Optimal. Leaf size=64 \[ \frac{e^c \text{Erf}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) \sqrt{\pi } b x\right )^2}{8 b}-\frac{1}{4} i b e^c x^2 \text{HypergeometricPFQ}\left (\{1,1\},\left \{\frac{3}{2},2\right \},-\frac{1}{2} i \pi b^2 x^2\right ) \]
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Rubi [A] time = 0.0681901, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {6436, 6373, 30, 6378} \[ \frac{e^c \text{Erf}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) \sqrt{\pi } b x\right )^2}{8 b}-\frac{1}{4} i b e^c x^2 \, _2F_2\left (1,1;\frac{3}{2},2;-\frac{1}{2} i b^2 \pi x^2\right ) \]
Antiderivative was successfully verified.
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Rule 6436
Rule 6373
Rule 30
Rule 6378
Rubi steps
\begin{align*} \int e^{c-\frac{1}{2} i b^2 \pi x^2} S(b x) \, dx &=\left (-\frac{1}{4}-\frac{i}{4}\right ) \int e^{c-\frac{1}{2} i b^2 \pi x^2} \text{erfi}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) b \sqrt{\pi } x\right ) \, dx+\left (\frac{1}{4}+\frac{i}{4}\right ) \int e^{c-\frac{1}{2} i b^2 \pi x^2} \text{erf}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) b \sqrt{\pi } x\right ) \, dx\\ &=-\frac{1}{4} i b e^c x^2 \, _2F_2\left (1,1;\frac{3}{2},2;-\frac{1}{2} i b^2 \pi x^2\right )+\frac{e^c \operatorname{Subst}\left (\int x \, dx,x,\text{erf}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) b \sqrt{\pi } x\right )\right )}{4 b}\\ &=\frac{e^c \text{erf}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) b \sqrt{\pi } x\right )^2}{8 b}-\frac{1}{4} i b e^c 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 = 0.0321957, size = 0, normalized size = 0. \[ \int e^{c-\frac{1}{2} i b^2 \pi x^2} S(b x) \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.069, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{c-{\frac{i}{2}}{b}^{2}\pi \,{x}^{2}}}{\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 e^{\left (-\frac{1}{2} i \, \pi b^{2} x^{2} + c\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 (e^{\left (-\frac{1}{2} i \, \pi b^{2} x^{2} + c\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*} e^{c} \int e^{- \frac{i \pi b^{2} x^{2}}{2}} 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 e^{\left (-\frac{1}{2} i \, \pi b^{2} x^{2} + c\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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