Optimal. Leaf size=124 \[ -\frac{105 \text{FresnelC}(b x)}{8 \pi ^4 b^8}-\frac{x^7 \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{8 \pi b}+\frac{35 x^3 \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{8 \pi ^3 b^5}-\frac{7 x^5 \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{8 \pi ^2 b^3}+\frac{105 x \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{8 \pi ^4 b^7}+\frac{1}{8} x^8 \text{FresnelC}(b x) \]
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Rubi [A] time = 0.0821382, antiderivative size = 124, 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, 3386, 3385, 3352} \[ -\frac{105 \text{FresnelC}(b x)}{8 \pi ^4 b^8}-\frac{x^7 \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{8 \pi b}+\frac{35 x^3 \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{8 \pi ^3 b^5}-\frac{7 x^5 \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{8 \pi ^2 b^3}+\frac{105 x \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{8 \pi ^4 b^7}+\frac{1}{8} x^8 \text{FresnelC}(b x) \]
Antiderivative was successfully verified.
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Rule 6427
Rule 3386
Rule 3385
Rule 3352
Rubi steps
\begin{align*} \int x^7 C(b x) \, dx &=\frac{1}{8} x^8 C(b x)-\frac{1}{8} b \int x^8 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) \, dx\\ &=\frac{1}{8} x^8 C(b x)-\frac{x^7 \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{8 b \pi }+\frac{7 \int x^6 \sin \left (\frac{1}{2} b^2 \pi x^2\right ) \, dx}{8 b \pi }\\ &=-\frac{7 x^5 \cos \left (\frac{1}{2} b^2 \pi x^2\right )}{8 b^3 \pi ^2}+\frac{1}{8} x^8 C(b x)-\frac{x^7 \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{8 b \pi }+\frac{35 \int x^4 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) \, dx}{8 b^3 \pi ^2}\\ &=-\frac{7 x^5 \cos \left (\frac{1}{2} b^2 \pi x^2\right )}{8 b^3 \pi ^2}+\frac{1}{8} x^8 C(b x)+\frac{35 x^3 \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{8 b^5 \pi ^3}-\frac{x^7 \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{8 b \pi }-\frac{105 \int x^2 \sin \left (\frac{1}{2} b^2 \pi x^2\right ) \, dx}{8 b^5 \pi ^3}\\ &=\frac{105 x \cos \left (\frac{1}{2} b^2 \pi x^2\right )}{8 b^7 \pi ^4}-\frac{7 x^5 \cos \left (\frac{1}{2} b^2 \pi x^2\right )}{8 b^3 \pi ^2}+\frac{1}{8} x^8 C(b x)+\frac{35 x^3 \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{8 b^5 \pi ^3}-\frac{x^7 \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{8 b \pi }-\frac{105 \int \cos \left (\frac{1}{2} b^2 \pi x^2\right ) \, dx}{8 b^7 \pi ^4}\\ &=\frac{105 x \cos \left (\frac{1}{2} b^2 \pi x^2\right )}{8 b^7 \pi ^4}-\frac{7 x^5 \cos \left (\frac{1}{2} b^2 \pi x^2\right )}{8 b^3 \pi ^2}-\frac{105 C(b x)}{8 b^8 \pi ^4}+\frac{1}{8} x^8 C(b x)+\frac{35 x^3 \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{8 b^5 \pi ^3}-\frac{x^7 \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{8 b \pi }\\ \end{align*}
Mathematica [A] time = 0.0712275, size = 89, normalized size = 0.72 \[ \frac{\left (\pi ^4 b^8 x^8-105\right ) \text{FresnelC}(b x)+\pi b^3 x^3 \left (35-\pi ^2 b^4 x^4\right ) \sin \left (\frac{1}{2} \pi b^2 x^2\right )-7 b x \left (\pi ^2 b^4 x^4-15\right ) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{8 \pi ^4 b^8} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 123, normalized size = 1. \begin{align*}{\frac{1}{{b}^{8}} \left ({\frac{{\it FresnelC} \left ( bx \right ){b}^{8}{x}^{8}}{8}}-{\frac{{b}^{7}{x}^{7}}{8\,\pi }\sin \left ({\frac{{b}^{2}\pi \,{x}^{2}}{2}} \right ) }+{\frac{7}{8\,\pi } \left ( -{\frac{{b}^{5}{x}^{5}}{\pi }\cos \left ({\frac{{b}^{2}\pi \,{x}^{2}}{2}} \right ) }+5\,{\frac{1}{\pi } \left ({\frac{{x}^{3}{b}^{3}\sin \left ( 1/2\,{b}^{2}\pi \,{x}^{2} \right ) }{\pi }}-3\,{\frac{1}{\pi } \left ( -{\frac{\cos \left ( 1/2\,{b}^{2}\pi \,{x}^{2} \right ) bx}{\pi }}+{\frac{{\it FresnelC} \left ( bx \right ) }{\pi }} \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 x^{7}{\rm fresnelc}\left (b x\right )\,{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 (x^{7}{\rm fresnelc}\left (b x\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.60003, size = 184, normalized size = 1.48 \begin{align*} \frac{45 x^{8} C\left (b x\right ) \Gamma \left (\frac{1}{4}\right )}{512 \Gamma \left (\frac{13}{4}\right )} - \frac{45 x^{7} \sin{\left (\frac{\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac{1}{4}\right )}{512 \pi b \Gamma \left (\frac{13}{4}\right )} - \frac{315 x^{5} \cos{\left (\frac{\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac{1}{4}\right )}{512 \pi ^{2} b^{3} \Gamma \left (\frac{13}{4}\right )} + \frac{1575 x^{3} \sin{\left (\frac{\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac{1}{4}\right )}{512 \pi ^{3} b^{5} \Gamma \left (\frac{13}{4}\right )} + \frac{4725 x \cos{\left (\frac{\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac{1}{4}\right )}{512 \pi ^{4} b^{7} \Gamma \left (\frac{13}{4}\right )} - \frac{4725 C\left (b x\right ) \Gamma \left (\frac{1}{4}\right )}{512 \pi ^{4} b^{8} \Gamma \left (\frac{13}{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 x^{7}{\rm fresnelc}\left (b x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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