3.3.16 \(\int \frac {\text {FresnelC}(b x) \sin (\frac {1}{2} b^2 \pi x^2)}{x^8} \, dx\) [216]

Optimal. Leaf size=202 \[ -\frac {b^3 \pi }{280 x^4}-\frac {b^3 \pi \cos \left (b^2 \pi x^2\right )}{105 x^4}-\frac {1}{84} b^7 \pi ^3 \text {CosIntegral}\left (b^2 \pi x^2\right )-\frac {b^2 \pi \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \text {FresnelC}(b x)}{35 x^5}-\frac {\text {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 x^7}+\frac {b^4 \pi ^2 \text {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{105 x^3}-\frac {b \sin \left (b^2 \pi x^2\right )}{84 x^6}+\frac {b^5 \pi ^2 \sin \left (b^2 \pi x^2\right )}{84 x^2}-\frac {1}{105} b^6 \pi ^3 \text {Int}\left (\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \text {FresnelC}(b x)}{x^2},x\right ) \]

[Out]

-1/280*b^3*Pi/x^4-1/84*b^7*Pi^3*Ci(b^2*Pi*x^2)-1/105*b^3*Pi*cos(b^2*Pi*x^2)/x^4-1/35*b^2*Pi*cos(1/2*b^2*Pi*x^2
)*FresnelC(b*x)/x^5-1/7*FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/x^7+1/105*b^4*Pi^2*FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)
/x^3-1/84*b*sin(b^2*Pi*x^2)/x^6+1/84*b^5*Pi^2*sin(b^2*Pi*x^2)/x^2-1/105*b^6*Pi^3*Unintegrable(cos(1/2*b^2*Pi*x
^2)*FresnelC(b*x)/x^2,x)

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Rubi [A]
time = 0.22, 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 {\text {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

-1/280*(b^3*Pi)/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 \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 0.02, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\text {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^8} \, dx \end {gather*}

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]

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Maple [A]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\FresnelC \left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{x^{8}}\, dx\]

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)

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

[Out]

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

[Out]

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

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )}{x^{8}}\, dx \end {gather*}

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]

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

[Out]

integrate(fresnel_cos(b*x)*sin(1/2*pi*b^2*x^2)/x^8, 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 )\,\sin \left (\frac {\Pi \,b^2\,x^2}{2}\right )}{x^8} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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