Optimal. Leaf size=13 \[ \frac{S(b x)^3}{3 b} \]
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Rubi [A] time = 0.0147598, 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} \[ \frac{S(b x)^3}{3 b} \]
Antiderivative was successfully verified.
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Rule 6440
Rule 30
Rubi steps
\begin{align*} \int S(b x)^2 \sin \left (\frac{1}{2} b^2 \pi x^2\right ) \, dx &=\frac{\operatorname{Subst}\left (\int x^2 \, dx,x,S(b x)\right )}{b}\\ &=\frac{S(b x)^3}{3 b}\\ \end{align*}
Mathematica [A] time = 0.0037452, size = 13, normalized size = 1. \[ \frac{S(b x)^3}{3 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 12, normalized size = 0.9 \begin{align*}{\frac{ \left ({\it FresnelS} \left ( bx \right ) \right ) ^{3}}{3\,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{\rm fresnels}\left (b x\right )^{2} \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 ({\rm fresnels}\left (b x\right )^{2} \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 [A] time = 1.4036, size = 10, normalized size = 0.77 \begin{align*} \begin{cases} \frac{S^{3}\left (b x\right )}{3 b} & \text{for}\: b \neq 0 \\0 & \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{\rm fresnels}\left (b x\right )^{2} \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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