Optimal. Leaf size=124 \[ -\frac{5 \text{FresnelC}\left (\sqrt{2} b x\right )}{6 \sqrt{2} \pi ^2 b^3}-\frac{4 S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{3 \pi ^2 b^3}+\frac{2 x^2 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{3 \pi b}+\frac{x \cos \left (\pi b^2 x^2\right )}{6 \pi ^2 b^2}+\frac{2 x}{3 \pi ^2 b^2}+\frac{1}{3} x^3 S(b x)^2 \]
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Rubi [A] time = 0.110876, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {6430, 6454, 6460, 3357, 3352, 3385} \[ -\frac{5 \text{FresnelC}\left (\sqrt{2} b x\right )}{6 \sqrt{2} \pi ^2 b^3}-\frac{4 S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{3 \pi ^2 b^3}+\frac{2 x^2 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{3 \pi b}+\frac{x \cos \left (\pi b^2 x^2\right )}{6 \pi ^2 b^2}+\frac{2 x}{3 \pi ^2 b^2}+\frac{1}{3} x^3 S(b x)^2 \]
Antiderivative was successfully verified.
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Rule 6430
Rule 6454
Rule 6460
Rule 3357
Rule 3352
Rule 3385
Rubi steps
\begin{align*} \int x^2 S(b x)^2 \, dx &=\frac{1}{3} x^3 S(b x)^2-\frac{1}{3} (2 b) \int x^3 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right ) \, dx\\ &=\frac{2 x^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{3 b \pi }+\frac{1}{3} x^3 S(b x)^2-\frac{\int x^2 \sin \left (b^2 \pi x^2\right ) \, dx}{3 \pi }-\frac{4 \int x \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x) \, dx}{3 b \pi }\\ &=\frac{x \cos \left (b^2 \pi x^2\right )}{6 b^2 \pi ^2}+\frac{2 x^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{3 b \pi }+\frac{1}{3} x^3 S(b x)^2-\frac{4 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{3 b^3 \pi ^2}-\frac{\int \cos \left (b^2 \pi x^2\right ) \, dx}{6 b^2 \pi ^2}+\frac{4 \int \sin ^2\left (\frac{1}{2} b^2 \pi x^2\right ) \, dx}{3 b^2 \pi ^2}\\ &=\frac{x \cos \left (b^2 \pi x^2\right )}{6 b^2 \pi ^2}-\frac{C\left (\sqrt{2} b x\right )}{6 \sqrt{2} b^3 \pi ^2}+\frac{2 x^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{3 b \pi }+\frac{1}{3} x^3 S(b x)^2-\frac{4 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{3 b^3 \pi ^2}+\frac{4 \int \left (\frac{1}{2}-\frac{1}{2} \cos \left (b^2 \pi x^2\right )\right ) \, dx}{3 b^2 \pi ^2}\\ &=\frac{2 x}{3 b^2 \pi ^2}+\frac{x \cos \left (b^2 \pi x^2\right )}{6 b^2 \pi ^2}-\frac{C\left (\sqrt{2} b x\right )}{6 \sqrt{2} b^3 \pi ^2}+\frac{2 x^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{3 b \pi }+\frac{1}{3} x^3 S(b x)^2-\frac{4 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{3 b^3 \pi ^2}-\frac{2 \int \cos \left (b^2 \pi x^2\right ) \, dx}{3 b^2 \pi ^2}\\ &=\frac{2 x}{3 b^2 \pi ^2}+\frac{x \cos \left (b^2 \pi x^2\right )}{6 b^2 \pi ^2}-\frac{C\left (\sqrt{2} b x\right )}{6 \sqrt{2} b^3 \pi ^2}-\frac{\sqrt{2} C\left (\sqrt{2} b x\right )}{3 b^3 \pi ^2}+\frac{2 x^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{3 b \pi }+\frac{1}{3} x^3 S(b x)^2-\frac{4 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{3 b^3 \pi ^2}\\ \end{align*}
Mathematica [A] time = 0.107693, size = 100, normalized size = 0.81 \[ \frac{4 \pi ^2 b^3 x^3 S(b x)^2+8 S(b x) \left (\pi b^2 x^2 \cos \left (\frac{1}{2} \pi b^2 x^2\right )-2 \sin \left (\frac{1}{2} \pi b^2 x^2\right )\right )+2 b x \left (\cos \left (\pi b^2 x^2\right )+4\right )-5 \sqrt{2} \text{FresnelC}\left (\sqrt{2} b x\right )}{12 \pi ^2 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.072, size = 122, normalized size = 1. \begin{align*}{\frac{1}{{b}^{3}} \left ({\frac{{b}^{3}{x}^{3} \left ({\it FresnelS} \left ( bx \right ) \right ) ^{2}}{3}}-2\,{\it FresnelS} \left ( bx \right ) \left ( -1/3\,{\frac{{b}^{2}{x}^{2}\cos \left ( 1/2\,{b}^{2}\pi \,{x}^{2} \right ) }{\pi }}+2/3\,{\frac{\sin \left ( 1/2\,{b}^{2}\pi \,{x}^{2} \right ) }{{\pi }^{2}}} \right ) +{\frac{2\,bx}{3\,{\pi }^{2}}}-{\frac{\sqrt{2}{\it FresnelC} \left ( bx\sqrt{2} \right ) }{3\,{\pi }^{2}}}-{\frac{1}{3\,\pi } \left ( -{\frac{bx\cos \left ({b}^{2}\pi \,{x}^{2} \right ) }{2\,\pi }}+{\frac{\sqrt{2}{\it FresnelC} \left ( bx\sqrt{2} \right ) }{4\,\pi }} \right ) } \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 x^{2}{\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 (x^{2}{\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 x^{2} 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 x^{2}{\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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