3.193 \(\int \frac{\cos (\frac{1}{2} b^2 \pi x^2) \text{FresnelC}(b x)}{x^5} \, dx\)

Optimal. Leaf size=152 \[ -\frac{1}{8} \pi ^2 b^4 \text{Unintegrable}\left (\frac{\text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{x},x\right )+\frac{\pi b^2 \text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{8 x^2}-\frac{\text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{4 x^4}-\frac{7 \pi ^2 b^4 \text{FresnelC}\left (\sqrt{2} b x\right )}{24 \sqrt{2}}+\frac{7 \pi b^3 \sin \left (\pi b^2 x^2\right )}{48 x}-\frac{b \cos \left (\pi b^2 x^2\right )}{24 x^3}-\frac{b}{24 x^3} \]

[Out]

-b/(24*x^3) - (b*Cos[b^2*Pi*x^2])/(24*x^3) - (Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(4*x^4) - (7*b^4*Pi^2*Fresnel
C[Sqrt[2]*b*x])/(24*Sqrt[2]) + (b^2*Pi*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(8*x^2) + (7*b^3*Pi*Sin[b^2*Pi*x^2])
/(48*x) - (b^4*Pi^2*Unintegrable[(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x, x])/8

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Rubi [A]  time = 0.115349, 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 = {} \[ \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) \text{FresnelC}(b x)}{x^5} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x^5,x]

[Out]

-b/(24*x^3) - (b*Cos[b^2*Pi*x^2])/(24*x^3) - (Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(4*x^4) - (7*b^4*Pi^2*Fresnel
C[Sqrt[2]*b*x])/(24*Sqrt[2]) + (b^2*Pi*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(8*x^2) + (7*b^3*Pi*Sin[b^2*Pi*x^2])
/(48*x) - (b^4*Pi^2*Defer[Int][(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x, x])/8

Rubi steps

\begin{align*} \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{x^5} \, dx &=-\frac{b}{24 x^3}-\frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{4 x^4}+\frac{1}{8} b \int \frac{\cos \left (b^2 \pi x^2\right )}{x^4} \, dx-\frac{1}{4} \left (b^2 \pi \right ) \int \frac{C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^3} \, dx\\ &=-\frac{b}{24 x^3}-\frac{b \cos \left (b^2 \pi x^2\right )}{24 x^3}-\frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{4 x^4}+\frac{b^2 \pi C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{8 x^2}-\frac{1}{16} \left (b^3 \pi \right ) \int \frac{\sin \left (b^2 \pi x^2\right )}{x^2} \, dx-\frac{1}{12} \left (b^3 \pi \right ) \int \frac{\sin \left (b^2 \pi x^2\right )}{x^2} \, dx-\frac{1}{8} \left (b^4 \pi ^2\right ) \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{x} \, dx\\ &=-\frac{b}{24 x^3}-\frac{b \cos \left (b^2 \pi x^2\right )}{24 x^3}-\frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{4 x^4}+\frac{b^2 \pi C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{8 x^2}+\frac{7 b^3 \pi \sin \left (b^2 \pi x^2\right )}{48 x}-\frac{1}{8} \left (b^4 \pi ^2\right ) \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{x} \, dx-\frac{1}{8} \left (b^5 \pi ^2\right ) \int \cos \left (b^2 \pi x^2\right ) \, dx-\frac{1}{6} \left (b^5 \pi ^2\right ) \int \cos \left (b^2 \pi x^2\right ) \, dx\\ &=-\frac{b}{24 x^3}-\frac{b \cos \left (b^2 \pi x^2\right )}{24 x^3}-\frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{4 x^4}-\frac{7 b^4 \pi ^2 C\left (\sqrt{2} b x\right )}{24 \sqrt{2}}+\frac{b^2 \pi C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{8 x^2}+\frac{7 b^3 \pi \sin \left (b^2 \pi x^2\right )}{48 x}-\frac{1}{8} \left (b^4 \pi ^2\right ) \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{x} \, dx\\ \end{align*}

Mathematica [A]  time = 0.0300349, size = 0, normalized size = 0. \[ \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) \text{FresnelC}(b x)}{x^5} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x^5,x]

[Out]

Integrate[(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x^5, x]

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Maple [A]  time = 0.065, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it FresnelC} \left ( bx \right ) }{{x}^{5}}\cos \left ({\frac{{b}^{2}\pi \,{x}^{2}}{2}} \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/x^5,x)

[Out]

int(cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/x^5,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (\frac{1}{2} \, \pi b^{2} x^{2}\right ){\rm fresnelc}\left (b x\right )}{x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(1/2*b^2*pi*x^2)*fresnelc(b*x)/x^5,x, algorithm="maxima")

[Out]

integrate(cos(1/2*pi*b^2*x^2)*fresnelc(b*x)/x^5, x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\cos \left (\frac{1}{2} \, \pi b^{2} x^{2}\right ){\rm fresnelc}\left (b x\right )}{x^{5}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(1/2*b^2*pi*x^2)*fresnelc(b*x)/x^5,x, algorithm="fricas")

[Out]

integral(cos(1/2*pi*b^2*x^2)*fresnelc(b*x)/x^5, x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos{\left (\frac{\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )}{x^{5}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(1/2*b**2*pi*x**2)*fresnelc(b*x)/x**5,x)

[Out]

Integral(cos(pi*b**2*x**2/2)*fresnelc(b*x)/x**5, x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (\frac{1}{2} \, \pi b^{2} x^{2}\right ){\rm fresnelc}\left (b x\right )}{x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(1/2*b^2*pi*x^2)*fresnelc(b*x)/x^5,x, algorithm="giac")

[Out]

integrate(cos(1/2*pi*b^2*x^2)*fresnelc(b*x)/x^5, x)