Optimal. Leaf size=285 \[ \frac{\pi ^4 b^9 \text{Unintegrable}\left (\frac{\text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{x},x\right )}{1728}-\frac{\pi ^3 b^7 \text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{1728 x^2}+\frac{\pi b^3 \text{FresnelC}(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{216 x^6}+\frac{\pi ^2 b^5 \text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{864 x^4}-\frac{b \text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{36 x^8}+\frac{853 \pi ^4 b^9 \text{FresnelC}\left (\sqrt{2} b x\right )}{181440 \sqrt{2}}+\frac{\pi ^2 b^6}{5184 x^3}-\frac{b^2}{504 x^7}-\frac{853 \pi ^3 b^8 \sin \left (\pi b^2 x^2\right )}{362880 x}+\frac{19 \pi b^4 \sin \left (\pi b^2 x^2\right )}{15120 x^5}+\frac{187 \pi ^2 b^6 \cos \left (\pi b^2 x^2\right )}{181440 x^3}-\frac{b^2 \cos \left (\pi b^2 x^2\right )}{504 x^7}-\frac{\text{FresnelC}(b x)^2}{9 x^9} \]
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Rubi [A] time = 0.342115, 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 = {} \[ \int \frac{\text{FresnelC}(b x)^2}{x^{10}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{C(b x)^2}{x^{10}} \, dx &=-\frac{C(b x)^2}{9 x^9}+\frac{1}{9} (2 b) \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{x^9} \, dx\\ &=-\frac{b^2}{504 x^7}-\frac{b \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{36 x^8}-\frac{C(b x)^2}{9 x^9}+\frac{1}{72} b^2 \int \frac{\cos \left (b^2 \pi x^2\right )}{x^8} \, dx-\frac{1}{36} \left (b^3 \pi \right ) \int \frac{C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^7} \, dx\\ &=-\frac{b^2}{504 x^7}-\frac{b^2 \cos \left (b^2 \pi x^2\right )}{504 x^7}-\frac{b \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{36 x^8}-\frac{C(b x)^2}{9 x^9}+\frac{b^3 \pi C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{216 x^6}-\frac{1}{432} \left (b^4 \pi \right ) \int \frac{\sin \left (b^2 \pi x^2\right )}{x^6} \, dx-\frac{1}{252} \left (b^4 \pi \right ) \int \frac{\sin \left (b^2 \pi x^2\right )}{x^6} \, dx-\frac{1}{216} \left (b^5 \pi ^2\right ) \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{x^5} \, dx\\ &=-\frac{b^2}{504 x^7}+\frac{b^6 \pi ^2}{5184 x^3}-\frac{b^2 \cos \left (b^2 \pi x^2\right )}{504 x^7}-\frac{b \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{36 x^8}+\frac{b^5 \pi ^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{864 x^4}-\frac{C(b x)^2}{9 x^9}+\frac{b^3 \pi C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{216 x^6}+\frac{19 b^4 \pi \sin \left (b^2 \pi x^2\right )}{15120 x^5}-\frac{\left (b^6 \pi ^2\right ) \int \frac{\cos \left (b^2 \pi x^2\right )}{x^4} \, dx}{1728}-\frac{\left (b^6 \pi ^2\right ) \int \frac{\cos \left (b^2 \pi x^2\right )}{x^4} \, dx}{1080}-\frac{1}{630} \left (b^6 \pi ^2\right ) \int \frac{\cos \left (b^2 \pi x^2\right )}{x^4} \, dx+\frac{1}{864} \left (b^7 \pi ^3\right ) \int \frac{C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^3} \, dx\\ &=-\frac{b^2}{504 x^7}+\frac{b^6 \pi ^2}{5184 x^3}-\frac{b^2 \cos \left (b^2 \pi x^2\right )}{504 x^7}+\frac{187 b^6 \pi ^2 \cos \left (b^2 \pi x^2\right )}{181440 x^3}-\frac{b \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{36 x^8}+\frac{b^5 \pi ^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{864 x^4}-\frac{C(b x)^2}{9 x^9}+\frac{b^3 \pi C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{216 x^6}-\frac{b^7 \pi ^3 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{1728 x^2}+\frac{19 b^4 \pi \sin \left (b^2 \pi x^2\right )}{15120 x^5}+\frac{\left (b^8 \pi ^3\right ) \int \frac{\sin \left (b^2 \pi x^2\right )}{x^2} \, dx}{3456}+\frac{\left (b^8 \pi ^3\right ) \int \frac{\sin \left (b^2 \pi x^2\right )}{x^2} \, dx}{2592}+\frac{\left (b^8 \pi ^3\right ) \int \frac{\sin \left (b^2 \pi x^2\right )}{x^2} \, dx}{1620}+\frac{1}{945} \left (b^8 \pi ^3\right ) \int \frac{\sin \left (b^2 \pi x^2\right )}{x^2} \, dx+\frac{\left (b^9 \pi ^4\right ) \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{x} \, dx}{1728}\\ &=-\frac{b^2}{504 x^7}+\frac{b^6 \pi ^2}{5184 x^3}-\frac{b^2 \cos \left (b^2 \pi x^2\right )}{504 x^7}+\frac{187 b^6 \pi ^2 \cos \left (b^2 \pi x^2\right )}{181440 x^3}-\frac{b \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{36 x^8}+\frac{b^5 \pi ^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{864 x^4}-\frac{C(b x)^2}{9 x^9}+\frac{b^3 \pi C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{216 x^6}-\frac{b^7 \pi ^3 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{1728 x^2}+\frac{19 b^4 \pi \sin \left (b^2 \pi x^2\right )}{15120 x^5}-\frac{853 b^8 \pi ^3 \sin \left (b^2 \pi x^2\right )}{362880 x}+\frac{\left (b^9 \pi ^4\right ) \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{x} \, dx}{1728}+\frac{\left (b^{10} \pi ^4\right ) \int \cos \left (b^2 \pi x^2\right ) \, dx}{1728}+\frac{\left (b^{10} \pi ^4\right ) \int \cos \left (b^2 \pi x^2\right ) \, dx}{1296}+\frac{1}{810} \left (b^{10} \pi ^4\right ) \int \cos \left (b^2 \pi x^2\right ) \, dx+\frac{1}{945} \left (2 b^{10} \pi ^4\right ) \int \cos \left (b^2 \pi x^2\right ) \, dx\\ &=-\frac{b^2}{504 x^7}+\frac{b^6 \pi ^2}{5184 x^3}-\frac{b^2 \cos \left (b^2 \pi x^2\right )}{504 x^7}+\frac{187 b^6 \pi ^2 \cos \left (b^2 \pi x^2\right )}{181440 x^3}-\frac{b \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{36 x^8}+\frac{b^5 \pi ^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{864 x^4}-\frac{C(b x)^2}{9 x^9}+\frac{67 b^9 \pi ^4 C\left (\sqrt{2} b x\right )}{25920 \sqrt{2}}+\frac{1}{945} \sqrt{2} b^9 \pi ^4 C\left (\sqrt{2} b x\right )+\frac{b^3 \pi C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{216 x^6}-\frac{b^7 \pi ^3 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{1728 x^2}+\frac{19 b^4 \pi \sin \left (b^2 \pi x^2\right )}{15120 x^5}-\frac{853 b^8 \pi ^3 \sin \left (b^2 \pi x^2\right )}{362880 x}+\frac{\left (b^9 \pi ^4\right ) \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{x} \, dx}{1728}\\ \end{align*}
Mathematica [A] time = 0.0265919, size = 0, normalized size = 0. \[ \int \frac{\text{FresnelC}(b x)^2}{x^{10}} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.052, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\it FresnelC} \left ( bx \right ) \right ) ^{2}}{{x}^{10}}}\, 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^{10}}\,{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^{10}}, 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^{10}}\, 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^{10}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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