Optimal. Leaf size=217 \[ \frac{x^6 S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac{24 x^2 S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi ^3 b^6}+\frac{6 x^4 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}-\frac{48 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi ^4 b^8}+\frac{531 S\left (\sqrt{2} b x\right )}{16 \sqrt{2} \pi ^4 b^8}+\frac{4 x^3}{\pi ^3 b^5}+\frac{x^5 \sin \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3}-\frac{147 x \sin \left (\pi b^2 x^2\right )}{16 \pi ^4 b^7}+\frac{17 x^3 \cos \left (\pi b^2 x^2\right )}{8 \pi ^3 b^5}-\frac{x^7}{14 \pi b} \]
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Rubi [A] time = 0.271429, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {6462, 3391, 30, 3386, 3385, 3351, 6454, 6452} \[ \frac{x^6 S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac{24 x^2 S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi ^3 b^6}+\frac{6 x^4 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}-\frac{48 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi ^4 b^8}+\frac{531 S\left (\sqrt{2} b x\right )}{16 \sqrt{2} \pi ^4 b^8}+\frac{4 x^3}{\pi ^3 b^5}+\frac{x^5 \sin \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3}-\frac{147 x \sin \left (\pi b^2 x^2\right )}{16 \pi ^4 b^7}+\frac{17 x^3 \cos \left (\pi b^2 x^2\right )}{8 \pi ^3 b^5}-\frac{x^7}{14 \pi b} \]
Antiderivative was successfully verified.
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Rule 6462
Rule 3391
Rule 30
Rule 3386
Rule 3385
Rule 3351
Rule 6454
Rule 6452
Rubi steps
\begin{align*} \int x^7 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x) \, dx &=\frac{x^6 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac{6 \int x^5 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right ) \, dx}{b^2 \pi }-\frac{\int x^6 \sin ^2\left (\frac{1}{2} b^2 \pi x^2\right ) \, dx}{b \pi }\\ &=\frac{6 x^4 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{b^4 \pi ^2}+\frac{x^6 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac{24 \int x^3 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x) \, dx}{b^4 \pi ^2}-\frac{3 \int x^4 \sin \left (b^2 \pi x^2\right ) \, dx}{b^3 \pi ^2}-\frac{\int x^6 \, dx}{2 b \pi }+\frac{\int x^6 \cos \left (b^2 \pi x^2\right ) \, dx}{2 b \pi }\\ &=-\frac{x^7}{14 b \pi }+\frac{3 x^3 \cos \left (b^2 \pi x^2\right )}{2 b^5 \pi ^3}+\frac{6 x^4 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{b^4 \pi ^2}-\frac{24 x^2 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac{x^6 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^2 \pi }+\frac{x^5 \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac{48 \int x S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right ) \, dx}{b^6 \pi ^3}-\frac{9 \int x^2 \cos \left (b^2 \pi x^2\right ) \, dx}{2 b^5 \pi ^3}+\frac{24 \int x^2 \sin ^2\left (\frac{1}{2} b^2 \pi x^2\right ) \, dx}{b^5 \pi ^3}-\frac{5 \int x^4 \sin \left (b^2 \pi x^2\right ) \, dx}{4 b^3 \pi ^2}\\ &=-\frac{x^7}{14 b \pi }+\frac{17 x^3 \cos \left (b^2 \pi x^2\right )}{8 b^5 \pi ^3}-\frac{48 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{b^8 \pi ^4}+\frac{6 x^4 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{b^4 \pi ^2}-\frac{24 x^2 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac{x^6 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac{9 x \sin \left (b^2 \pi x^2\right )}{4 b^7 \pi ^4}+\frac{x^5 \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac{9 \int \sin \left (b^2 \pi x^2\right ) \, dx}{4 b^7 \pi ^4}+\frac{24 \int \sin \left (b^2 \pi x^2\right ) \, dx}{b^7 \pi ^4}-\frac{15 \int x^2 \cos \left (b^2 \pi x^2\right ) \, dx}{8 b^5 \pi ^3}+\frac{12 \int x^2 \, dx}{b^5 \pi ^3}-\frac{12 \int x^2 \cos \left (b^2 \pi x^2\right ) \, dx}{b^5 \pi ^3}\\ &=\frac{4 x^3}{b^5 \pi ^3}-\frac{x^7}{14 b \pi }+\frac{17 x^3 \cos \left (b^2 \pi x^2\right )}{8 b^5 \pi ^3}-\frac{48 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{b^8 \pi ^4}+\frac{6 x^4 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{b^4 \pi ^2}+\frac{9 S\left (\sqrt{2} b x\right )}{4 \sqrt{2} b^8 \pi ^4}+\frac{12 \sqrt{2} S\left (\sqrt{2} b x\right )}{b^8 \pi ^4}-\frac{24 x^2 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac{x^6 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac{147 x \sin \left (b^2 \pi x^2\right )}{16 b^7 \pi ^4}+\frac{x^5 \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac{15 \int \sin \left (b^2 \pi x^2\right ) \, dx}{16 b^7 \pi ^4}+\frac{6 \int \sin \left (b^2 \pi x^2\right ) \, dx}{b^7 \pi ^4}\\ &=\frac{4 x^3}{b^5 \pi ^3}-\frac{x^7}{14 b \pi }+\frac{17 x^3 \cos \left (b^2 \pi x^2\right )}{8 b^5 \pi ^3}-\frac{48 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{b^8 \pi ^4}+\frac{6 x^4 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{b^4 \pi ^2}+\frac{51 S\left (\sqrt{2} b x\right )}{16 \sqrt{2} b^8 \pi ^4}+\frac{15 \sqrt{2} S\left (\sqrt{2} b x\right )}{b^8 \pi ^4}-\frac{24 x^2 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac{x^6 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac{147 x \sin \left (b^2 \pi x^2\right )}{16 b^7 \pi ^4}+\frac{x^5 \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}\\ \end{align*}
Mathematica [A] time = 0.175812, size = 163, normalized size = 0.75 \[ \frac{224 S(b x) \left (\pi b^2 x^2 \left (\pi ^2 b^4 x^4-24\right ) \sin \left (\frac{1}{2} \pi b^2 x^2\right )+6 \left (\pi ^2 b^4 x^4-8\right ) \cos \left (\frac{1}{2} \pi b^2 x^2\right )\right )-16 \pi ^3 b^7 x^7+896 \pi b^3 x^3+56 \pi ^2 b^5 x^5 \sin \left (\pi b^2 x^2\right )-2058 b x \sin \left (\pi b^2 x^2\right )+476 \pi b^3 x^3 \cos \left (\pi b^2 x^2\right )+3717 \sqrt{2} S\left (\sqrt{2} b x\right )}{224 \pi ^4 b^8} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.084, size = 321, normalized size = 1.5 \begin{align*}{\frac{1}{b} \left ({\frac{{\it FresnelS} \left ( bx \right ) }{{b}^{7}} \left ({\frac{{b}^{6}{x}^{6}}{\pi }\sin \left ({\frac{{b}^{2}\pi \,{x}^{2}}{2}} \right ) }-6\,{\frac{1}{\pi } \left ( -{\frac{{x}^{4}{b}^{4}\cos \left ( 1/2\,{b}^{2}\pi \,{x}^{2} \right ) }{\pi }}+4\,{\frac{1}{\pi } \left ({\frac{{b}^{2}{x}^{2}\sin \left ( 1/2\,{b}^{2}\pi \,{x}^{2} \right ) }{\pi }}+2\,{\frac{\cos \left ( 1/2\,{b}^{2}\pi \,{x}^{2} \right ) }{{\pi }^{2}}} \right ) } \right ) } \right ) }-{\frac{1}{{b}^{7}} \left ({\frac{1}{2\,{\pi }^{3}} \left ({\frac{{\pi }^{2}{b}^{7}{x}^{7}}{7}}-8\,{x}^{3}{b}^{3} \right ) }+3\,{\frac{1}{{\pi }^{4}} \left ( -1/2\,\pi \,{b}^{3}{x}^{3}\cos \left ({b}^{2}\pi \,{x}^{2} \right ) +3/2\,\pi \, \left ( 1/2\,{\frac{bx\sin \left ({b}^{2}\pi \,{x}^{2} \right ) }{\pi }}-1/4\,{\frac{\sqrt{2}{\it FresnelS} \left ( bx\sqrt{2} \right ) }{\pi }} \right ) -4\,\sqrt{2}{\it FresnelS} \left ( bx\sqrt{2} \right ) \right ) }-{\frac{1}{2\,{\pi }^{3}} \left ({\frac{\pi \,{b}^{5}{x}^{5}\sin \left ({b}^{2}\pi \,{x}^{2} \right ) }{2}}-{\frac{5\,\pi }{2} \left ( -{\frac{{x}^{3}{b}^{3}\cos \left ({b}^{2}\pi \,{x}^{2} \right ) }{2\,\pi }}+{\frac{3}{2\,\pi } \left ({\frac{bx\sin \left ({b}^{2}\pi \,{x}^{2} \right ) }{2\,\pi }}-{\frac{\sqrt{2}{\it FresnelS} \left ( bx\sqrt{2} \right ) }{4\,\pi }} \right ) } \right ) }-12\,{\frac{bx\sin \left ({b}^{2}\pi \,{x}^{2} \right ) }{\pi }}+6\,{\frac{\sqrt{2}{\it FresnelS} \left ( bx\sqrt{2} \right ) }{\pi }} \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} \cos \left (\frac{1}{2} \, \pi b^{2} x^{2}\right ){\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^{7} \cos \left (\frac{1}{2} \, \pi b^{2} x^{2}\right ){\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 [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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} \cos \left (\frac{1}{2} \, \pi b^{2} x^{2}\right ){\rm fresnels}\left (b x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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