Optimal. Leaf size=166 \[ -\frac{43 \text{FresnelC}\left (\sqrt{2} b x\right )}{8 \sqrt{2} \pi ^3 b^6}+\frac{x^4 S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac{8 S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi ^3 b^6}+\frac{4 x^2 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}+\frac{x^3 \sin \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3}+\frac{11 x \cos \left (\pi b^2 x^2\right )}{8 \pi ^3 b^5}+\frac{4 x}{\pi ^3 b^5}-\frac{x^5}{10 \pi b} \]
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Rubi [A] time = 0.168881, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45, Rules used = {6462, 3391, 30, 3386, 3385, 3352, 6454, 6460, 3357} \[ -\frac{43 \text{FresnelC}\left (\sqrt{2} b x\right )}{8 \sqrt{2} \pi ^3 b^6}+\frac{x^4 S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac{8 S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi ^3 b^6}+\frac{4 x^2 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}+\frac{x^3 \sin \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3}+\frac{11 x \cos \left (\pi b^2 x^2\right )}{8 \pi ^3 b^5}+\frac{4 x}{\pi ^3 b^5}-\frac{x^5}{10 \pi b} \]
Antiderivative was successfully verified.
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Rule 6462
Rule 3391
Rule 30
Rule 3386
Rule 3385
Rule 3352
Rule 6454
Rule 6460
Rule 3357
Rubi steps
\begin{align*} \int x^5 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x) \, dx &=\frac{x^4 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac{4 \int x^3 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right ) \, dx}{b^2 \pi }-\frac{\int x^4 \sin ^2\left (\frac{1}{2} b^2 \pi x^2\right ) \, dx}{b \pi }\\ &=\frac{4 x^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{b^4 \pi ^2}+\frac{x^4 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac{8 \int x \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x) \, dx}{b^4 \pi ^2}-\frac{2 \int x^2 \sin \left (b^2 \pi x^2\right ) \, dx}{b^3 \pi ^2}-\frac{\int x^4 \, dx}{2 b \pi }+\frac{\int x^4 \cos \left (b^2 \pi x^2\right ) \, dx}{2 b \pi }\\ &=-\frac{x^5}{10 b \pi }+\frac{x \cos \left (b^2 \pi x^2\right )}{b^5 \pi ^3}+\frac{4 x^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{b^4 \pi ^2}-\frac{8 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac{x^4 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^2 \pi }+\frac{x^3 \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}-\frac{\int \cos \left (b^2 \pi x^2\right ) \, dx}{b^5 \pi ^3}+\frac{8 \int \sin ^2\left (\frac{1}{2} b^2 \pi x^2\right ) \, dx}{b^5 \pi ^3}-\frac{3 \int x^2 \sin \left (b^2 \pi x^2\right ) \, dx}{4 b^3 \pi ^2}\\ &=-\frac{x^5}{10 b \pi }+\frac{11 x \cos \left (b^2 \pi x^2\right )}{8 b^5 \pi ^3}-\frac{C\left (\sqrt{2} b x\right )}{\sqrt{2} b^6 \pi ^3}+\frac{4 x^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{b^4 \pi ^2}-\frac{8 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac{x^4 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^2 \pi }+\frac{x^3 \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}-\frac{3 \int \cos \left (b^2 \pi x^2\right ) \, dx}{8 b^5 \pi ^3}+\frac{8 \int \left (\frac{1}{2}-\frac{1}{2} \cos \left (b^2 \pi x^2\right )\right ) \, dx}{b^5 \pi ^3}\\ &=\frac{4 x}{b^5 \pi ^3}-\frac{x^5}{10 b \pi }+\frac{11 x \cos \left (b^2 \pi x^2\right )}{8 b^5 \pi ^3}-\frac{11 C\left (\sqrt{2} b x\right )}{8 \sqrt{2} b^6 \pi ^3}+\frac{4 x^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{b^4 \pi ^2}-\frac{8 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac{x^4 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^2 \pi }+\frac{x^3 \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}-\frac{4 \int \cos \left (b^2 \pi x^2\right ) \, dx}{b^5 \pi ^3}\\ &=\frac{4 x}{b^5 \pi ^3}-\frac{x^5}{10 b \pi }+\frac{11 x \cos \left (b^2 \pi x^2\right )}{8 b^5 \pi ^3}-\frac{11 C\left (\sqrt{2} b x\right )}{8 \sqrt{2} b^6 \pi ^3}-\frac{2 \sqrt{2} C\left (\sqrt{2} b x\right )}{b^6 \pi ^3}+\frac{4 x^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{b^4 \pi ^2}-\frac{8 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac{x^4 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^2 \pi }+\frac{x^3 \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}\\ \end{align*}
Mathematica [A] time = 0.173342, size = 126, normalized size = 0.76 \[ \frac{80 S(b x) \left (\left (\pi ^2 b^4 x^4-8\right ) \sin \left (\frac{1}{2} \pi b^2 x^2\right )+4 \pi b^2 x^2 \cos \left (\frac{1}{2} \pi b^2 x^2\right )\right )+2 b x \left (-4 \pi ^2 b^4 x^4+10 \pi b^2 x^2 \sin \left (\pi b^2 x^2\right )+55 \cos \left (\pi b^2 x^2\right )+160\right )-215 \sqrt{2} \text{FresnelC}\left (\sqrt{2} b x\right )}{80 \pi ^3 b^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.079, size = 212, normalized size = 1.3 \begin{align*}{\frac{1}{b} \left ({\frac{{\it FresnelS} \left ( bx \right ) }{{b}^{5}} \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 ) }-{\frac{1}{{b}^{5}} \left ({\frac{1}{2\,{\pi }^{3}} \left ({\frac{{\pi }^{2}{b}^{5}{x}^{5}}{5}}-8\,bx \right ) }+2\,{\frac{1}{{\pi }^{2}} \left ( -1/2\,{\frac{bx\cos \left ({b}^{2}\pi \,{x}^{2} \right ) }{\pi }}+1/4\,{\frac{\sqrt{2}{\it FresnelC} \left ( bx\sqrt{2} \right ) }{\pi }} \right ) }-{\frac{1}{2\,{\pi }^{3}} \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 ) } \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^{5} \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^{5} \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{5} \cos{\left (\frac{\pi b^{2} x^{2}}{2} \right )} S\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^{5} \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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