Optimal. Leaf size=80 \[ -\frac{1}{8} i b x^2 \text{HypergeometricPFQ}\left (\{1,1\},\left \{\frac{3}{2},2\right \},-\frac{1}{2} i \pi b^2 x^2\right )+\frac{1}{8} i b x^2 \text{HypergeometricPFQ}\left (\{1,1\},\left \{\frac{3}{2},2\right \},\frac{1}{2} i \pi b^2 x^2\right )+\frac{\text{FresnelC}(b x) S(b x)}{2 b} \]
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Rubi [A] time = 0.0167872, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {6446} \[ -\frac{1}{8} i b x^2 \, _2F_2\left (1,1;\frac{3}{2},2;-\frac{1}{2} i b^2 \pi x^2\right )+\frac{1}{8} i b x^2 \, _2F_2\left (1,1;\frac{3}{2},2;\frac{1}{2} i b^2 \pi x^2\right )+\frac{\text{FresnelC}(b x) S(b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 6446
Rubi steps
\begin{align*} \int \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x) \, dx &=\frac{C(b x) S(b x)}{2 b}-\frac{1}{8} i b x^2 \, _2F_2\left (1,1;\frac{3}{2},2;-\frac{1}{2} i b^2 \pi x^2\right )+\frac{1}{8} i b x^2 \, _2F_2\left (1,1;\frac{3}{2},2;\frac{1}{2} i b^2 \pi x^2\right )\\ \end{align*}
Mathematica [F] time = 0.0139307, size = 0, normalized size = 0. \[ \int \cos \left (\frac{1}{2} b^2 \pi x^2\right ) S(b x) \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.058, size = 0, normalized size = 0. \begin{align*} \int \cos \left ({\frac{{b}^{2}\pi \,{x}^{2}}{2}} \right ){\it FresnelS} \left ( bx \right ) \, dx \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 \cos \left (\frac{1}{2} \, \pi b^{2} x^{2}\right ){\rm fresnels}\left (b x\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 (\cos \left (\frac{1}{2} \, \pi b^{2} x^{2}\right ){\rm fresnels}\left (b x\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 \cos{\left (\frac{\pi b^{2} x^{2}}{2} \right )} S\left (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 \cos \left (\frac{1}{2} \, \pi b^{2} x^{2}\right ){\rm fresnels}\left (b x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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