Optimal. Leaf size=258 \[ \frac{1}{168} \pi ^3 b^7 \text{Unintegrable}\left (\frac{\text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{x},x\right )+\frac{\pi b^3 \text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{84 x^4}+\frac{\pi ^2 b^5 \text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{168 x^2}-\frac{b \text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{21 x^6}+\frac{2}{315} \sqrt{2} \pi ^3 b^7 S\left (\sqrt{2} b x\right )+\frac{\pi ^3 b^7 S\left (\sqrt{2} b x\right )}{72 \sqrt{2}}-\frac{b^2}{210 x^5}+\frac{13 \pi b^4 \sin \left (\pi b^2 x^2\right )}{2520 x^3}+\frac{67 \pi ^2 b^6 \cos \left (\pi b^2 x^2\right )}{5040 x}-\frac{b^2 \cos \left (\pi b^2 x^2\right )}{210 x^5}+\frac{\pi ^2 b^6}{336 x}-\frac{\text{FresnelC}(b x)^2}{7 x^7} \]
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Rubi [A] time = 0.239743, 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 = {} \[ \int \frac{\text{FresnelC}(b x)^2}{x^8} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{C(b x)^2}{x^8} \, dx &=-\frac{C(b x)^2}{7 x^7}+\frac{1}{7} (2 b) \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{x^7} \, dx\\ &=-\frac{b^2}{210 x^5}-\frac{b \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{21 x^6}-\frac{C(b x)^2}{7 x^7}+\frac{1}{42} b^2 \int \frac{\cos \left (b^2 \pi x^2\right )}{x^6} \, dx-\frac{1}{21} \left (b^3 \pi \right ) \int \frac{C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^5} \, dx\\ &=-\frac{b^2}{210 x^5}-\frac{b^2 \cos \left (b^2 \pi x^2\right )}{210 x^5}-\frac{b \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{21 x^6}-\frac{C(b x)^2}{7 x^7}+\frac{b^3 \pi C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{84 x^4}-\frac{1}{168} \left (b^4 \pi \right ) \int \frac{\sin \left (b^2 \pi x^2\right )}{x^4} \, dx-\frac{1}{105} \left (b^4 \pi \right ) \int \frac{\sin \left (b^2 \pi x^2\right )}{x^4} \, dx-\frac{1}{84} \left (b^5 \pi ^2\right ) \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{x^3} \, dx\\ &=-\frac{b^2}{210 x^5}+\frac{b^6 \pi ^2}{336 x}-\frac{b^2 \cos \left (b^2 \pi x^2\right )}{210 x^5}-\frac{b \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{21 x^6}+\frac{b^5 \pi ^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{168 x^2}-\frac{C(b x)^2}{7 x^7}+\frac{b^3 \pi C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{84 x^4}+\frac{13 b^4 \pi \sin \left (b^2 \pi x^2\right )}{2520 x^3}-\frac{1}{336} \left (b^6 \pi ^2\right ) \int \frac{\cos \left (b^2 \pi x^2\right )}{x^2} \, dx-\frac{1}{252} \left (b^6 \pi ^2\right ) \int \frac{\cos \left (b^2 \pi x^2\right )}{x^2} \, dx-\frac{1}{315} \left (2 b^6 \pi ^2\right ) \int \frac{\cos \left (b^2 \pi x^2\right )}{x^2} \, dx+\frac{1}{168} \left (b^7 \pi ^3\right ) \int \frac{C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x} \, dx\\ &=-\frac{b^2}{210 x^5}+\frac{b^6 \pi ^2}{336 x}-\frac{b^2 \cos \left (b^2 \pi x^2\right )}{210 x^5}+\frac{67 b^6 \pi ^2 \cos \left (b^2 \pi x^2\right )}{5040 x}-\frac{b \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{21 x^6}+\frac{b^5 \pi ^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{168 x^2}-\frac{C(b x)^2}{7 x^7}+\frac{b^3 \pi C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{84 x^4}+\frac{13 b^4 \pi \sin \left (b^2 \pi x^2\right )}{2520 x^3}+\frac{1}{168} \left (b^7 \pi ^3\right ) \int \frac{C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x} \, dx+\frac{1}{168} \left (b^8 \pi ^3\right ) \int \sin \left (b^2 \pi x^2\right ) \, dx+\frac{1}{126} \left (b^8 \pi ^3\right ) \int \sin \left (b^2 \pi x^2\right ) \, dx+\frac{1}{315} \left (4 b^8 \pi ^3\right ) \int \sin \left (b^2 \pi x^2\right ) \, dx\\ &=-\frac{b^2}{210 x^5}+\frac{b^6 \pi ^2}{336 x}-\frac{b^2 \cos \left (b^2 \pi x^2\right )}{210 x^5}+\frac{67 b^6 \pi ^2 \cos \left (b^2 \pi x^2\right )}{5040 x}-\frac{b \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{21 x^6}+\frac{b^5 \pi ^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{168 x^2}-\frac{C(b x)^2}{7 x^7}+\frac{b^7 \pi ^3 S\left (\sqrt{2} b x\right )}{72 \sqrt{2}}+\frac{2}{315} \sqrt{2} b^7 \pi ^3 S\left (\sqrt{2} b x\right )+\frac{b^3 \pi C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{84 x^4}+\frac{13 b^4 \pi \sin \left (b^2 \pi x^2\right )}{2520 x^3}+\frac{1}{168} \left (b^7 \pi ^3\right ) \int \frac{C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x} \, dx\\ \end{align*}
Mathematica [A] time = 0.0270006, size = 0, normalized size = 0. \[ \int \frac{\text{FresnelC}(b x)^2}{x^8} \, dx \]
Verification is Not applicable to the result.
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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}^{8}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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^{8}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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^{8}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{C^{2}\left (b x\right )}{x^{8}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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^{8}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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