Optimal. Leaf size=37 \[ \frac{(a+b x) \text{FresnelC}(a+b x)}{b}-\frac{\sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{\pi b} \]
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Rubi [A] time = 0.0067759, antiderivative size = 37, 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 = {6419} \[ \frac{(a+b x) \text{FresnelC}(a+b x)}{b}-\frac{\sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{\pi b} \]
Antiderivative was successfully verified.
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Rule 6419
Rubi steps
\begin{align*} \int C(a+b x) \, dx &=\frac{(a+b x) C(a+b x)}{b}-\frac{\sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{b \pi }\\ \end{align*}
Mathematica [B] time = 0.0290167, size = 90, normalized size = 2.43 \[ -\frac{\sin \left (\frac{\pi a^2}{2}\right ) \cos \left (\pi a b x+\frac{1}{2} \pi b^2 x^2\right )}{\pi b}-\frac{\cos \left (\frac{\pi a^2}{2}\right ) \sin \left (\pi a b x+\frac{1}{2} \pi b^2 x^2\right )}{\pi b}+x \text{FresnelC}(a+b x)+\frac{a \text{FresnelC}(a+b x)}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 34, normalized size = 0.9 \begin{align*}{\frac{1}{b} \left ({\it FresnelC} \left ( bx+a \right ) \left ( bx+a \right ) -{\frac{1}{\pi }\sin \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 fresnelc}\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 fresnelc}\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 C\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 fresnelc}\left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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