3.2.94 \(\int \frac {\cos (\frac {1}{2} b^2 \pi x^2) \text {FresnelC}(b x)}{x^6} \, dx\) [194]

Optimal. Leaf size=148 \[ -\frac {b}{40 x^4}-\frac {b \cos \left (b^2 \pi x^2\right )}{40 x^4}-\frac {1}{24} b^5 \pi ^2 \text {CosIntegral}\left (b^2 \pi x^2\right )-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \text {FresnelC}(b x)}{5 x^5}+\frac {b^2 \pi \text {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{15 x^3}+\frac {b^3 \pi \sin \left (b^2 \pi x^2\right )}{24 x^2}-\frac {1}{15} b^4 \pi ^2 \text {Int}\left (\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \text {FresnelC}(b x)}{x^2},x\right ) \]

[Out]

-1/40*b/x^4-1/24*b^5*Pi^2*Ci(b^2*Pi*x^2)-1/40*b*cos(b^2*Pi*x^2)/x^4-1/5*cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/x^5+
1/15*b^2*Pi*FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/x^3+1/24*b^3*Pi*sin(b^2*Pi*x^2)/x^2-1/15*b^4*Pi^2*Unintegrable(c
os(1/2*b^2*Pi*x^2)*FresnelC(b*x)/x^2,x)

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Rubi [A]
time = 0.13, 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^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x^6,x]

[Out]

-1/40*b/x^4 - (b*Cos[b^2*Pi*x^2])/(40*x^4) - (b^5*Pi^2*CosIntegral[b^2*Pi*x^2])/24 - (Cos[(b^2*Pi*x^2)/2]*Fres
nelC[b*x])/(5*x^5) + (b^2*Pi*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(15*x^3) + (b^3*Pi*Sin[b^2*Pi*x^2])/(24*x^2) -
(b^4*Pi^2*Defer[Int][(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x^2, x])/15

Rubi steps

\begin {align*} \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{x^6} \, dx &=-\frac {b}{40 x^4}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{5 x^5}+\frac {1}{10} b \int \frac {\cos \left (b^2 \pi x^2\right )}{x^5} \, dx-\frac {1}{5} \left (b^2 \pi \right ) \int \frac {C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{x^4} \, dx\\ &=-\frac {b}{40 x^4}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{5 x^5}+\frac {b^2 \pi C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{15 x^3}+\frac {1}{20} b \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )-\frac {1}{30} \left (b^3 \pi \right ) \int \frac {\sin \left (b^2 \pi x^2\right )}{x^3} \, dx-\frac {1}{15} \left (b^4 \pi ^2\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{x^2} \, dx\\ &=-\frac {b}{40 x^4}-\frac {b \cos \left (b^2 \pi x^2\right )}{40 x^4}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{5 x^5}+\frac {b^2 \pi C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{15 x^3}-\frac {1}{60} \left (b^3 \pi \right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )-\frac {1}{40} \left (b^3 \pi \right ) \text {Subst}\left (\int \frac {\sin \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )-\frac {1}{15} \left (b^4 \pi ^2\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{x^2} \, dx\\ &=-\frac {b}{40 x^4}-\frac {b \cos \left (b^2 \pi x^2\right )}{40 x^4}-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{5 x^5}+\frac {b^2 \pi C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{15 x^3}+\frac {b^3 \pi \sin \left (b^2 \pi x^2\right )}{24 x^2}-\frac {1}{15} \left (b^4 \pi ^2\right ) \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{x^2} \, dx-\frac {1}{60} \left (b^5 \pi ^2\right ) \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )-\frac {1}{40} \left (b^5 \pi ^2\right ) \text {Subst}\left (\int \frac {\cos \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )\\ &=-\frac {b}{40 x^4}-\frac {b \cos \left (b^2 \pi x^2\right )}{40 x^4}-\frac {1}{24} b^5 \pi ^2 \text {Ci}\left (b^2 \pi x^2\right )-\frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{5 x^5}+\frac {b^2 \pi C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{15 x^3}+\frac {b^3 \pi \sin \left (b^2 \pi x^2\right )}{24 x^2}-\frac {1}{15} \left (b^4 \pi ^2\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 {\cos \left (\frac {1}{2} b^2 \pi x^2\right ) \text {FresnelC}(b x)}{x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x^6,x]

[Out]

Integrate[(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x^6, x]

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Maple [A]
time = 0.25, size = 0, normalized size = 0.00 \[\int \frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) \FresnelC \left (b x \right )}{x^{6}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/x^6,x)

[Out]

int(cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/x^6,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^6,x, algorithm="maxima")

[Out]

integrate(cos(1/2*pi*b^2*x^2)*fresnel_cos(b*x)/x^6, 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^6,x, algorithm="fricas")

[Out]

integral(cos(1/2*pi*b^2*x^2)*fresnel_cos(b*x)/x^6, 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^{6}}\, 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**6,x)

[Out]

Integral(cos(pi*b**2*x**2/2)*fresnelc(b*x)/x**6, 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^6,x, algorithm="giac")

[Out]

integrate(cos(1/2*pi*b^2*x^2)*fresnel_cos(b*x)/x^6, x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\mathrm {FresnelC}\left (b\,x\right )\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right )}{x^6} \,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^6,x)

[Out]

int((FresnelC(b*x)*cos((Pi*b^2*x^2)/2))/x^6, x)

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