Optimal. Leaf size=74 \[ -\frac{x \text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac{\text{FresnelC}(b x)^2}{2 \pi b^3}+\frac{\sin \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3}+\frac{x^2}{4 \pi b} \]
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Rubi [A] time = 0.0570508, antiderivative size = 74, 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 = {6463, 6441, 30, 3380, 2634} \[ -\frac{x \text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac{\text{FresnelC}(b x)^2}{2 \pi b^3}+\frac{\sin \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3}+\frac{x^2}{4 \pi b} \]
Antiderivative was successfully verified.
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Rule 6463
Rule 6441
Rule 30
Rule 3380
Rule 2634
Rubi steps
\begin{align*} \int x^2 C(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right ) \, dx &=-\frac{x \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{b^2 \pi }+\frac{\int \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x) \, dx}{b^2 \pi }+\frac{\int x \cos ^2\left (\frac{1}{2} b^2 \pi x^2\right ) \, dx}{b \pi }\\ &=-\frac{x \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{b^2 \pi }+\frac{\operatorname{Subst}(\int x \, dx,x,C(b x))}{b^3 \pi }+\frac{\operatorname{Subst}\left (\int \cos ^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{x \cos \left (\frac{1}{2} b^2 \pi x^2\right ) C(b x)}{b^2 \pi }+\frac{C(b x)^2}{2 b^3 \pi }+\frac{\sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}\\ \end{align*}
Mathematica [A] time = 0.007526, size = 74, normalized size = 1. \[ -\frac{x \text{FresnelC}(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac{\text{FresnelC}(b x)^2}{2 \pi b^3}+\frac{\sin \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3}+\frac{x^2}{4 \pi b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.086, size = 0, normalized size = 0. \begin{align*} \int{x}^{2}{\it FresnelC} \left ( bx \right ) \sin \left ({\frac{{b}^{2}\pi \,{x}^{2}}{2}} \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}{\rm fresnelc}\left (b x\right ) \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 (x^{2}{\rm fresnelc}\left (b x\right ) \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 = 4.37083, size = 114, normalized size = 1.54 \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 \cos{\left (\frac{\pi b^{2} x^{2}}{2} \right )} C\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{C^{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}{\rm fresnelc}\left (b x\right ) \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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