Optimal. Leaf size=140 \[ \frac {3 S(b x)^2}{4 \pi ^2 b^4}+\frac {x^3 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{2 \pi b}+\frac {3 x^2}{8 \pi ^2 b^2}+\frac {x^2 \cos \left (\pi b^2 x^2\right )}{8 \pi ^2 b^2}-\frac {\sin \left (\pi b^2 x^2\right )}{2 \pi ^3 b^4}-\frac {3 x S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{2 \pi ^2 b^3}+\frac {1}{4} x^4 S(b x)^2 \]
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Rubi [A] time = 0.15, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, 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 30
Rule 2634
Rule 2637
Rule 3296
Rule 3379
Rule 6430
Rule 6440
Rule 6454
Rule 6462
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*}
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Mathematica [A] time = 0.01, size = 140, normalized size = 1.00 \[ \frac {3 S(b x)^2}{4 \pi ^2 b^4}+\frac {x^3 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{2 \pi b}+\frac {3 x^2}{8 \pi ^2 b^2}+\frac {x^2 \cos \left (\pi b^2 x^2\right )}{8 \pi ^2 b^2}-\frac {\sin \left (\pi b^2 x^2\right )}{2 \pi ^3 b^4}-\frac {3 x S(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{2 \pi ^2 b^3}+\frac {1}{4} x^4 S(b x)^2 \]
Antiderivative was successfully verified.
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fricas [F] time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{3} {\rm fresnels}\left (b x\right )^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} {\rm fresnels}\left (b x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[ \int x^{3} \mathrm {S}\left (b x \right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} {\rm fresnels}\left (b x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^3\,{\mathrm {FresnelS}\left (b\,x\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} S^{2}\left (b x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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