Optimal. Leaf size=27 \[ x \text{FresnelC}(b x)-\frac{\sin \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi b} \]
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Rubi [A] time = 0.0047574, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6419} \[ x \text{FresnelC}(b x)-\frac{\sin \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi b} \]
Antiderivative was successfully verified.
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Rule 6419
Rubi steps
\begin{align*} \int C(b x) \, dx &=x C(b x)-\frac{\sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b \pi }\\ \end{align*}
Mathematica [A] time = 0.0021411, size = 27, normalized size = 1. \[ x \text{FresnelC}(b x)-\frac{\sin \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 28, normalized size = 1. \begin{align*}{\frac{1}{b} \left ( bx{\it FresnelC} \left ( bx \right ) -{\frac{1}{\pi }\sin \left ({\frac{{b}^{2}\pi \,{x}^{2}}{2}} \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{\rm fresnelc}\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 ({\rm fresnelc}\left (b x\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.633405, size = 44, normalized size = 1.63 \begin{align*} \frac{x C\left (b x\right ) \Gamma \left (\frac{1}{4}\right )}{4 \Gamma \left (\frac{5}{4}\right )} - \frac{\sin{\left (\frac{\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac{1}{4}\right )}{4 \pi b \Gamma \left (\frac{5}{4}\right )} \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{\rm fresnelc}\left (b x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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