Optimal. Leaf size=73 \[ \frac{x S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac{S(b x)^2}{2 \pi b^3}+\frac{\sin \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3}-\frac{x^2}{4 \pi b} \]
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Rubi [A] time = 0.0547892, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6462, 3379, 2634, 6440, 30} \[ \frac{x S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac{S(b x)^2}{2 \pi b^3}+\frac{\sin \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3}-\frac{x^2}{4 \pi b} \]
Antiderivative was successfully verified.
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Rule 6462
Rule 3379
Rule 2634
Rule 6440
Rule 30
Rubi steps
\begin{align*} \int x^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x) \, dx &=\frac{x S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac{\int S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right ) \, dx}{b^2 \pi }-\frac{\int x \sin ^2\left (\frac{1}{2} b^2 \pi x^2\right ) \, dx}{b \pi }\\ &=\frac{x S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac{\operatorname{Subst}(\int x \, dx,x,S(b x))}{b^3 \pi }-\frac{\operatorname{Subst}\left (\int \sin ^2\left (\frac{1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b \pi }\\ &=-\frac{x^2}{4 b \pi }-\frac{S(b x)^2}{2 b^3 \pi }+\frac{x S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^2 \pi }+\frac{\sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}\\ \end{align*}
Mathematica [A] time = 0.0045326, size = 73, normalized size = 1. \[ \frac{x S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac{S(b x)^2}{2 \pi b^3}+\frac{\sin \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3}-\frac{x^2}{4 \pi b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.086, size = 0, normalized size = 0. \begin{align*} \int{x}^{2}\cos \left ({\frac{{b}^{2}\pi \,{x}^{2}}{2}} \right ){\it FresnelS} \left ( bx \right ) \, dx \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 x^{2} \cos \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 (x^{2} \cos \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 = 4.71556, size = 114, normalized size = 1.56 \begin{align*} \begin{cases} - \frac{x^{2} \sin ^{2}{\left (\frac{\pi b^{2} x^{2}}{2} \right )}}{4 \pi b} - \frac{x^{2} \cos ^{2}{\left (\frac{\pi b^{2} x^{2}}{2} \right )}}{4 \pi b} + \frac{x \sin{\left (\frac{\pi b^{2} x^{2}}{2} \right )} S\left (b x\right )}{\pi b^{2}} + \frac{\sin{\left (\frac{\pi b^{2} x^{2}}{2} \right )} \cos{\left (\frac{\pi b^{2} x^{2}}{2} \right )}}{2 \pi ^{2} b^{3}} - \frac{S^{2}\left (b x\right )}{2 \pi b^{3}} & \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 x^{2} \cos \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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