Optimal. Leaf size=60 \[ -\frac{\text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac{\text{FresnelC}\left (\sqrt{2} b x\right )}{2 \sqrt{2} \pi b^2}+\frac{x}{2 \pi b} \]
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Rubi [A] time = 0.0317032, antiderivative size = 60, 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 = {6461, 3358, 3352} \[ -\frac{\text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac{\text{FresnelC}\left (\sqrt{2} b x\right )}{2 \sqrt{2} \pi b^2}+\frac{x}{2 \pi b} \]
Antiderivative was successfully verified.
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Rule 6461
Rule 3358
Rule 3352
Rubi steps
\begin{align*} \int x C(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 ) C(b x)}{b^2 \pi }+\frac{\int \cos ^2\left (\frac{1}{2} b^2 \pi x^2\right ) \, dx}{b \pi }\\ &=-\frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{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{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{b^2 \pi }+\frac{\int \cos \left (b^2 \pi x^2\right ) \, dx}{2 b \pi }\\ &=\frac{x}{2 b \pi }-\frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{b^2 \pi }+\frac{C\left (\sqrt{2} b x\right )}{2 \sqrt{2} b^2 \pi }\\ \end{align*}
Mathematica [A] time = 0.0263026, size = 48, normalized size = 0.8 \[ \frac{-4 \text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )+\sqrt{2} \text{FresnelC}\left (\sqrt{2} b x\right )+2 b x}{4 \pi b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.061, size = 52, normalized size = 0.9 \begin{align*}{\frac{1}{b} \left ( -{\frac{{\it FresnelC} \left ( bx \right ) }{b\pi }\cos \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{\rm fresnelc}\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 fresnelc}\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 )} C\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 fresnelc}\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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