Optimal. Leaf size=55 \[ \frac{2 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi b}+x S(b x)^2-\frac{S\left (\sqrt{2} b x\right )}{\sqrt{2} \pi b} \]
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Rubi [A] time = 0.0383974, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {6420, 12, 6452, 3351} \[ \frac{2 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi b}+x S(b x)^2-\frac{S\left (\sqrt{2} b x\right )}{\sqrt{2} \pi b} \]
Antiderivative was successfully verified.
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Rule 6420
Rule 12
Rule 6452
Rule 3351
Rubi steps
\begin{align*} \int S(b x)^2 \, dx &=x S(b x)^2-2 \int b x S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right ) \, dx\\ &=x S(b x)^2-(2 b) \int x S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right ) \, dx\\ &=\frac{2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{b \pi }+x S(b x)^2-\frac{\int \sin \left (b^2 \pi x^2\right ) \, dx}{\pi }\\ &=\frac{2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{b \pi }+x S(b x)^2-\frac{S\left (\sqrt{2} b x\right )}{\sqrt{2} b \pi }\\ \end{align*}
Mathematica [A] time = 0.0093133, size = 55, normalized size = 1. \[ \frac{2 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi b}+x S(b x)^2-\frac{S\left (\sqrt{2} b x\right )}{\sqrt{2} \pi b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 49, normalized size = 0.9 \begin{align*}{\frac{1}{b} \left ( bx \left ({\it FresnelS} \left ( bx \right ) \right ) ^{2}+2\,{\frac{\cos \left ( 1/2\,{b}^{2}\pi \,{x}^{2} \right ){\it FresnelS} \left ( bx \right ) }{\pi }}-{\frac{\sqrt{2}{\it FresnelS} \left ( bx\sqrt{2} \right ) }{2\,\pi }} \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 )^{2}\,{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}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int S^{2}\left (b x\right )\, dx \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}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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