Optimal. Leaf size=109 \[ -\frac{6 x^4 \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{7 \pi ^2 b^3}+\frac{48 \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{7 \pi ^4 b^7}+\frac{x^6 \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{7 \pi b}-\frac{24 x^2 \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{7 \pi ^3 b^5}+\frac{1}{7} x^7 S(b x) \]
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Rubi [A] time = 0.110693, antiderivative size = 109, 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 = {6426, 3379, 3296, 2637} \[ -\frac{6 x^4 \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{7 \pi ^2 b^3}+\frac{48 \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{7 \pi ^4 b^7}+\frac{x^6 \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{7 \pi b}-\frac{24 x^2 \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{7 \pi ^3 b^5}+\frac{1}{7} x^7 S(b x) \]
Antiderivative was successfully verified.
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Rule 6426
Rule 3379
Rule 3296
Rule 2637
Rubi steps
\begin{align*} \int x^6 S(b x) \, dx &=\frac{1}{7} x^7 S(b x)-\frac{1}{7} b \int x^7 \sin \left (\frac{1}{2} b^2 \pi x^2\right ) \, dx\\ &=\frac{1}{7} x^7 S(b x)-\frac{1}{14} b \operatorname{Subst}\left (\int x^3 \sin \left (\frac{1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )\\ &=\frac{x^6 \cos \left (\frac{1}{2} b^2 \pi x^2\right )}{7 b \pi }+\frac{1}{7} x^7 S(b x)-\frac{3 \operatorname{Subst}\left (\int x^2 \cos \left (\frac{1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )}{7 b \pi }\\ &=\frac{x^6 \cos \left (\frac{1}{2} b^2 \pi x^2\right )}{7 b \pi }+\frac{1}{7} x^7 S(b x)-\frac{6 x^4 \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{7 b^3 \pi ^2}+\frac{12 \operatorname{Subst}\left (\int x \sin \left (\frac{1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )}{7 b^3 \pi ^2}\\ &=-\frac{24 x^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right )}{7 b^5 \pi ^3}+\frac{x^6 \cos \left (\frac{1}{2} b^2 \pi x^2\right )}{7 b \pi }+\frac{1}{7} x^7 S(b x)-\frac{6 x^4 \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{7 b^3 \pi ^2}+\frac{24 \operatorname{Subst}\left (\int \cos \left (\frac{1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )}{7 b^5 \pi ^3}\\ &=-\frac{24 x^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right )}{7 b^5 \pi ^3}+\frac{x^6 \cos \left (\frac{1}{2} b^2 \pi x^2\right )}{7 b \pi }+\frac{1}{7} x^7 S(b x)+\frac{48 \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{7 b^7 \pi ^4}-\frac{6 x^4 \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{7 b^3 \pi ^2}\\ \end{align*}
Mathematica [A] time = 0.0516547, size = 83, normalized size = 0.76 \[ -\frac{6 \left (\pi ^2 b^4 x^4-8\right ) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{7 \pi ^4 b^7}+\frac{x^2 \left (\pi ^2 b^4 x^4-24\right ) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{7 \pi ^3 b^5}+\frac{1}{7} x^7 S(b x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 107, normalized size = 1. \begin{align*}{\frac{1}{{b}^{7}} \left ({\frac{{b}^{7}{x}^{7}{\it FresnelS} \left ( bx \right ) }{7}}+{\frac{{b}^{6}{x}^{6}}{7\,\pi }\cos \left ({\frac{{b}^{2}\pi \,{x}^{2}}{2}} \right ) }-{\frac{6}{7\,\pi } \left ({\frac{{x}^{4}{b}^{4}}{\pi }\sin \left ({\frac{{b}^{2}\pi \,{x}^{2}}{2}} \right ) }-4\,{\frac{1}{\pi } \left ( -{\frac{{b}^{2}{x}^{2}\cos \left ( 1/2\,{b}^{2}\pi \,{x}^{2} \right ) }{\pi }}+2\,{\frac{\sin \left ( 1/2\,{b}^{2}\pi \,{x}^{2} \right ) }{{\pi }^{2}}} \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^{6}{\rm fresnels}\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^{6}{\rm fresnels}\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 = 1.76553, size = 156, normalized size = 1.43 \begin{align*} \frac{3 x^{7} S\left (b x\right ) \Gamma \left (\frac{3}{4}\right )}{28 \Gamma \left (\frac{7}{4}\right )} + \frac{3 x^{6} \cos{\left (\frac{\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac{3}{4}\right )}{28 \pi b \Gamma \left (\frac{7}{4}\right )} - \frac{9 x^{4} \sin{\left (\frac{\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac{3}{4}\right )}{14 \pi ^{2} b^{3} \Gamma \left (\frac{7}{4}\right )} - \frac{18 x^{2} \cos{\left (\frac{\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac{3}{4}\right )}{7 \pi ^{3} b^{5} \Gamma \left (\frac{7}{4}\right )} + \frac{36 \sin{\left (\frac{\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac{3}{4}\right )}{7 \pi ^{4} b^{7} \Gamma \left (\frac{7}{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^{6}{\rm fresnels}\left (b x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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