Optimal. Leaf size=119 \[ \frac {1}{840} \pi ^4 b^8 C(b x)-\frac {b \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{56 x^7}-\frac {\pi ^3 b^7 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{840 x}+\frac {\pi ^2 b^5 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{840 x^3}+\frac {\pi b^3 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{280 x^5}-\frac {C(b x)}{8 x^8} \]
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Rubi [A] time = 0.08, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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 3352
Rule 3387
Rule 3388
Rule 6427
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*}
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Mathematica [A] time = 0.07, size = 85, normalized size = 0.71 \[ \frac {\left (\pi ^4 b^8 x^8-105\right ) C(b x)+b x \left (\pi ^2 b^4 x^4-15\right ) \cos \left (\frac {1}{2} \pi b^2 x^2\right )+\pi b^3 x^3 \left (3-\pi ^2 b^4 x^4\right ) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{840 x^8} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\rm fresnelc}\left (b x\right )}{x^{9}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm fresnelc}\left (b x\right )}{x^{9}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 108, normalized size = 0.91 \[ b^{8} \left (-\frac {\FresnelC \left (b x \right )}{8 b^{8} x^{8}}-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{56 b^{7} x^{7}}-\frac {\pi \left (-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{5 b^{5} x^{5}}+\frac {\pi \left (-\frac {\cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{3 b^{3} x^{3}}-\frac {\pi \left (-\frac {\sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b x}+\pi \FresnelC \left (b x \right )\right )}{3}\right )}{5}\right )}{56}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm fresnelc}\left (b x\right )}{x^{9}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {FresnelC}\left (b\,x\right )}{x^9} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.91, size = 185, normalized size = 1.55 \[ \frac {\pi ^{4} b^{8} C\left (b x\right ) \Gamma \left (- \frac {7}{4}\right )}{2560 \Gamma \left (\frac {5}{4}\right )} - \frac {\pi ^{3} b^{7} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (- \frac {7}{4}\right )}{2560 x \Gamma \left (\frac {5}{4}\right )} + \frac {\pi ^{2} b^{5} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (- \frac {7}{4}\right )}{2560 x^{3} \Gamma \left (\frac {5}{4}\right )} + \frac {3 \pi b^{3} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (- \frac {7}{4}\right )}{2560 x^{5} \Gamma \left (\frac {5}{4}\right )} - \frac {3 b \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (- \frac {7}{4}\right )}{512 x^{7} \Gamma \left (\frac {5}{4}\right )} - \frac {21 C\left (b x\right ) \Gamma \left (- \frac {7}{4}\right )}{512 x^{8} \Gamma \left (\frac {5}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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