Optimal. Leaf size=127 \[ \frac{\pi ^4 b^9 \text{CosIntegral}\left (\frac{1}{2} \pi b^2 x^2\right )}{6912}-\frac{\pi ^3 b^7 \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{3456 x^2}+\frac{\pi b^3 \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{432 x^6}+\frac{\pi ^2 b^5 \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{1728 x^4}-\frac{b \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{72 x^8}-\frac{\text{FresnelC}(b x)}{9 x^9} \]
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Rubi [A] time = 0.143552, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6427, 3380, 3297, 3302} \[ \frac{\pi ^4 b^9 \text{CosIntegral}\left (\frac{1}{2} \pi b^2 x^2\right )}{6912}-\frac{\pi ^3 b^7 \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{3456 x^2}+\frac{\pi b^3 \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{432 x^6}+\frac{\pi ^2 b^5 \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{1728 x^4}-\frac{b \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{72 x^8}-\frac{\text{FresnelC}(b x)}{9 x^9} \]
Antiderivative was successfully verified.
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Rule 6427
Rule 3380
Rule 3297
Rule 3302
Rubi steps
\begin{align*} \int \frac{C(b x)}{x^{10}} \, dx &=-\frac{C(b x)}{9 x^9}+\frac{1}{9} b \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right )}{x^9} \, dx\\ &=-\frac{C(b x)}{9 x^9}+\frac{1}{18} b \operatorname{Subst}\left (\int \frac{\cos \left (\frac{1}{2} b^2 \pi x\right )}{x^5} \, dx,x,x^2\right )\\ &=-\frac{b \cos \left (\frac{1}{2} b^2 \pi x^2\right )}{72 x^8}-\frac{C(b x)}{9 x^9}-\frac{1}{144} \left (b^3 \pi \right ) \operatorname{Subst}\left (\int \frac{\sin \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 )}{72 x^8}-\frac{C(b x)}{9 x^9}+\frac{b^3 \pi \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{432 x^6}-\frac{1}{864} \left (b^5 \pi ^2\right ) \operatorname{Subst}\left (\int \frac{\cos \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 )}{72 x^8}+\frac{b^5 \pi ^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right )}{1728 x^4}-\frac{C(b x)}{9 x^9}+\frac{b^3 \pi \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{432 x^6}+\frac{\left (b^7 \pi ^3\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{1}{2} b^2 \pi x\right )}{x^2} \, dx,x,x^2\right )}{3456}\\ &=-\frac{b \cos \left (\frac{1}{2} b^2 \pi x^2\right )}{72 x^8}+\frac{b^5 \pi ^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right )}{1728 x^4}-\frac{C(b x)}{9 x^9}+\frac{b^3 \pi \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{432 x^6}-\frac{b^7 \pi ^3 \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{3456 x^2}+\frac{\left (b^9 \pi ^4\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{1}{2} b^2 \pi x\right )}{x} \, dx,x,x^2\right )}{6912}\\ &=-\frac{b \cos \left (\frac{1}{2} b^2 \pi x^2\right )}{72 x^8}+\frac{b^5 \pi ^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right )}{1728 x^4}+\frac{b^9 \pi ^4 \text{Ci}\left (\frac{1}{2} b^2 \pi x^2\right )}{6912}-\frac{C(b x)}{9 x^9}+\frac{b^3 \pi \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{432 x^6}-\frac{b^7 \pi ^3 \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{3456 x^2}\\ \end{align*}
Mathematica [A] time = 0.0805378, size = 96, normalized size = 0.76 \[ \frac{\pi ^4 b^9 \text{CosIntegral}\left (\frac{1}{2} \pi b^2 x^2\right )-\frac{2 \pi b^3 \left (\pi ^2 b^4 x^4-8\right ) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{x^6}+\frac{4 b \left (\pi ^2 b^4 x^4-24\right ) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{x^8}-\frac{768 \text{FresnelC}(b x)}{x^9}}{6912} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 115, normalized size = 0.9 \begin{align*}{b}^{9} \left ( -{\frac{{\it FresnelC} \left ( bx \right ) }{9\,{b}^{9}{x}^{9}}}-{\frac{1}{72\,{b}^{8}{x}^{8}}\cos \left ({\frac{{b}^{2}\pi \,{x}^{2}}{2}} \right ) }-{\frac{\pi }{72} \left ( -{\frac{1}{6\,{b}^{6}{x}^{6}}\sin \left ({\frac{{b}^{2}\pi \,{x}^{2}}{2}} \right ) }+{\frac{\pi }{6} \left ( -{\frac{1}{4\,{x}^{4}{b}^{4}}\cos \left ({\frac{{b}^{2}\pi \,{x}^{2}}{2}} \right ) }-{\frac{\pi }{4} \left ( -{\frac{1}{2\,{b}^{2}{x}^{2}}\sin \left ({\frac{{b}^{2}\pi \,{x}^{2}}{2}} \right ) }+{\frac{\pi }{4}{\it Ci} \left ({\frac{{b}^{2}\pi \,{x}^{2}}{2}} \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^{10}}\,{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^{10}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.86518, size = 76, normalized size = 0.6 \begin{align*} - \frac{\pi ^{6} b^{13} x^{4} \Gamma \left (\frac{13}{4}\right ){{}_{3}F_{4}\left (\begin{matrix} 1, 1, \frac{13}{4} \\ 2, \frac{7}{2}, 4, \frac{17}{4} \end{matrix}\middle |{- \frac{\pi ^{2} b^{4} x^{4}}{16}} \right )}}{737280 \Gamma \left (\frac{17}{4}\right )} + \frac{\pi ^{4} b^{9} \log{\left (b^{4} x^{4} \right )}}{13824} + \frac{\pi ^{2} b^{5}}{160 x^{4}} - \frac{b}{8 x^{8}} \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^{10}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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