3.216 \(\int \frac{\text{FresnelC}(b x) \sin (\frac{1}{2} b^2 \pi x^2)}{x^8} \, dx\)

Optimal. Leaf size=201 \[ -\frac{1}{105} \pi ^3 b^6 \text{Unintegrable}\left (\frac{\text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{x^2},x\right )-\frac{1}{84} \pi ^3 b^7 \text{CosIntegral}\left (\pi b^2 x^2\right )+\frac{\pi ^2 b^4 \text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{105 x^3}-\frac{\text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{7 x^7}-\frac{\pi b^2 \text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{35 x^5}-\frac{\pi b^3}{280 x^4}+\frac{\pi ^2 b^5 \sin \left (\pi b^2 x^2\right )}{84 x^2}-\frac{b \sin \left (\pi b^2 x^2\right )}{84 x^6}-\frac{\pi b^3 \cos \left (\pi b^2 x^2\right )}{105 x^4} \]

[Out]

-(b^3*Pi)/(280*x^4) - (b^3*Pi*Cos[b^2*Pi*x^2])/(105*x^4) - (b^7*Pi^3*CosIntegral[b^2*Pi*x^2])/84 - (b^2*Pi*Cos
[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(35*x^5) - (FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(7*x^7) + (b^4*Pi^2*FresnelC[b*
x]*Sin[(b^2*Pi*x^2)/2])/(105*x^3) - (b*Sin[b^2*Pi*x^2])/(84*x^6) + (b^5*Pi^2*Sin[b^2*Pi*x^2])/(84*x^2) - (b^6*
Pi^3*Unintegrable[(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x^2, x])/105

________________________________________________________________________________________

Rubi [A]  time = 0.314121, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\text{FresnelC}(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^8} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/x^8,x]

[Out]

-(b^3*Pi)/(280*x^4) - (b^3*Pi*Cos[b^2*Pi*x^2])/(105*x^4) - (b^7*Pi^3*CosIntegral[b^2*Pi*x^2])/84 - (b^2*Pi*Cos
[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(35*x^5) - (FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(7*x^7) + (b^4*Pi^2*FresnelC[b*
x]*Sin[(b^2*Pi*x^2)/2])/(105*x^3) - (b*Sin[b^2*Pi*x^2])/(84*x^6) + (b^5*Pi^2*Sin[b^2*Pi*x^2])/(84*x^2) - (b^6*
Pi^3*Defer[Int][(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x^2, x])/105

Rubi steps

\begin{align*} \int \frac{C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^8} \, dx &=-\frac{C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{7 x^7}+\frac{1}{14} b \int \frac{\sin \left (b^2 \pi x^2\right )}{x^7} \, dx+\frac{1}{7} \left (b^2 \pi \right ) \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{x^6} \, dx\\ &=-\frac{b^3 \pi }{280 x^4}-\frac{b^2 \pi \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{35 x^5}-\frac{C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{7 x^7}+\frac{1}{28} b \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x^4} \, dx,x,x^2\right )+\frac{1}{70} \left (b^3 \pi \right ) \int \frac{\cos \left (b^2 \pi x^2\right )}{x^5} \, dx-\frac{1}{35} \left (b^4 \pi ^2\right ) \int \frac{C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^4} \, dx\\ &=-\frac{b^3 \pi }{280 x^4}-\frac{b^2 \pi \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{35 x^5}-\frac{C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{7 x^7}+\frac{b^4 \pi ^2 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{105 x^3}-\frac{b \sin \left (b^2 \pi x^2\right )}{84 x^6}+\frac{1}{140} \left (b^3 \pi \right ) \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )+\frac{1}{84} \left (b^3 \pi \right ) \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )-\frac{1}{210} \left (b^5 \pi ^2\right ) \int \frac{\sin \left (b^2 \pi x^2\right )}{x^3} \, dx-\frac{1}{105} \left (b^6 \pi ^3\right ) \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{x^2} \, dx\\ &=-\frac{b^3 \pi }{280 x^4}-\frac{b^3 \pi \cos \left (b^2 \pi x^2\right )}{105 x^4}-\frac{b^2 \pi \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{35 x^5}-\frac{C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{7 x^7}+\frac{b^4 \pi ^2 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{105 x^3}-\frac{b \sin \left (b^2 \pi x^2\right )}{84 x^6}-\frac{1}{420} \left (b^5 \pi ^2\right ) \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )-\frac{1}{280} \left (b^5 \pi ^2\right ) \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )-\frac{1}{168} \left (b^5 \pi ^2\right ) \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )-\frac{1}{105} \left (b^6 \pi ^3\right ) \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{x^2} \, dx\\ &=-\frac{b^3 \pi }{280 x^4}-\frac{b^3 \pi \cos \left (b^2 \pi x^2\right )}{105 x^4}-\frac{b^2 \pi \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{35 x^5}-\frac{C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{7 x^7}+\frac{b^4 \pi ^2 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{105 x^3}-\frac{b \sin \left (b^2 \pi x^2\right )}{84 x^6}+\frac{b^5 \pi ^2 \sin \left (b^2 \pi x^2\right )}{84 x^2}-\frac{1}{105} \left (b^6 \pi ^3\right ) \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{x^2} \, dx-\frac{1}{420} \left (b^7 \pi ^3\right ) \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )-\frac{1}{280} \left (b^7 \pi ^3\right ) \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )-\frac{1}{168} \left (b^7 \pi ^3\right ) \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )\\ &=-\frac{b^3 \pi }{280 x^4}-\frac{b^3 \pi \cos \left (b^2 \pi x^2\right )}{105 x^4}-\frac{1}{84} b^7 \pi ^3 \text{Ci}\left (b^2 \pi x^2\right )-\frac{b^2 \pi \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{35 x^5}-\frac{C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{7 x^7}+\frac{b^4 \pi ^2 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{105 x^3}-\frac{b \sin \left (b^2 \pi x^2\right )}{84 x^6}+\frac{b^5 \pi ^2 \sin \left (b^2 \pi x^2\right )}{84 x^2}-\frac{1}{105} \left (b^6 \pi ^3\right ) \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{x^2} \, dx\\ \end{align*}

Mathematica [A]  time = 0.0328026, size = 0, normalized size = 0. \[ \int \frac{\text{FresnelC}(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^8} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/x^8,x]

[Out]

Integrate[(FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/x^8, x]

________________________________________________________________________________________

Maple [A]  time = 0.069, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it FresnelC} \left ( bx \right ) }{{x}^{8}}\sin \left ({\frac{{b}^{2}\pi \,{x}^{2}}{2}} \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/x^8,x)

[Out]

int(FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/x^8,x)

________________________________________________________________________________________

Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm fresnelc}\left (b x\right ) \sin \left (\frac{1}{2} \, \pi b^{2} x^{2}\right )}{x^{8}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnelc(b*x)*sin(1/2*b^2*pi*x^2)/x^8,x, algorithm="maxima")

[Out]

integrate(fresnelc(b*x)*sin(1/2*pi*b^2*x^2)/x^8, x)

________________________________________________________________________________________

Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\rm fresnelc}\left (b x\right ) \sin \left (\frac{1}{2} \, \pi b^{2} x^{2}\right )}{x^{8}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnelc(b*x)*sin(1/2*b^2*pi*x^2)/x^8,x, algorithm="fricas")

[Out]

integral(fresnelc(b*x)*sin(1/2*pi*b^2*x^2)/x^8, x)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnelc(b*x)*sin(1/2*b**2*pi*x**2)/x**8,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm fresnelc}\left (b x\right ) \sin \left (\frac{1}{2} \, \pi b^{2} x^{2}\right )}{x^{8}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnelc(b*x)*sin(1/2*b^2*pi*x^2)/x^8,x, algorithm="giac")

[Out]

integrate(fresnelc(b*x)*sin(1/2*pi*b^2*x^2)/x^8, x)