Optimal. Leaf size=140 \[ -\frac{3 x S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{2 \pi ^2 b^3}+\frac{x^3 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{2 \pi b}+\frac{3 S(b x)^2}{4 \pi ^2 b^4}+\frac{3 x^2}{8 \pi ^2 b^2}-\frac{\sin \left (\pi b^2 x^2\right )}{2 \pi ^3 b^4}+\frac{x^2 \cos \left (\pi b^2 x^2\right )}{8 \pi ^2 b^2}+\frac{1}{4} x^4 S(b x)^2 \]
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Rubi [A] time = 0.150878, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.9, Rules used = {6430, 6454, 6462, 3379, 2634, 6440, 30, 3296, 2637} \[ -\frac{3 x S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{2 \pi ^2 b^3}+\frac{x^3 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{2 \pi b}+\frac{3 S(b x)^2}{4 \pi ^2 b^4}+\frac{3 x^2}{8 \pi ^2 b^2}-\frac{\sin \left (\pi b^2 x^2\right )}{2 \pi ^3 b^4}+\frac{x^2 \cos \left (\pi b^2 x^2\right )}{8 \pi ^2 b^2}+\frac{1}{4} x^4 S(b x)^2 \]
Antiderivative was successfully verified.
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Rule 6430
Rule 6454
Rule 6462
Rule 3379
Rule 2634
Rule 6440
Rule 30
Rule 3296
Rule 2637
Rubi steps
\begin{align*} \int x^3 S(b x)^2 \, dx &=\frac{1}{4} x^4 S(b x)^2-\frac{1}{2} b \int x^4 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right ) \, dx\\ &=\frac{x^3 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{2 b \pi }+\frac{1}{4} x^4 S(b x)^2-\frac{\int x^3 \sin \left (b^2 \pi x^2\right ) \, dx}{4 \pi }-\frac{3 \int x^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x) \, dx}{2 b \pi }\\ &=\frac{x^3 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{2 b \pi }+\frac{1}{4} x^4 S(b x)^2-\frac{3 x S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{2 b^3 \pi ^2}+\frac{3 \int S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right ) \, dx}{2 b^3 \pi ^2}+\frac{3 \int x \sin ^2\left (\frac{1}{2} b^2 \pi x^2\right ) \, dx}{2 b^2 \pi ^2}-\frac{\operatorname{Subst}\left (\int x \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{8 \pi }\\ &=\frac{x^2 \cos \left (b^2 \pi x^2\right )}{8 b^2 \pi ^2}+\frac{x^3 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{2 b \pi }+\frac{1}{4} x^4 S(b x)^2-\frac{3 x S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{2 b^3 \pi ^2}+\frac{3 \operatorname{Subst}(\int x \, dx,x,S(b x))}{2 b^4 \pi ^2}-\frac{\operatorname{Subst}\left (\int \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{8 b^2 \pi ^2}+\frac{3 \operatorname{Subst}\left (\int \sin ^2\left (\frac{1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b^2 \pi ^2}\\ &=\frac{3 x^2}{8 b^2 \pi ^2}+\frac{x^2 \cos \left (b^2 \pi x^2\right )}{8 b^2 \pi ^2}+\frac{x^3 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{2 b \pi }+\frac{3 S(b x)^2}{4 b^4 \pi ^2}+\frac{1}{4} x^4 S(b x)^2-\frac{3 x S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{2 b^3 \pi ^2}-\frac{\sin \left (b^2 \pi x^2\right )}{2 b^4 \pi ^3}\\ \end{align*}
Mathematica [A] time = 0.0055243, size = 140, normalized size = 1. \[ -\frac{3 x S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{2 \pi ^2 b^3}+\frac{x^3 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{2 \pi b}+\frac{3 S(b x)^2}{4 \pi ^2 b^4}+\frac{3 x^2}{8 \pi ^2 b^2}-\frac{\sin \left (\pi b^2 x^2\right )}{2 \pi ^3 b^4}+\frac{x^2 \cos \left (\pi b^2 x^2\right )}{8 \pi ^2 b^2}+\frac{1}{4} x^4 S(b x)^2 \]
Antiderivative was successfully verified.
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Maple [F] time = 0.051, size = 0, normalized size = 0. \begin{align*} \int{x}^{3} \left ({\it FresnelS} \left ( bx \right ) \right ) ^{2}\, 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^{3}{\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^{3}{\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^{3} 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^{3}{\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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