Optimal. Leaf size=239 \[ \frac{531 \text{FresnelC}\left (\sqrt{2} b x\right )}{56 \sqrt{2} \pi ^4 b^7}-\frac{12 x^4 S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{7 \pi ^2 b^3}+\frac{96 S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{7 \pi ^4 b^7}+\frac{2 x^6 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{7 \pi b}-\frac{48 x^2 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{7 \pi ^3 b^5}+\frac{6 x^5}{35 \pi ^2 b^2}-\frac{17 x^3 \sin \left (\pi b^2 x^2\right )}{28 \pi ^3 b^4}+\frac{x^5 \cos \left (\pi b^2 x^2\right )}{14 \pi ^2 b^2}-\frac{21 x \cos \left (\pi b^2 x^2\right )}{8 \pi ^4 b^6}-\frac{48 x}{7 \pi ^4 b^6}+\frac{1}{7} x^7 S(b x)^2 \]
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Rubi [A] time = 0.317892, antiderivative size = 239, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 10, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {6430, 6454, 6462, 3391, 30, 3386, 3385, 3352, 6460, 3357} \[ \frac{531 \text{FresnelC}\left (\sqrt{2} b x\right )}{56 \sqrt{2} \pi ^4 b^7}-\frac{12 x^4 S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{7 \pi ^2 b^3}+\frac{96 S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{7 \pi ^4 b^7}+\frac{2 x^6 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{7 \pi b}-\frac{48 x^2 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{7 \pi ^3 b^5}+\frac{6 x^5}{35 \pi ^2 b^2}-\frac{17 x^3 \sin \left (\pi b^2 x^2\right )}{28 \pi ^3 b^4}+\frac{x^5 \cos \left (\pi b^2 x^2\right )}{14 \pi ^2 b^2}-\frac{21 x \cos \left (\pi b^2 x^2\right )}{8 \pi ^4 b^6}-\frac{48 x}{7 \pi ^4 b^6}+\frac{1}{7} x^7 S(b x)^2 \]
Antiderivative was successfully verified.
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Rule 6430
Rule 6454
Rule 6462
Rule 3391
Rule 30
Rule 3386
Rule 3385
Rule 3352
Rule 6460
Rule 3357
Rubi steps
\begin{align*} \int x^6 S(b x)^2 \, dx &=\frac{1}{7} x^7 S(b x)^2-\frac{1}{7} (2 b) \int x^7 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right ) \, dx\\ &=\frac{2 x^6 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{7 b \pi }+\frac{1}{7} x^7 S(b x)^2-\frac{\int x^6 \sin \left (b^2 \pi x^2\right ) \, dx}{7 \pi }-\frac{12 \int x^5 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x) \, dx}{7 b \pi }\\ &=\frac{x^5 \cos \left (b^2 \pi x^2\right )}{14 b^2 \pi ^2}+\frac{2 x^6 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{7 b \pi }+\frac{1}{7} x^7 S(b x)^2-\frac{12 x^4 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{7 b^3 \pi ^2}+\frac{48 \int x^3 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right ) \, dx}{7 b^3 \pi ^2}-\frac{5 \int x^4 \cos \left (b^2 \pi x^2\right ) \, dx}{14 b^2 \pi ^2}+\frac{12 \int x^4 \sin ^2\left (\frac{1}{2} b^2 \pi x^2\right ) \, dx}{7 b^2 \pi ^2}\\ &=\frac{x^5 \cos \left (b^2 \pi x^2\right )}{14 b^2 \pi ^2}-\frac{48 x^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{7 b^5 \pi ^3}+\frac{2 x^6 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{7 b \pi }+\frac{1}{7} x^7 S(b x)^2-\frac{12 x^4 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{7 b^3 \pi ^2}-\frac{5 x^3 \sin \left (b^2 \pi x^2\right )}{28 b^4 \pi ^3}+\frac{96 \int x \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x) \, dx}{7 b^5 \pi ^3}+\frac{15 \int x^2 \sin \left (b^2 \pi x^2\right ) \, dx}{28 b^4 \pi ^3}+\frac{24 \int x^2 \sin \left (b^2 \pi x^2\right ) \, dx}{7 b^4 \pi ^3}+\frac{6 \int x^4 \, dx}{7 b^2 \pi ^2}-\frac{6 \int x^4 \cos \left (b^2 \pi x^2\right ) \, dx}{7 b^2 \pi ^2}\\ &=\frac{6 x^5}{35 b^2 \pi ^2}-\frac{111 x \cos \left (b^2 \pi x^2\right )}{56 b^6 \pi ^4}+\frac{x^5 \cos \left (b^2 \pi x^2\right )}{14 b^2 \pi ^2}-\frac{48 x^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{7 b^5 \pi ^3}+\frac{2 x^6 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{7 b \pi }+\frac{1}{7} x^7 S(b x)^2+\frac{96 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{7 b^7 \pi ^4}-\frac{12 x^4 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{7 b^3 \pi ^2}-\frac{17 x^3 \sin \left (b^2 \pi x^2\right )}{28 b^4 \pi ^3}+\frac{15 \int \cos \left (b^2 \pi x^2\right ) \, dx}{56 b^6 \pi ^4}+\frac{12 \int \cos \left (b^2 \pi x^2\right ) \, dx}{7 b^6 \pi ^4}-\frac{96 \int \sin ^2\left (\frac{1}{2} b^2 \pi x^2\right ) \, dx}{7 b^6 \pi ^4}+\frac{9 \int x^2 \sin \left (b^2 \pi x^2\right ) \, dx}{7 b^4 \pi ^3}\\ &=\frac{6 x^5}{35 b^2 \pi ^2}-\frac{21 x \cos \left (b^2 \pi x^2\right )}{8 b^6 \pi ^4}+\frac{x^5 \cos \left (b^2 \pi x^2\right )}{14 b^2 \pi ^2}+\frac{15 C\left (\sqrt{2} b x\right )}{56 \sqrt{2} b^7 \pi ^4}+\frac{6 \sqrt{2} C\left (\sqrt{2} b x\right )}{7 b^7 \pi ^4}-\frac{48 x^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{7 b^5 \pi ^3}+\frac{2 x^6 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{7 b \pi }+\frac{1}{7} x^7 S(b x)^2+\frac{96 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{7 b^7 \pi ^4}-\frac{12 x^4 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{7 b^3 \pi ^2}-\frac{17 x^3 \sin \left (b^2 \pi x^2\right )}{28 b^4 \pi ^3}+\frac{9 \int \cos \left (b^2 \pi x^2\right ) \, dx}{14 b^6 \pi ^4}-\frac{96 \int \left (\frac{1}{2}-\frac{1}{2} \cos \left (b^2 \pi x^2\right )\right ) \, dx}{7 b^6 \pi ^4}\\ &=-\frac{48 x}{7 b^6 \pi ^4}+\frac{6 x^5}{35 b^2 \pi ^2}-\frac{21 x \cos \left (b^2 \pi x^2\right )}{8 b^6 \pi ^4}+\frac{x^5 \cos \left (b^2 \pi x^2\right )}{14 b^2 \pi ^2}+\frac{51 C\left (\sqrt{2} b x\right )}{56 \sqrt{2} b^7 \pi ^4}+\frac{6 \sqrt{2} C\left (\sqrt{2} b x\right )}{7 b^7 \pi ^4}-\frac{48 x^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{7 b^5 \pi ^3}+\frac{2 x^6 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{7 b \pi }+\frac{1}{7} x^7 S(b x)^2+\frac{96 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{7 b^7 \pi ^4}-\frac{12 x^4 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{7 b^3 \pi ^2}-\frac{17 x^3 \sin \left (b^2 \pi x^2\right )}{28 b^4 \pi ^3}+\frac{48 \int \cos \left (b^2 \pi x^2\right ) \, dx}{7 b^6 \pi ^4}\\ &=-\frac{48 x}{7 b^6 \pi ^4}+\frac{6 x^5}{35 b^2 \pi ^2}-\frac{21 x \cos \left (b^2 \pi x^2\right )}{8 b^6 \pi ^4}+\frac{x^5 \cos \left (b^2 \pi x^2\right )}{14 b^2 \pi ^2}+\frac{51 C\left (\sqrt{2} b x\right )}{56 \sqrt{2} b^7 \pi ^4}+\frac{30 \sqrt{2} C\left (\sqrt{2} b x\right )}{7 b^7 \pi ^4}-\frac{48 x^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{7 b^5 \pi ^3}+\frac{2 x^6 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{7 b \pi }+\frac{1}{7} x^7 S(b x)^2+\frac{96 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{7 b^7 \pi ^4}-\frac{12 x^4 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{7 b^3 \pi ^2}-\frac{17 x^3 \sin \left (b^2 \pi x^2\right )}{28 b^4 \pi ^3}\\ \end{align*}
Mathematica [A] time = 0.289179, size = 171, normalized size = 0.72 \[ \frac{80 \pi ^4 b^7 x^7 S(b x)^2+160 S(b x) \left (\pi b^2 x^2 \left (\pi ^2 b^4 x^4-24\right ) \cos \left (\frac{1}{2} \pi b^2 x^2\right )-6 \left (\pi ^2 b^4 x^4-8\right ) \sin \left (\frac{1}{2} \pi b^2 x^2\right )\right )+2 b x \left (5 \left (4 \pi ^2 b^4 x^4-147\right ) \cos \left (\pi b^2 x^2\right )-2 \left (-24 \pi ^2 b^4 x^4+85 \pi b^2 x^2 \sin \left (\pi b^2 x^2\right )+960\right )\right )+2655 \sqrt{2} \text{FresnelC}\left (\sqrt{2} b x\right )}{560 \pi ^4 b^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.077, size = 324, normalized size = 1.4 \begin{align*}{\frac{1}{{b}^{7}} \left ({\frac{{b}^{7}{x}^{7} \left ({\it FresnelS} \left ( bx \right ) \right ) ^{2}}{7}}-2\,{\it FresnelS} \left ( bx \right ) \left ( -1/7\,{\frac{{b}^{6}{x}^{6}\cos \left ( 1/2\,{b}^{2}\pi \,{x}^{2} \right ) }{\pi }}+6/7\,{\frac{1}{\pi } \left ({\frac{{x}^{4}{b}^{4}\sin \left ( 1/2\,{b}^{2}\pi \,{x}^{2} \right ) }{\pi }}-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 ) +{\frac{6}{7\,{\pi }^{4}} \left ({\frac{{\pi }^{2}{b}^{5}{x}^{5}}{5}}-8\,bx \right ) }-{\frac{6}{7\,{\pi }^{4}} \left ({\frac{\pi \,{b}^{3}{x}^{3}\sin \left ({b}^{2}\pi \,{x}^{2} \right ) }{2}}-{\frac{3\,\pi }{2} \left ( -{\frac{bx\cos \left ({b}^{2}\pi \,{x}^{2} \right ) }{2\,\pi }}+{\frac{\sqrt{2}{\it FresnelC} \left ( bx\sqrt{2} \right ) }{4\,\pi }} \right ) }-4\,\sqrt{2}{\it FresnelC} \left ( bx\sqrt{2} \right ) \right ) }-{\frac{1}{7\,{\pi }^{3}} \left ( -{\frac{\pi \,{b}^{5}{x}^{5}\cos \left ({b}^{2}\pi \,{x}^{2} \right ) }{2}}+{\frac{5\,\pi }{2} \left ({\frac{{x}^{3}{b}^{3}\sin \left ({b}^{2}\pi \,{x}^{2} \right ) }{2\,\pi }}-{\frac{3}{2\,\pi } \left ( -{\frac{bx\cos \left ({b}^{2}\pi \,{x}^{2} \right ) }{2\,\pi }}+{\frac{\sqrt{2}{\it FresnelC} \left ( bx\sqrt{2} \right ) }{4\,\pi }} \right ) } \right ) }+12\,{\frac{bx\cos \left ({b}^{2}\pi \,{x}^{2} \right ) }{\pi }}-6\,{\frac{\sqrt{2}{\it FresnelC} \left ( bx\sqrt{2} \right ) }{\pi }} \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 )^{2}\,{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 )^{2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{6} S^{2}\left (b x\right )\, dx \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 )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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