Optimal. Leaf size=232 \[ \frac{7 x^5 S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}-\frac{105 x S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi ^4 b^8}-\frac{x^7 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac{35 x^3 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi ^3 b^6}+\frac{105 S(b x)^2}{2 \pi ^4 b^9}-\frac{7 x^6}{12 \pi ^2 b^3}+\frac{105 x^2}{4 \pi ^4 b^7}+\frac{5 x^4 \sin \left (\pi b^2 x^2\right )}{2 \pi ^3 b^5}-\frac{40 \sin \left (\pi b^2 x^2\right )}{\pi ^5 b^9}-\frac{x^6 \cos \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3}+\frac{55 x^2 \cos \left (\pi b^2 x^2\right )}{4 \pi ^4 b^7} \]
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Rubi [A] time = 0.384108, antiderivative size = 232, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45, Rules used = {6454, 6462, 3379, 3309, 30, 3296, 2637, 2634, 6440} \[ \frac{7 x^5 S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}-\frac{105 x S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi ^4 b^8}-\frac{x^7 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac{35 x^3 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi ^3 b^6}+\frac{105 S(b x)^2}{2 \pi ^4 b^9}-\frac{7 x^6}{12 \pi ^2 b^3}+\frac{105 x^2}{4 \pi ^4 b^7}+\frac{5 x^4 \sin \left (\pi b^2 x^2\right )}{2 \pi ^3 b^5}-\frac{40 \sin \left (\pi b^2 x^2\right )}{\pi ^5 b^9}-\frac{x^6 \cos \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3}+\frac{55 x^2 \cos \left (\pi b^2 x^2\right )}{4 \pi ^4 b^7} \]
Antiderivative was successfully verified.
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Rule 6454
Rule 6462
Rule 3379
Rule 3309
Rule 30
Rule 3296
Rule 2637
Rule 2634
Rule 6440
Rubi steps
\begin{align*} \int x^8 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right ) \, dx &=-\frac{x^7 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{b^2 \pi }+\frac{7 \int x^6 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x) \, dx}{b^2 \pi }+\frac{\int x^7 \sin \left (b^2 \pi x^2\right ) \, dx}{2 b \pi }\\ &=-\frac{x^7 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{b^2 \pi }+\frac{7 x^5 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}-\frac{35 \int x^4 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right ) \, dx}{b^4 \pi ^2}-\frac{7 \int x^5 \sin ^2\left (\frac{1}{2} b^2 \pi x^2\right ) \, dx}{b^3 \pi ^2}+\frac{\operatorname{Subst}\left (\int x^3 \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b \pi }\\ &=-\frac{x^6 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac{35 x^3 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{b^6 \pi ^3}-\frac{x^7 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{b^2 \pi }+\frac{7 x^5 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}-\frac{105 \int x^2 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x) \, dx}{b^6 \pi ^3}-\frac{35 \int x^3 \sin \left (b^2 \pi x^2\right ) \, dx}{2 b^5 \pi ^3}+\frac{3 \operatorname{Subst}\left (\int x^2 \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b^3 \pi ^2}-\frac{7 \operatorname{Subst}\left (\int x^2 \sin ^2\left (\frac{1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b^3 \pi ^2}\\ &=-\frac{x^6 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac{35 x^3 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{b^6 \pi ^3}-\frac{x^7 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{b^2 \pi }-\frac{105 x S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^8 \pi ^4}+\frac{7 x^5 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}+\frac{3 x^4 \sin \left (b^2 \pi x^2\right )}{4 b^5 \pi ^3}+\frac{105 \int S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right ) \, dx}{b^8 \pi ^4}+\frac{105 \int x \sin ^2\left (\frac{1}{2} b^2 \pi x^2\right ) \, dx}{b^7 \pi ^4}-\frac{3 \operatorname{Subst}\left (\int x \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b^5 \pi ^3}-\frac{35 \operatorname{Subst}\left (\int x \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b^5 \pi ^3}-\frac{7 \operatorname{Subst}\left (\int x^2 \, dx,x,x^2\right )}{4 b^3 \pi ^2}+\frac{7 \operatorname{Subst}\left (\int x^2 \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b^3 \pi ^2}\\ &=-\frac{7 x^6}{12 b^3 \pi ^2}+\frac{41 x^2 \cos \left (b^2 \pi x^2\right )}{4 b^7 \pi ^4}-\frac{x^6 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac{35 x^3 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{b^6 \pi ^3}-\frac{x^7 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{b^2 \pi }-\frac{105 x S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^8 \pi ^4}+\frac{7 x^5 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}+\frac{5 x^4 \sin \left (b^2 \pi x^2\right )}{2 b^5 \pi ^3}+\frac{105 \operatorname{Subst}(\int x \, dx,x,S(b x))}{b^9 \pi ^4}-\frac{3 \operatorname{Subst}\left (\int \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b^7 \pi ^4}-\frac{35 \operatorname{Subst}\left (\int \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b^7 \pi ^4}+\frac{105 \operatorname{Subst}\left (\int \sin ^2\left (\frac{1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b^7 \pi ^4}-\frac{7 \operatorname{Subst}\left (\int x \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b^5 \pi ^3}\\ &=\frac{105 x^2}{4 b^7 \pi ^4}-\frac{7 x^6}{12 b^3 \pi ^2}+\frac{55 x^2 \cos \left (b^2 \pi x^2\right )}{4 b^7 \pi ^4}-\frac{x^6 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac{35 x^3 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{b^6 \pi ^3}-\frac{x^7 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{b^2 \pi }+\frac{105 S(b x)^2}{2 b^9 \pi ^4}-\frac{105 x S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^8 \pi ^4}+\frac{7 x^5 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}-\frac{73 \sin \left (b^2 \pi x^2\right )}{2 b^9 \pi ^5}+\frac{5 x^4 \sin \left (b^2 \pi x^2\right )}{2 b^5 \pi ^3}-\frac{7 \operatorname{Subst}\left (\int \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{2 b^7 \pi ^4}\\ &=\frac{105 x^2}{4 b^7 \pi ^4}-\frac{7 x^6}{12 b^3 \pi ^2}+\frac{55 x^2 \cos \left (b^2 \pi x^2\right )}{4 b^7 \pi ^4}-\frac{x^6 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac{35 x^3 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{b^6 \pi ^3}-\frac{x^7 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{b^2 \pi }+\frac{105 S(b x)^2}{2 b^9 \pi ^4}-\frac{105 x S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^8 \pi ^4}+\frac{7 x^5 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}-\frac{40 \sin \left (b^2 \pi x^2\right )}{b^9 \pi ^5}+\frac{5 x^4 \sin \left (b^2 \pi x^2\right )}{2 b^5 \pi ^3}\\ \end{align*}
Mathematica [A] time = 0.0130934, size = 232, normalized size = 1. \[ \frac{7 x^5 S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}-\frac{105 x S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi ^4 b^8}-\frac{x^7 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac{35 x^3 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi ^3 b^6}+\frac{105 S(b x)^2}{2 \pi ^4 b^9}-\frac{7 x^6}{12 \pi ^2 b^3}+\frac{105 x^2}{4 \pi ^4 b^7}+\frac{5 x^4 \sin \left (\pi b^2 x^2\right )}{2 \pi ^3 b^5}-\frac{40 \sin \left (\pi b^2 x^2\right )}{\pi ^5 b^9}-\frac{x^6 \cos \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3}+\frac{55 x^2 \cos \left (\pi b^2 x^2\right )}{4 \pi ^4 b^7} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.077, size = 0, normalized size = 0. \begin{align*} \int{x}^{8}{\it FresnelS} \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^{8}{\rm fresnels}\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^{8}{\rm fresnels}\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 [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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^{8}{\rm fresnels}\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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