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 = 0.0063385, antiderivative size = 36, normalized size of antiderivative = 1., 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*}
Mathematica [B] time = 0.0286256, 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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Maple [A] time = 0.046, size = 33, normalized size = 0.9 \begin{align*}{\frac{1}{b} \left ( \left ( bx+a \right ){\it FresnelS} \left ( bx+a \right ) +{\frac{1}{\pi }\cos \left ({\frac{\pi \, \left ( bx+a \right ) ^{2}}{2}} \right ) } \right ) } \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{\rm fresnels}\left (b x + a\right )\,{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 ({\rm fresnels}\left (b x + a\right ), 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 S\left (a + 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{\rm fresnels}\left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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