Optimal. Leaf size=64 \[ \frac{1}{4} b e^c x^2 \text{HypergeometricPFQ}\left (\{1,1\},\left \{\frac{3}{2},2\right \},\frac{1}{2} i \pi b^2 x^2\right )-\frac{i e^c \text{Erfi}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) \sqrt{\pi } b x\right )^2}{8 b} \]
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Rubi [A] time = 0.0693593, 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 = {6437, 6376, 6375, 30} \[ \frac{1}{4} b e^c x^2 \, _2F_2\left (1,1;\frac{3}{2},2;\frac{1}{2} i b^2 \pi x^2\right )-\frac{i e^c \text{Erfi}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) \sqrt{\pi } b x\right )^2}{8 b} \]
Antiderivative was successfully verified.
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Rule 6437
Rule 6376
Rule 6375
Rule 30
Rubi steps
\begin{align*} \int e^{c+\frac{1}{2} i b^2 \pi x^2} C(b x) \, 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+\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\\ &=\frac{1}{4} 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{\left (i e^c\right ) \operatorname{Subst}\left (\int x \, dx,x,\text{erfi}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) b \sqrt{\pi } x\right )\right )}{4 b}\\ &=-\frac{i e^c \text{erfi}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) b \sqrt{\pi } x\right )^2}{8 b}+\frac{1}{4} 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 = 0.0291736, size = 0, normalized size = 0. \[ \int e^{c+\frac{1}{2} i b^2 \pi x^2} \text{FresnelC}(b x) \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.056, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{c+{\frac{i}{2}}{b}^{2}\pi \,{x}^{2}}}{\it FresnelC} \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 fresnelc}\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 fresnelc}\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}} C\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 fresnelc}\left (b x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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