3.2.97 \(\int \frac {\cos (\frac {1}{2} b^2 \pi x^2) \text {FresnelC}(b x)}{x^9} \, dx\) [197]

Optimal. Leaf size=268 \[ -\frac {b}{112 x^7}+\frac {b^5 \pi ^2}{1152 x^3}-\frac {b \cos \left (b^2 \pi x^2\right )}{112 x^7}+\frac {187 b^5 \pi ^2 \cos \left (b^2 \pi x^2\right )}{40320 x^3}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \text {FresnelC}(b x)}{8 x^8}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \text {FresnelC}(b x)}{192 x^4}+\frac {853 b^8 \pi ^4 \text {FresnelC}\left (\sqrt {2} b x\right )}{40320 \sqrt {2}}+\frac {b^2 \pi \text {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{48 x^6}-\frac {b^6 \pi ^3 \text {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{384 x^2}+\frac {19 b^3 \pi \sin \left (b^2 \pi x^2\right )}{3360 x^5}-\frac {853 b^7 \pi ^3 \sin \left (b^2 \pi x^2\right )}{80640 x}+\frac {1}{384} b^8 \pi ^4 \text {Int}\left (\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \text {FresnelC}(b x)}{x},x\right ) \]

[Out]

-1/112*b/x^7+1/1152*b^5*Pi^2/x^3-1/112*b*cos(b^2*Pi*x^2)/x^7+187/40320*b^5*Pi^2*cos(b^2*Pi*x^2)/x^3-1/8*cos(1/
2*b^2*Pi*x^2)*FresnelC(b*x)/x^8+1/192*b^4*Pi^2*cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/x^4+1/48*b^2*Pi*FresnelC(b*x)
*sin(1/2*b^2*Pi*x^2)/x^6-1/384*b^6*Pi^3*FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/x^2+19/3360*b^3*Pi*sin(b^2*Pi*x^2)/x
^5-853/80640*b^7*Pi^3*sin(b^2*Pi*x^2)/x+853/80640*b^8*Pi^4*FresnelC(b*x*2^(1/2))*2^(1/2)+1/384*b^8*Pi^4*Uninte
grable(cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/x,x)

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Rubi [A]
time = 0.21, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \text {FresnelC}(b x)}{x^9} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

-1/112*b/x^7 + (b^5*Pi^2)/(1152*x^3) - (b*Cos[b^2*Pi*x^2])/(112*x^7) + (187*b^5*Pi^2*Cos[b^2*Pi*x^2])/(40320*x
^3) - (Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(8*x^8) + (b^4*Pi^2*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(192*x^4) + (
853*b^8*Pi^4*FresnelC[Sqrt[2]*b*x])/(40320*Sqrt[2]) + (b^2*Pi*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(48*x^6) - (b
^6*Pi^3*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(384*x^2) + (19*b^3*Pi*Sin[b^2*Pi*x^2])/(3360*x^5) - (853*b^7*Pi^3*
Sin[b^2*Pi*x^2])/(80640*x) + (b^8*Pi^4*Defer[Int][(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x, x])/384

Rubi steps

\begin {align*} \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{x^9} \, dx &=-\frac {b}{112 x^7}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{8 x^8}+\frac {1}{16} b \int \frac {\cos \left (b^2 \pi x^2\right )}{x^8} \, dx-\frac {1}{8} \left (b^2 \pi \right ) \int \frac {C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^7} \, dx\\ &=-\frac {b}{112 x^7}-\frac {b \cos \left (b^2 \pi x^2\right )}{112 x^7}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{8 x^8}+\frac {b^2 \pi C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{48 x^6}-\frac {1}{96} \left (b^3 \pi \right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^6} \, dx-\frac {1}{56} \left (b^3 \pi \right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^6} \, dx-\frac {1}{48} \left (b^4 \pi ^2\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{x^5} \, dx\\ &=-\frac {b}{112 x^7}+\frac {b^5 \pi ^2}{1152 x^3}-\frac {b \cos \left (b^2 \pi x^2\right )}{112 x^7}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{8 x^8}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{192 x^4}+\frac {b^2 \pi C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{48 x^6}+\frac {19 b^3 \pi \sin \left (b^2 \pi x^2\right )}{3360 x^5}-\frac {1}{384} \left (b^5 \pi ^2\right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^4} \, dx-\frac {1}{240} \left (b^5 \pi ^2\right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^4} \, dx-\frac {1}{140} \left (b^5 \pi ^2\right ) \int \frac {\cos \left (b^2 \pi x^2\right )}{x^4} \, dx+\frac {1}{192} \left (b^6 \pi ^3\right ) \int \frac {C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^3} \, dx\\ &=-\frac {b}{112 x^7}+\frac {b^5 \pi ^2}{1152 x^3}-\frac {b \cos \left (b^2 \pi x^2\right )}{112 x^7}+\frac {187 b^5 \pi ^2 \cos \left (b^2 \pi x^2\right )}{40320 x^3}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{8 x^8}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{192 x^4}+\frac {b^2 \pi C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{48 x^6}-\frac {b^6 \pi ^3 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{384 x^2}+\frac {19 b^3 \pi \sin \left (b^2 \pi x^2\right )}{3360 x^5}+\frac {1}{768} \left (b^7 \pi ^3\right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2} \, dx+\frac {1}{576} \left (b^7 \pi ^3\right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2} \, dx+\frac {1}{360} \left (b^7 \pi ^3\right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2} \, dx+\frac {1}{210} \left (b^7 \pi ^3\right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^2} \, dx+\frac {1}{384} \left (b^8 \pi ^4\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{x} \, dx\\ &=-\frac {b}{112 x^7}+\frac {b^5 \pi ^2}{1152 x^3}-\frac {b \cos \left (b^2 \pi x^2\right )}{112 x^7}+\frac {187 b^5 \pi ^2 \cos \left (b^2 \pi x^2\right )}{40320 x^3}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{8 x^8}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{192 x^4}+\frac {b^2 \pi C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{48 x^6}-\frac {b^6 \pi ^3 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{384 x^2}+\frac {19 b^3 \pi \sin \left (b^2 \pi x^2\right )}{3360 x^5}-\frac {853 b^7 \pi ^3 \sin \left (b^2 \pi x^2\right )}{80640 x}+\frac {1}{384} \left (b^8 \pi ^4\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{x} \, dx+\frac {1}{384} \left (b^9 \pi ^4\right ) \int \cos \left (b^2 \pi x^2\right ) \, dx+\frac {1}{288} \left (b^9 \pi ^4\right ) \int \cos \left (b^2 \pi x^2\right ) \, dx+\frac {1}{180} \left (b^9 \pi ^4\right ) \int \cos \left (b^2 \pi x^2\right ) \, dx+\frac {1}{105} \left (b^9 \pi ^4\right ) \int \cos \left (b^2 \pi x^2\right ) \, dx\\ &=-\frac {b}{112 x^7}+\frac {b^5 \pi ^2}{1152 x^3}-\frac {b \cos \left (b^2 \pi x^2\right )}{112 x^7}+\frac {187 b^5 \pi ^2 \cos \left (b^2 \pi x^2\right )}{40320 x^3}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{8 x^8}+\frac {b^4 \pi ^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{192 x^4}+\frac {853 b^8 \pi ^4 C\left (\sqrt {2} b x\right )}{40320 \sqrt {2}}+\frac {b^2 \pi C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{48 x^6}-\frac {b^6 \pi ^3 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{384 x^2}+\frac {19 b^3 \pi \sin \left (b^2 \pi x^2\right )}{3360 x^5}-\frac {853 b^7 \pi ^3 \sin \left (b^2 \pi x^2\right )}{80640 x}+\frac {1}{384} \left (b^8 \pi ^4\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{x} \, dx\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \text {FresnelC}(b x)}{x^9} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.23, size = 0, normalized size = 0.00 \[\int \frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \FresnelC \left (b x \right )}{x^{9}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )}{x^{9}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\mathrm {FresnelC}\left (b\,x\right )\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right )}{x^9} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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