Optimal. Leaf size=102 \[ \frac{1}{672} \pi ^3 b^7 \text{Si}\left (\frac{1}{2} b^2 \pi x^2\right )+\frac{\pi b^3 \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{168 x^4}+\frac{\pi ^2 b^5 \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{336 x^2}-\frac{b \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{42 x^6}-\frac{\text{FresnelC}(b x)}{7 x^7} \]
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Rubi [A] time = 0.118397, antiderivative size = 102, 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, 3380, 3297, 3299} \[ \frac{1}{672} \pi ^3 b^7 \text{Si}\left (\frac{1}{2} b^2 \pi x^2\right )+\frac{\pi b^3 \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{168 x^4}+\frac{\pi ^2 b^5 \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{336 x^2}-\frac{b \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{42 x^6}-\frac{\text{FresnelC}(b x)}{7 x^7} \]
Antiderivative was successfully verified.
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Rule 6427
Rule 3380
Rule 3297
Rule 3299
Rubi steps
\begin{align*} \int \frac{C(b x)}{x^8} \, dx &=-\frac{C(b x)}{7 x^7}+\frac{1}{7} b \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right )}{x^7} \, dx\\ &=-\frac{C(b x)}{7 x^7}+\frac{1}{14} b \operatorname{Subst}\left (\int \frac{\cos \left (\frac{1}{2} b^2 \pi x\right )}{x^4} \, dx,x,x^2\right )\\ &=-\frac{b \cos \left (\frac{1}{2} b^2 \pi x^2\right )}{42 x^6}-\frac{C(b x)}{7 x^7}-\frac{1}{84} \left (b^3 \pi \right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{1}{2} b^2 \pi x\right )}{x^3} \, dx,x,x^2\right )\\ &=-\frac{b \cos \left (\frac{1}{2} b^2 \pi x^2\right )}{42 x^6}-\frac{C(b x)}{7 x^7}+\frac{b^3 \pi \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{168 x^4}-\frac{1}{336} \left (b^5 \pi ^2\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{1}{2} b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )\\ &=-\frac{b \cos \left (\frac{1}{2} b^2 \pi x^2\right )}{42 x^6}+\frac{b^5 \pi ^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right )}{336 x^2}-\frac{C(b x)}{7 x^7}+\frac{b^3 \pi \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{168 x^4}+\frac{1}{672} \left (b^7 \pi ^3\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{1}{2} b^2 \pi x\right )}{x} \, dx,x,x^2\right )\\ &=-\frac{b \cos \left (\frac{1}{2} b^2 \pi x^2\right )}{42 x^6}+\frac{b^5 \pi ^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right )}{336 x^2}-\frac{C(b x)}{7 x^7}+\frac{b^3 \pi \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{168 x^4}+\frac{1}{672} b^7 \pi ^3 \text{Si}\left (\frac{1}{2} b^2 \pi x^2\right )\\ \end{align*}
Mathematica [A] time = 0.126535, size = 84, normalized size = 0.82 \[ \frac{1}{672} \left (\pi ^3 b^7 \text{Si}\left (\frac{1}{2} b^2 \pi x^2\right )+\frac{4 \pi b^3 \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{x^4}+\frac{2 b \left (\pi ^2 b^4 x^4-8\right ) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{x^6}-\frac{96 \text{FresnelC}(b x)}{x^7}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 93, normalized size = 0.9 \begin{align*}{b}^{7} \left ( -{\frac{{\it FresnelC} \left ( bx \right ) }{7\,{b}^{7}{x}^{7}}}-{\frac{1}{42\,{b}^{6}{x}^{6}}\cos \left ({\frac{{b}^{2}\pi \,{x}^{2}}{2}} \right ) }-{\frac{\pi }{42} \left ( -{\frac{1}{4\,{x}^{4}{b}^{4}}\sin \left ({\frac{{b}^{2}\pi \,{x}^{2}}{2}} \right ) }+{\frac{\pi }{4} \left ( -{\frac{1}{2\,{b}^{2}{x}^{2}}\cos \left ({\frac{{b}^{2}\pi \,{x}^{2}}{2}} \right ) }-{\frac{\pi }{4}{\it Si} \left ({\frac{{b}^{2}\pi \,{x}^{2}}{2}} \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^{8}}\,{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^{8}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.12267, size = 44, normalized size = 0.43 \begin{align*} - \frac{b \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{3}\left (\begin{matrix} - \frac{3}{2}, \frac{1}{4} \\ - \frac{1}{2}, \frac{1}{2}, \frac{5}{4} \end{matrix}\middle |{- \frac{\pi ^{2} b^{4} x^{4}}{16}} \right )}}{24 x^{6} \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^{8}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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