Optimal. Leaf size=36 \[ \frac {(a+b x) S(a+b x)}{b}+\frac {\cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{\pi b} \]
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Rubi [A] time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6418} \[ \frac {(a+b x) S(a+b x)}{b}+\frac {\cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{\pi b} \]
Antiderivative was successfully verified.
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Rule 6418
Rubi steps
\begin {align*} \int S(a+b x) \, dx &=\frac {\cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{b \pi }+\frac {(a+b x) S(a+b x)}{b}\\ \end {align*}
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Mathematica [B] time = 0.04, size = 89, normalized size = 2.47 \[ -\frac {\sin \left (\frac {\pi a^2}{2}\right ) \sin \left (\pi a b x+\frac {1}{2} \pi b^2 x^2\right )}{\pi b}+\frac {\cos \left (\frac {\pi a^2}{2}\right ) \cos \left (\pi a b x+\frac {1}{2} \pi b^2 x^2\right )}{\pi b}+x S(a+b x)+\frac {a S(a+b x)}{b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\rm fresnels}\left (b x + a\right ), x\right ) \]
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 \[ \int {\rm fresnels}\left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 33, normalized size = 0.92 \[ \frac {\left (b x +a \right ) \mathrm {S}\left (b x +a \right )+\frac {\cos \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }}{b} \]
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 \[ \int {\rm fresnels}\left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \mathrm {FresnelS}\left (a+b\,x\right ) \,d x \]
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 \[ \int S\left (a + b x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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