Optimal. Leaf size=13 \[ -\frac{1}{2 b S(b x)^2} \]
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Rubi [A] time = 0.0149547, antiderivative size = 13, 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, 30} \[ -\frac{1}{2 b S(b x)^2} \]
Antiderivative was successfully verified.
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Rule 6440
Rule 30
Rubi steps
\begin{align*} \int \frac{\sin \left (\frac{1}{2} b^2 \pi x^2\right )}{S(b x)^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^3} \, dx,x,S(b x)\right )}{b}\\ &=-\frac{1}{2 b S(b x)^2}\\ \end{align*}
Mathematica [A] time = 0.0040066, size = 13, normalized size = 1. \[ -\frac{1}{2 b S(b x)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 12, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,b \left ({\it FresnelS} \left ( bx \right ) \right ) ^{2}}} \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 )^{3}}\,{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 )^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.93431, size = 14, normalized size = 1.08 \begin{align*} \begin{cases} - \frac{1}{2 b S^{2}\left (b x\right )} & \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 )^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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