3.154 \(\int \frac{\text{FresnelC}(b x)^2}{x^7} \, dx\)

Optimal. Leaf size=165 \[ -\frac{1}{45} \pi ^2 b^5 \text{Unintegrable}\left (\frac{\text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{x^2},x\right )-\frac{1}{72} \pi ^2 b^6 \text{CosIntegral}\left (\pi b^2 x^2\right )+\frac{\pi b^3 \text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{45 x^3}-\frac{b \text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{15 x^5}-\frac{b^2}{120 x^4}+\frac{\pi b^4 \sin \left (\pi b^2 x^2\right )}{72 x^2}-\frac{b^2 \cos \left (\pi b^2 x^2\right )}{120 x^4}-\frac{\text{FresnelC}(b x)^2}{6 x^6} \]

[Out]

-b^2/(120*x^4) - (b^2*Cos[b^2*Pi*x^2])/(120*x^4) - (b^6*Pi^2*CosIntegral[b^2*Pi*x^2])/72 - (b*Cos[(b^2*Pi*x^2)
/2]*FresnelC[b*x])/(15*x^5) - FresnelC[b*x]^2/(6*x^6) + (b^3*Pi*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(45*x^3) +
(b^4*Pi*Sin[b^2*Pi*x^2])/(72*x^2) - (b^5*Pi^2*Unintegrable[(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x^2, x])/45

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Rubi [A]  time = 0.220258, 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 = {} \[ \int \frac{\text{FresnelC}(b x)^2}{x^7} \, dx \]

Verification is Not applicable to the result.

[In]

Int[FresnelC[b*x]^2/x^7,x]

[Out]

-b^2/(120*x^4) - (b^2*Cos[b^2*Pi*x^2])/(120*x^4) - (b^6*Pi^2*CosIntegral[b^2*Pi*x^2])/72 - (b*Cos[(b^2*Pi*x^2)
/2]*FresnelC[b*x])/(15*x^5) - FresnelC[b*x]^2/(6*x^6) + (b^3*Pi*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(45*x^3) +
(b^4*Pi*Sin[b^2*Pi*x^2])/(72*x^2) - (b^5*Pi^2*Defer[Int][(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x^2, x])/45

Rubi steps

\begin{align*} \int \frac{C(b x)^2}{x^7} \, dx &=-\frac{C(b x)^2}{6 x^6}+\frac{1}{3} b \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{x^6} \, dx\\ &=-\frac{b^2}{120 x^4}-\frac{b \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{15 x^5}-\frac{C(b x)^2}{6 x^6}+\frac{1}{30} b^2 \int \frac{\cos \left (b^2 \pi x^2\right )}{x^5} \, dx-\frac{1}{15} \left (b^3 \pi \right ) \int \frac{C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^4} \, dx\\ &=-\frac{b^2}{120 x^4}-\frac{b \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{15 x^5}-\frac{C(b x)^2}{6 x^6}+\frac{b^3 \pi C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{45 x^3}+\frac{1}{60} b^2 \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )-\frac{1}{90} \left (b^4 \pi \right ) \int \frac{\sin \left (b^2 \pi x^2\right )}{x^3} \, dx-\frac{1}{45} \left (b^5 \pi ^2\right ) \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{x^2} \, dx\\ &=-\frac{b^2}{120 x^4}-\frac{b^2 \cos \left (b^2 \pi x^2\right )}{120 x^4}-\frac{b \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{15 x^5}-\frac{C(b x)^2}{6 x^6}+\frac{b^3 \pi C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{45 x^3}-\frac{1}{180} \left (b^4 \pi \right ) \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )-\frac{1}{120} \left (b^4 \pi \right ) \operatorname{Subst}\left (\int \frac{\sin \left (b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )-\frac{1}{45} \left (b^5 \pi ^2\right ) \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{x^2} \, dx\\ &=-\frac{b^2}{120 x^4}-\frac{b^2 \cos \left (b^2 \pi x^2\right )}{120 x^4}-\frac{b \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{15 x^5}-\frac{C(b x)^2}{6 x^6}+\frac{b^3 \pi C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{45 x^3}+\frac{b^4 \pi \sin \left (b^2 \pi x^2\right )}{72 x^2}-\frac{1}{45} \left (b^5 \pi ^2\right ) \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{x^2} \, dx-\frac{1}{180} \left (b^6 \pi ^2\right ) \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )-\frac{1}{120} \left (b^6 \pi ^2\right ) \operatorname{Subst}\left (\int \frac{\cos \left (b^2 \pi x\right )}{x} \, dx,x,x^2\right )\\ &=-\frac{b^2}{120 x^4}-\frac{b^2 \cos \left (b^2 \pi x^2\right )}{120 x^4}-\frac{1}{72} b^6 \pi ^2 \text{Ci}\left (b^2 \pi x^2\right )-\frac{b \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{15 x^5}-\frac{C(b x)^2}{6 x^6}+\frac{b^3 \pi C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{45 x^3}+\frac{b^4 \pi \sin \left (b^2 \pi x^2\right )}{72 x^2}-\frac{1}{45} \left (b^5 \pi ^2\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 = 0.0201974, size = 0, normalized size = 0. \[ \int \frac{\text{FresnelC}(b x)^2}{x^7} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[FresnelC[b*x]^2/x^7,x]

[Out]

Integrate[FresnelC[b*x]^2/x^7, x]

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Maple [A]  time = 0.053, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\it FresnelC} \left ( bx \right ) \right ) ^{2}}{{x}^{7}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(FresnelC(b*x)^2/x^7,x)

[Out]

int(FresnelC(b*x)^2/x^7,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm fresnelc}\left (b x\right )^{2}}{x^{7}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnelc(b*x)^2/x^7,x, algorithm="maxima")

[Out]

integrate(fresnelc(b*x)^2/x^7, x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\rm fresnelc}\left (b x\right )^{2}}{x^{7}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnelc(b*x)^2/x^7,x, algorithm="fricas")

[Out]

integral(fresnelc(b*x)^2/x^7, x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{C^{2}\left (b x\right )}{x^{7}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnelc(b*x)**2/x**7,x)

[Out]

Integral(fresnelc(b*x)**2/x**7, x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm fresnelc}\left (b x\right )^{2}}{x^{7}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnelc(b*x)^2/x^7,x, algorithm="giac")

[Out]

integrate(fresnelc(b*x)^2/x^7, x)