Optimal. Leaf size=26 \[ \frac{\cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi b}+x S(b x) \]
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Rubi [A] time = 0.0048741, antiderivative size = 26, 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 = {6418} \[ \frac{\cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi b}+x S(b x) \]
Antiderivative was successfully verified.
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Rule 6418
Rubi steps
\begin{align*} \int S(b x) \, dx &=\frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right )}{b \pi }+x S(b x)\\ \end{align*}
Mathematica [A] time = 0.0020637, size = 26, normalized size = 1. \[ \frac{\cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi b}+x S(b x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 27, normalized size = 1. \begin{align*}{\frac{1}{b} \left ( bx{\it FresnelS} \left ( bx \right ) +{\frac{1}{\pi }\cos \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 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 ({\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 [B] time = 0.83294, size = 48, normalized size = 1.85 \begin{align*} \frac{3 x S\left (b x\right ) \Gamma \left (\frac{3}{4}\right )}{4 \Gamma \left (\frac{7}{4}\right )} + \frac{3 \cos{\left (\frac{\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac{3}{4}\right )}{4 \pi b \Gamma \left (\frac{7}{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 fresnels}\left (b x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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