Pi*x^2])/(112*x^7) + (187*b^5*Pi^
2*Sin[b^2*Pi*x^2])/(40320*x^3) + (b^8*Pi^4*Defer[Int][(FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/x, x])/384

Rubi steps \begin{align*} \text {integral}& = -\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{8 x^8}+\frac {1}{16} b \int \frac {\sin \left (b^2 \pi x^2\right )}{x^8} \, dx+\frac {1}{8} \left (b^2 \pi \right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^7} \, dx \\ & = -\frac {b^3 \pi }{480 x^5}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{48 x^6}-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{8 x^8}-\frac {b \sin \left (b^2 \pi x^2\right )}{112 x^7}+\frac {1}{96} \left (b^3 \pi \right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^6} \, dx+\frac {1}{56} \left (b^3 \pi \right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^6} \, dx-\frac {1}{48} \left (b^4 \pi ^2\right ) \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^5} \, dx \\ & = -\frac {b^3 \pi }{480 x^5}-\frac {19 b^3 \pi \cos \left (b^2 \pi x^2\right )}{3360 x^5}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{48 x^6}-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{8 x^8}+\frac {b^4 \pi ^2 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{192 x^4}-\frac {b \sin \left (b^2 \pi x^2\right )}{112 x^7}-\frac {1}{384} \left (b^5 \pi ^2\right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^4} \, dx-\frac {1}{240} \left (b^5 \pi ^2\right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^4} \, dx-\frac {1}{140} \left (b^5 \pi ^2\right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^4} \, dx-\frac {1}{192} \left (b^6 \pi ^3\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{x^3} \, dx \\ & = -\frac {b^3 \pi }{480 x^5}+\frac {b^7 \pi ^3}{768 x}-\frac {19 b^3 \pi \cos \left (b^2 \pi x^2\right )}{3360 x^5}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{48 x^6}+\frac {b^6 \pi ^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{384 x^2}-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{8 x^8}+\frac {b^4 \pi ^2 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{192 x^4}-\frac {b \sin \left (b^2 \pi x^2\right )}{112 x^7}+\frac {187 b^5 \pi ^2 \sin \left (b^2 \pi x^2\right )}{40320 x^3}-\frac {1}{768} \left (b^7 \pi ^3\right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^2} \, dx-\frac {1}{576} \left (b^7 \pi ^3\right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^2} \, dx-\frac {1}{360} \left (b^7 \pi ^3\right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^2} \, dx-\frac {1}{210} \left (b^7 \pi ^3\right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^2} \, dx+\frac {1}{384} \left (b^8 \pi ^4\right ) \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x} \, dx \\ & = -\frac {b^3 \pi }{480 x^5}+\frac {b^7 \pi ^3}{768 x}-\frac {19 b^3 \pi \cos \left (b^2 \pi x^2\right )}{3360 x^5}+\frac {853 b^7 \pi ^3 \cos \left (b^2 \pi x^2\right )}{80640 x}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{48 x^6}+\frac {b^6 \pi ^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{384 x^2}-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{8 x^8}+\frac {b^4 \pi ^2 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{192 x^4}-\frac {b \sin \left (b^2 \pi x^2\right )}{112 x^7}+\frac {187 b^5 \pi ^2 \sin \left (b^2 \pi x^2\right )}{40320 x^3}+\frac {1}{384} \left (b^8 \pi ^4\right ) \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x} \, dx+\frac {1}{384} \left (b^9 \pi ^4\right ) \int \sin \left (b^2 \pi x^2\right ) \, dx+\frac {1}{288} \left (b^9 \pi ^4\right ) \int \sin \left (b^2 \pi x^2\right ) \, dx+\frac {1}{180} \left (b^9 \pi ^4\right ) \int \sin \left (b^2 \pi x^2\right ) \, dx+\frac {1}{105} \left (b^9 \pi ^4\right ) \int \sin \left (b^2 \pi x^2\right ) \, dx \\ & = -\frac {b^3 \pi }{480 x^5}+\frac {b^7 \pi ^3}{768 x}-\frac {19 b^3 \pi \cos \left (b^2 \pi x^2\right )}{3360 x^5}+\frac {853 b^7 \pi ^3 \cos \left (b^2 \pi x^2\right )}{80640 x}-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{48 x^6}+\frac {b^6 \pi ^3 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{384 x^2}+\frac {853 b^8 \pi ^4 \operatorname {FresnelS}\left (\sqrt {2} b x\right )}{40320 \sqrt {2}}-\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{8 x^8}+\frac {b^4 \pi ^2 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{192 x^4}-\frac {b \sin \left (b^2 \pi x^2\right )}{112 x^7}+\frac {187 b^5 \pi ^2 \sin \left (b^2 \pi x^2\right )}{40320 x^3}+\frac {1}{384} \left (b^8 \pi ^4\right ) \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^9} \, dx=\int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^9} \, dx \]

[In]

Integrate[(FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/x^9,x]

[Out]

Integrate[(FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/x^9, x]

Maple [N/A] (verified)

Not integrable

Time = 0.13 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90

\[\int \frac {\operatorname {FresnelC}\left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{x^{9}}d x\]

[In]

int(FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/x^9,x)

[Out]

int(FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/x^9,x)

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^9} \, dx=\int { \frac {\operatorname {C}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{x^{9}} \,d x } \]

[In]

integrate(fresnel_cos(b*x)*sin(1/2*b^2*pi*x^2)/x^9,x, algorithm="fricas")

[Out]

integral(fresnel_cos(b*x)*sin(1/2*pi*b^2*x^2)/x^9, x)

Sympy [N/A]

Not integrable

Time = 36.67 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^9} \, dx=\int \frac {\sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )}{x^{9}}\, dx \]

[In]

integrate(fresnelc(b*x)*sin(1/2*b**2*pi*x**2)/x**9,x)

[Out]

Integral(sin(pi*b**2*x**2/2)*fresnelc(b*x)/x**9, x)

Maxima [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^9} \, dx=\int { \frac {\operatorname {C}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{x^{9}} \,d x } \]

[In]

integrate(fresnel_cos(b*x)*sin(1/2*b^2*pi*x^2)/x^9,x, algorithm="maxima")

[Out]

integrate(fresnel_cos(b*x)*sin(1/2*pi*b^2*x^2)/x^9, x)

Giac [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^9} \, dx=\int { \frac {\operatorname {C}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{x^{9}} \,d x } \]

[In]

integrate(fresnel_cos(b*x)*sin(1/2*b^2*pi*x^2)/x^9,x, algorithm="giac")

[Out]

integrate(fresnel_cos(b*x)*sin(1/2*pi*b^2*x^2)/x^9, x)

Mupad [N/A]

Not integrable

Time = 4.77 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^9} \, dx=\int \frac {\mathrm {FresnelC}\left (b\,x\right )\,\sin \left (\frac {\Pi \,b^2\,x^2}{2}\right )}{x^9} \,d x \]

[In]

int((FresnelC(b*x)*sin((Pi*b^2*x^2)/2))/x^9,x)

[Out]

int((FresnelC(b*x)*sin((Pi*b^2*x^2)/2))/x^9, x)