Optimal. Leaf size=119 \[ \frac{1}{840} \pi ^4 b^8 \text{FresnelC}(b x)-\frac{\pi ^3 b^7 \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{840 x}+\frac{\pi b^3 \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{280 x^5}+\frac{\pi ^2 b^5 \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{840 x^3}-\frac{b \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{56 x^7}-\frac{\text{FresnelC}(b x)}{8 x^8} \]
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Rubi [A] time = 0.0784687, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6427, 3388, 3387, 3352} \[ \frac{1}{840} \pi ^4 b^8 \text{FresnelC}(b x)-\frac{\pi ^3 b^7 \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{840 x}+\frac{\pi b^3 \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{280 x^5}+\frac{\pi ^2 b^5 \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{840 x^3}-\frac{b \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{56 x^7}-\frac{\text{FresnelC}(b x)}{8 x^8} \]
Antiderivative was successfully verified.
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Rule 6427
Rule 3388
Rule 3387
Rule 3352
Rubi steps
\begin{align*} \int \frac{C(b x)}{x^9} \, dx &=-\frac{C(b x)}{8 x^8}+\frac{1}{8} b \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right )}{x^8} \, dx\\ &=-\frac{b \cos \left (\frac{1}{2} b^2 \pi x^2\right )}{56 x^7}-\frac{C(b x)}{8 x^8}-\frac{1}{56} \left (b^3 \pi \right ) \int \frac{\sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^6} \, dx\\ &=-\frac{b \cos \left (\frac{1}{2} b^2 \pi x^2\right )}{56 x^7}-\frac{C(b x)}{8 x^8}+\frac{b^3 \pi \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{280 x^5}-\frac{1}{280} \left (b^5 \pi ^2\right ) \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right )}{x^4} \, dx\\ &=-\frac{b \cos \left (\frac{1}{2} b^2 \pi x^2\right )}{56 x^7}+\frac{b^5 \pi ^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right )}{840 x^3}-\frac{C(b x)}{8 x^8}+\frac{b^3 \pi \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{280 x^5}+\frac{1}{840} \left (b^7 \pi ^3\right ) \int \frac{\sin \left (\frac{1}{2} b^2 \pi x^2\right )}{x^2} \, dx\\ &=-\frac{b \cos \left (\frac{1}{2} b^2 \pi x^2\right )}{56 x^7}+\frac{b^5 \pi ^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right )}{840 x^3}-\frac{C(b x)}{8 x^8}+\frac{b^3 \pi \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{280 x^5}-\frac{b^7 \pi ^3 \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{840 x}+\frac{1}{840} \left (b^9 \pi ^4\right ) \int \cos \left (\frac{1}{2} b^2 \pi x^2\right ) \, dx\\ &=-\frac{b \cos \left (\frac{1}{2} b^2 \pi x^2\right )}{56 x^7}+\frac{b^5 \pi ^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right )}{840 x^3}+\frac{1}{840} b^8 \pi ^4 C(b x)-\frac{C(b x)}{8 x^8}+\frac{b^3 \pi \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{280 x^5}-\frac{b^7 \pi ^3 \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{840 x}\\ \end{align*}
Mathematica [A] time = 0.0577638, size = 85, normalized size = 0.71 \[ \frac{\left (\pi ^4 b^8 x^8-105\right ) \text{FresnelC}(b x)+\pi b^3 x^3 \left (3-\pi ^2 b^4 x^4\right ) \sin \left (\frac{1}{2} \pi b^2 x^2\right )+b x \left (\pi ^2 b^4 x^4-15\right ) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{840 x^8} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 108, normalized size = 0.9 \begin{align*}{b}^{8} \left ( -{\frac{{\it FresnelC} \left ( bx \right ) }{8\,{b}^{8}{x}^{8}}}-{\frac{1}{56\,{b}^{7}{x}^{7}}\cos \left ({\frac{{b}^{2}\pi \,{x}^{2}}{2}} \right ) }-{\frac{\pi }{56} \left ( -{\frac{1}{5\,{b}^{5}{x}^{5}}\sin \left ({\frac{{b}^{2}\pi \,{x}^{2}}{2}} \right ) }+{\frac{\pi }{5} \left ( -{\frac{1}{3\,{x}^{3}{b}^{3}}\cos \left ({\frac{{b}^{2}\pi \,{x}^{2}}{2}} \right ) }-{\frac{\pi }{3} \left ( -{\frac{1}{bx}\sin \left ({\frac{{b}^{2}\pi \,{x}^{2}}{2}} \right ) }+\pi \,{\it FresnelC} \left ( bx \right ) \right ) } \right ) } \right ) } \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm fresnelc}\left (b x\right )}{x^{9}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\rm fresnelc}\left (b x\right )}{x^{9}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.70601, size = 185, normalized size = 1.55 \begin{align*} \frac{\pi ^{4} b^{8} C\left (b x\right ) \Gamma \left (- \frac{7}{4}\right )}{2560 \Gamma \left (\frac{5}{4}\right )} - \frac{\pi ^{3} b^{7} \sin{\left (\frac{\pi b^{2} x^{2}}{2} \right )} \Gamma \left (- \frac{7}{4}\right )}{2560 x \Gamma \left (\frac{5}{4}\right )} + \frac{\pi ^{2} b^{5} \cos{\left (\frac{\pi b^{2} x^{2}}{2} \right )} \Gamma \left (- \frac{7}{4}\right )}{2560 x^{3} \Gamma \left (\frac{5}{4}\right )} + \frac{3 \pi b^{3} \sin{\left (\frac{\pi b^{2} x^{2}}{2} \right )} \Gamma \left (- \frac{7}{4}\right )}{2560 x^{5} \Gamma \left (\frac{5}{4}\right )} - \frac{3 b \cos{\left (\frac{\pi b^{2} x^{2}}{2} \right )} \Gamma \left (- \frac{7}{4}\right )}{512 x^{7} \Gamma \left (\frac{5}{4}\right )} - \frac{21 C\left (b x\right ) \Gamma \left (- \frac{7}{4}\right )}{512 x^{8} \Gamma \left (\frac{5}{4}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm fresnelc}\left (b x\right )}{x^{9}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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