Optimal. Leaf size=9 \[ \frac{\log (S(b x))}{b} \]
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Rubi [A] time = 0.0147908, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {6440, 29} \[ \frac{\log (S(b x))}{b} \]
Antiderivative was successfully verified.
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Rule 6440
Rule 29
Rubi steps
\begin{align*} \int \frac{\sin \left (\frac{1}{2} b^2 \pi x^2\right )}{S(b x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,S(b x)\right )}{b}\\ &=\frac{\log (S(b x))}{b}\\ \end{align*}
Mathematica [A] time = 0.010098, size = 9, normalized size = 1. \[ \frac{\log (S(b x))}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.118, size = 10, normalized size = 1.1 \begin{align*}{\frac{\ln \left ({\it FresnelS} \left ( bx \right ) \right ) }{b}} \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 \frac{\sin \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 (\frac{\sin \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 [A] time = 0.235245, size = 8, normalized size = 0.89 \begin{align*} \begin{cases} \frac{\log{\left (S\left (b x\right ) \right )}}{b} & \text{for}\: b \neq 0 \\\text{NaN} & \text{otherwise} \end{cases} \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 \frac{\sin \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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