Optimal. Leaf size=49 \[ \frac{S\left (\sqrt{2} b x\right )}{2 \sqrt{2} \pi b^2}-\frac{S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi b^2} \]
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Rubi [A] time = 0.0216188, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {6452, 3351} \[ \frac{S\left (\sqrt{2} b x\right )}{2 \sqrt{2} \pi b^2}-\frac{S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi b^2} \]
Antiderivative was successfully verified.
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Rule 6452
Rule 3351
Rubi steps
\begin{align*} \int x S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right ) \, dx &=-\frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{b^2 \pi }+\frac{\int \sin \left (b^2 \pi x^2\right ) \, dx}{2 b \pi }\\ &=-\frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{b^2 \pi }+\frac{S\left (\sqrt{2} b x\right )}{2 \sqrt{2} b^2 \pi }\\ \end{align*}
Mathematica [A] time = 0.0244937, size = 44, normalized size = 0.9 \[ \frac{\sqrt{2} S\left (\sqrt{2} b x\right )-4 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{4 \pi b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.063, size = 46, normalized size = 0.9 \begin{align*}{\frac{1}{b} \left ( -{\frac{{\it FresnelS} \left ( bx \right ) }{b\pi }\cos \left ({\frac{{b}^{2}\pi \,{x}^{2}}{2}} \right ) }+{\frac{\sqrt{2}{\it FresnelS} \left ( bx\sqrt{2} \right ) }{4\,b\pi }} \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{\rm fresnels}\left (b x\right ) \sin \left (\frac{1}{2} \, \pi b^{2} x^{2}\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{\rm fresnels}\left (b x\right ) \sin \left (\frac{1}{2} \, \pi b^{2} x^{2}\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 \sin{\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{\rm fresnels}\left (b x\right ) \sin \left (\frac{1}{2} \, \pi b^{2} x^{2}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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