Optimal. Leaf size=55 \[ \frac {2 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b}+x S(b x)^2-\frac {S\left (\sqrt {2} b x\right )}{\sqrt {2} \pi b} \]
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Rubi [A] time = 0.04, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6420, 12, 6452, 3351} \[ \frac {2 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b}+x S(b x)^2-\frac {S\left (\sqrt {2} b x\right )}{\sqrt {2} \pi b} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3351
Rule 6420
Rule 6452
Rubi steps
\begin {align*} \int S(b x)^2 \, dx &=x S(b x)^2-2 \int b x S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx\\ &=x S(b x)^2-(2 b) \int x S(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx\\ &=\frac {2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b \pi }+x S(b x)^2-\frac {\int \sin \left (b^2 \pi x^2\right ) \, dx}{\pi }\\ &=\frac {2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) S(b x)}{b \pi }+x S(b x)^2-\frac {S\left (\sqrt {2} b x\right )}{\sqrt {2} b \pi }\\ \end {align*}
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Mathematica [A] time = 0.02, size = 55, normalized size = 1.00 \[ \frac {2 S(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b}+x S(b x)^2-\frac {S\left (\sqrt {2} b x\right )}{\sqrt {2} \pi b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\rm fresnels}\left (b x\right )^{2}, x\right ) \]
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 \[ \int {\rm fresnels}\left (b x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 49, normalized size = 0.89 \[ \frac {b x \mathrm {S}\left (b x \right )^{2}+\frac {2 \,\mathrm {S}\left (b x \right ) \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }-\frac {\sqrt {2}\, \mathrm {S}\left (b x \sqrt {2}\right )}{2 \pi }}{b} \]
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 \[ \int {\rm fresnels}\left (b x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\mathrm {FresnelS}\left (b\,x\right )}^2 \,d x \]
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 \[ \int S^{2}\left (b x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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