Optimal. Leaf size=9 \[ \frac {\log (S(b x))}{b} \]
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Rubi [A] time = 0.01, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {6440, 29} \[ \frac {\log (S(b x))}{b} \]
Antiderivative was successfully verified.
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Rule 29
Rule 6440
Rubi steps
\begin {align*} \int \frac {\sin \left (\frac {1}{2} b^2 \pi x^2\right )}{S(b x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,S(b x)\right )}{b}\\ &=\frac {\log (S(b x))}{b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 9, normalized size = 1.00 \[ \frac {\log (S(b x))}{b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{{\rm fresnels}\left (b x\right )}, x\right ) \]
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 \[ \int \frac {\sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{{\rm fresnels}\left (b x\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 10, normalized size = 1.11 \[ \frac {\ln \left (\mathrm {S}\left (b x \right )\right )}{b} \]
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 \[ \int \frac {\sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{{\rm fresnels}\left (b x\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.11 \[ \int \frac {\sin \left (\frac {\Pi \,b^2\,x^2}{2}\right )}{\mathrm {FresnelS}\left (b\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.22, size = 8, normalized size = 0.89 \[ \begin {cases} \frac {\log {\left (S\left (b x\right ) \right )}}{b} & \text {for}\: b \neq 0 \\\text {NaN} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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