Optimal. Leaf size=59 \[ \frac{\text{FresnelC}\left (\sqrt{2} b x\right )}{2 \sqrt{2} \pi b^2}+\frac{S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac{x}{2 \pi b} \]
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Rubi [A] time = 0.0321775, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6460, 3357, 3352} \[ \frac{\text{FresnelC}\left (\sqrt{2} b x\right )}{2 \sqrt{2} \pi b^2}+\frac{S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac{x}{2 \pi b} \]
Antiderivative was successfully verified.
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Rule 6460
Rule 3357
Rule 3352
Rubi steps
\begin{align*} \int x \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x) \, dx &=\frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac{\int \sin ^2\left (\frac{1}{2} b^2 \pi x^2\right ) \, dx}{b \pi }\\ &=\frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac{\int \left (\frac{1}{2}-\frac{1}{2} \cos \left (b^2 \pi x^2\right )\right ) \, dx}{b \pi }\\ &=-\frac{x}{2 b \pi }+\frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^2 \pi }+\frac{\int \cos \left (b^2 \pi x^2\right ) \, dx}{2 b \pi }\\ &=-\frac{x}{2 b \pi }+\frac{C\left (\sqrt{2} b x\right )}{2 \sqrt{2} b^2 \pi }+\frac{S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^2 \pi }\\ \end{align*}
Mathematica [A] time = 0.0279266, size = 48, normalized size = 0.81 \[ \frac{4 S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )+\sqrt{2} \text{FresnelC}\left (\sqrt{2} b x\right )-2 b x}{4 \pi b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.066, size = 52, normalized size = 0.9 \begin{align*}{\frac{1}{b} \left ({\frac{{\it FresnelS} \left ( bx \right ) }{b\pi }\sin \left ({\frac{{b}^{2}\pi \,{x}^{2}}{2}} \right ) }-{\frac{1}{b\pi } \left ({\frac{bx}{2}}-{\frac{\sqrt{2}{\it FresnelC} \left ( bx\sqrt{2} \right ) }{4}} \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 \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 \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 \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 \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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