Optimal. Leaf size=265 \[ -\frac{5 i x^2 \text{HypergeometricPFQ}\left (\{1,1\},\left \{\frac{3}{2},2\right \},-\frac{1}{2} i \pi b^2 x^2\right )}{8 \pi ^3 b^4}+\frac{5 i x^2 \text{HypergeometricPFQ}\left (\{1,1\},\left \{\frac{3}{2},2\right \},\frac{1}{2} i \pi b^2 x^2\right )}{8 \pi ^3 b^4}+\frac{5 \text{FresnelC}(b x) S(b x)}{2 \pi ^3 b^6}-\frac{5 x^3 S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{3 \pi ^2 b^3}+\frac{x^5 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{3 \pi b}-\frac{5 x S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi ^3 b^5}+\frac{5 x^4}{24 \pi ^2 b^2}-\frac{7 x^2 \sin \left (\pi b^2 x^2\right )}{12 \pi ^3 b^4}+\frac{x^4 \cos \left (\pi b^2 x^2\right )}{12 \pi ^2 b^2}-\frac{11 \cos \left (\pi b^2 x^2\right )}{6 \pi ^4 b^6}+\frac{1}{6} x^6 S(b x)^2 \]
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Rubi [A] time = 0.300067, antiderivative size = 265, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 9, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.9, Rules used = {6430, 6454, 6462, 3379, 3309, 30, 3296, 2638, 6446} \[ -\frac{5 i x^2 \, _2F_2\left (1,1;\frac{3}{2},2;-\frac{1}{2} i b^2 \pi x^2\right )}{8 \pi ^3 b^4}+\frac{5 i x^2 \, _2F_2\left (1,1;\frac{3}{2},2;\frac{1}{2} i b^2 \pi x^2\right )}{8 \pi ^3 b^4}+\frac{5 \text{FresnelC}(b x) S(b x)}{2 \pi ^3 b^6}-\frac{5 x^3 S(b x) \sin \left (\frac{1}{2} \pi b^2 x^2\right )}{3 \pi ^2 b^3}+\frac{x^5 S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{3 \pi b}-\frac{5 x S(b x) \cos \left (\frac{1}{2} \pi b^2 x^2\right )}{\pi ^3 b^5}+\frac{5 x^4}{24 \pi ^2 b^2}-\frac{7 x^2 \sin \left (\pi b^2 x^2\right )}{12 \pi ^3 b^4}+\frac{x^4 \cos \left (\pi b^2 x^2\right )}{12 \pi ^2 b^2}-\frac{11 \cos \left (\pi b^2 x^2\right )}{6 \pi ^4 b^6}+\frac{1}{6} x^6 S(b x)^2 \]
Antiderivative was successfully verified.
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Rule 6430
Rule 6454
Rule 6462
Rule 3379
Rule 3309
Rule 30
Rule 3296
Rule 2638
Rule 6446
Rubi steps
\begin{align*} \int x^5 S(b x)^2 \, dx &=\frac{1}{6} x^6 S(b x)^2-\frac{1}{3} b \int x^6 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right ) \, dx\\ &=\frac{x^5 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{3 b \pi }+\frac{1}{6} x^6 S(b x)^2-\frac{\int x^5 \sin \left (b^2 \pi x^2\right ) \, dx}{6 \pi }-\frac{5 \int x^4 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x) \, dx}{3 b \pi }\\ &=\frac{x^5 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{3 b \pi }+\frac{1}{6} x^6 S(b x)^2-\frac{5 x^3 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{3 b^3 \pi ^2}+\frac{5 \int x^2 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right ) \, dx}{b^3 \pi ^2}+\frac{5 \int x^3 \sin ^2\left (\frac{1}{2} b^2 \pi x^2\right ) \, dx}{3 b^2 \pi ^2}-\frac{\operatorname{Subst}\left (\int x^2 \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{12 \pi }\\ &=\frac{x^4 \cos \left (b^2 \pi x^2\right )}{12 b^2 \pi ^2}-\frac{5 x \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{b^5 \pi ^3}+\frac{x^5 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{3 b \pi }+\frac{1}{6} x^6 S(b x)^2-\frac{5 x^3 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{3 b^3 \pi ^2}+\frac{5 \int \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x) \, dx}{b^5 \pi ^3}+\frac{5 \int x \sin \left (b^2 \pi x^2\right ) \, dx}{2 b^4 \pi ^3}-\frac{\operatorname{Subst}\left (\int x \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{6 b^2 \pi ^2}+\frac{5 \operatorname{Subst}\left (\int x \sin ^2\left (\frac{1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )}{6 b^2 \pi ^2}\\ &=\frac{x^4 \cos \left (b^2 \pi x^2\right )}{12 b^2 \pi ^2}-\frac{5 x \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{b^5 \pi ^3}+\frac{x^5 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{3 b \pi }+\frac{5 C(b x) S(b x)}{2 b^6 \pi ^3}+\frac{1}{6} x^6 S(b x)^2-\frac{5 i x^2 \, _2F_2\left (1,1;\frac{3}{2},2;-\frac{1}{2} i b^2 \pi x^2\right )}{8 b^4 \pi ^3}+\frac{5 i x^2 \, _2F_2\left (1,1;\frac{3}{2},2;\frac{1}{2} i b^2 \pi x^2\right )}{8 b^4 \pi ^3}-\frac{5 x^3 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{3 b^3 \pi ^2}-\frac{x^2 \sin \left (b^2 \pi x^2\right )}{6 b^4 \pi ^3}+\frac{\operatorname{Subst}\left (\int \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{6 b^4 \pi ^3}+\frac{5 \operatorname{Subst}\left (\int \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{4 b^4 \pi ^3}+\frac{5 \operatorname{Subst}\left (\int x \, dx,x,x^2\right )}{12 b^2 \pi ^2}-\frac{5 \operatorname{Subst}\left (\int x \cos \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{12 b^2 \pi ^2}\\ &=\frac{5 x^4}{24 b^2 \pi ^2}-\frac{17 \cos \left (b^2 \pi x^2\right )}{12 b^6 \pi ^4}+\frac{x^4 \cos \left (b^2 \pi x^2\right )}{12 b^2 \pi ^2}-\frac{5 x \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{b^5 \pi ^3}+\frac{x^5 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{3 b \pi }+\frac{5 C(b x) S(b x)}{2 b^6 \pi ^3}+\frac{1}{6} x^6 S(b x)^2-\frac{5 i x^2 \, _2F_2\left (1,1;\frac{3}{2},2;-\frac{1}{2} i b^2 \pi x^2\right )}{8 b^4 \pi ^3}+\frac{5 i x^2 \, _2F_2\left (1,1;\frac{3}{2},2;\frac{1}{2} i b^2 \pi x^2\right )}{8 b^4 \pi ^3}-\frac{5 x^3 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{3 b^3 \pi ^2}-\frac{7 x^2 \sin \left (b^2 \pi x^2\right )}{12 b^4 \pi ^3}+\frac{5 \operatorname{Subst}\left (\int \sin \left (b^2 \pi x\right ) \, dx,x,x^2\right )}{12 b^4 \pi ^3}\\ &=\frac{5 x^4}{24 b^2 \pi ^2}-\frac{11 \cos \left (b^2 \pi x^2\right )}{6 b^6 \pi ^4}+\frac{x^4 \cos \left (b^2 \pi x^2\right )}{12 b^2 \pi ^2}-\frac{5 x \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{b^5 \pi ^3}+\frac{x^5 \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x)}{3 b \pi }+\frac{5 C(b x) S(b x)}{2 b^6 \pi ^3}+\frac{1}{6} x^6 S(b x)^2-\frac{5 i x^2 \, _2F_2\left (1,1;\frac{3}{2},2;-\frac{1}{2} i b^2 \pi x^2\right )}{8 b^4 \pi ^3}+\frac{5 i x^2 \, _2F_2\left (1,1;\frac{3}{2},2;\frac{1}{2} i b^2 \pi x^2\right )}{8 b^4 \pi ^3}-\frac{5 x^3 S(b x) \sin \left (\frac{1}{2} b^2 \pi x^2\right )}{3 b^3 \pi ^2}-\frac{7 x^2 \sin \left (b^2 \pi x^2\right )}{12 b^4 \pi ^3}\\ \end{align*}
Mathematica [F] time = 0.207061, size = 0, normalized size = 0. \[ \int x^5 S(b x)^2 \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.053, size = 0, normalized size = 0. \begin{align*} \int{x}^{5} \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^{5}{\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^{5}{\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^{5} 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^{5}{\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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