Optimal. Leaf size=100 \[ \frac{\sqrt{\pi } \cos \left (2 a+\frac{b^2}{2 c}\right ) \text{FresnelC}\left (\frac{b-2 c x}{\sqrt{\pi } \sqrt{c}}\right )}{4 \sqrt{c}}+\frac{\sqrt{\pi } \sin \left (2 a+\frac{b^2}{2 c}\right ) S\left (\frac{b-2 c x}{\sqrt{c} \sqrt{\pi }}\right )}{4 \sqrt{c}}+\frac{x}{2} \]
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Rubi [A] time = 0.0478904, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3449, 3448, 3352, 3351} \[ \frac{\sqrt{\pi } \cos \left (2 a+\frac{b^2}{2 c}\right ) \text{FresnelC}\left (\frac{b-2 c x}{\sqrt{\pi } \sqrt{c}}\right )}{4 \sqrt{c}}+\frac{\sqrt{\pi } \sin \left (2 a+\frac{b^2}{2 c}\right ) S\left (\frac{b-2 c x}{\sqrt{c} \sqrt{\pi }}\right )}{4 \sqrt{c}}+\frac{x}{2} \]
Antiderivative was successfully verified.
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Rule 3449
Rule 3448
Rule 3352
Rule 3351
Rubi steps
\begin{align*} \int \sin ^2\left (a+b x-c x^2\right ) \, dx &=\int \left (\frac{1}{2}-\frac{1}{2} \cos \left (2 a+2 b x-2 c x^2\right )\right ) \, dx\\ &=\frac{x}{2}-\frac{1}{2} \int \cos \left (2 a+2 b x-2 c x^2\right ) \, dx\\ &=\frac{x}{2}-\frac{1}{2} \cos \left (2 a+\frac{b^2}{2 c}\right ) \int \cos \left (\frac{(2 b-4 c x)^2}{8 c}\right ) \, dx-\frac{1}{2} \sin \left (2 a+\frac{b^2}{2 c}\right ) \int \sin \left (\frac{(2 b-4 c x)^2}{8 c}\right ) \, dx\\ &=\frac{x}{2}+\frac{\sqrt{\pi } \cos \left (2 a+\frac{b^2}{2 c}\right ) C\left (\frac{b-2 c x}{\sqrt{c} \sqrt{\pi }}\right )}{4 \sqrt{c}}+\frac{\sqrt{\pi } S\left (\frac{b-2 c x}{\sqrt{c} \sqrt{\pi }}\right ) \sin \left (2 a+\frac{b^2}{2 c}\right )}{4 \sqrt{c}}\\ \end{align*}
Mathematica [A] time = 0.109744, size = 102, normalized size = 1.02 \[ \frac{-\sqrt{\pi } \cos \left (2 a+\frac{b^2}{2 c}\right ) \text{FresnelC}\left (\frac{2 c x-b}{\sqrt{\pi } \sqrt{c}}\right )-\sqrt{\pi } \sin \left (2 a+\frac{b^2}{2 c}\right ) S\left (\frac{2 c x-b}{\sqrt{c} \sqrt{\pi }}\right )+2 \sqrt{c} x}{4 \sqrt{c}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 76, normalized size = 0.8 \begin{align*}{\frac{x}{2}}-{\frac{\sqrt{\pi }}{4} \left ( \cos \left ({\frac{4\,ca+{b}^{2}}{2\,c}} \right ){\it FresnelC} \left ({\frac{2\,cx-b}{\sqrt{\pi }}{\frac{1}{\sqrt{c}}}} \right ) +\sin \left ({\frac{4\,ca+{b}^{2}}{2\,c}} \right ){\it FresnelS} \left ({\frac{2\,cx-b}{\sqrt{\pi }}{\frac{1}{\sqrt{c}}}} \right ) \right ){\frac{1}{\sqrt{c}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 2.07347, size = 410, normalized size = 4.1 \begin{align*} -\frac{\sqrt{2} \sqrt{\pi }{\left ({\left ({\left (\cos \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, c\right )\right ) + \cos \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, c\right )\right ) - i \, \sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, c\right )\right ) + i \, \sin \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, c\right )\right )\right )} \cos \left (\frac{b^{2} + 4 \, a c}{2 \, c}\right ) +{\left (i \, \cos \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, c\right )\right ) + i \, \cos \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, c\right )\right ) + \sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, c\right )\right ) - \sin \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, c\right )\right )\right )} \sin \left (\frac{b^{2} + 4 \, a c}{2 \, c}\right )\right )} \operatorname{erf}\left (\frac{2 i \, c x - i \, b}{\sqrt{2 i \, c}}\right ) -{\left ({\left (\cos \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, c\right )\right ) + \cos \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, c\right )\right ) + i \, \sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, c\right )\right ) - i \, \sin \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, c\right )\right )\right )} \cos \left (\frac{b^{2} + 4 \, a c}{2 \, c}\right ) -{\left (i \, \cos \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, c\right )\right ) + i \, \cos \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, c\right )\right ) - \sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, c\right )\right ) + \sin \left (-\frac{1}{4} \, \pi + \frac{1}{2} \, \arctan \left (0, c\right )\right )\right )} \sin \left (\frac{b^{2} + 4 \, a c}{2 \, c}\right )\right )} \operatorname{erf}\left (\frac{2 i \, c x - i \, b}{\sqrt{-2 i \, c}}\right )\right )} \sqrt{{\left | c \right |}} - 16 \, x{\left | c \right |}}{32 \,{\left | c \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.536, size = 235, normalized size = 2.35 \begin{align*} -\frac{\pi \sqrt{\frac{c}{\pi }} \cos \left (\frac{b^{2} + 4 \, a c}{2 \, c}\right ) \operatorname{C}\left (\frac{{\left (2 \, c x - b\right )} \sqrt{\frac{c}{\pi }}}{c}\right ) + \pi \sqrt{\frac{c}{\pi }} \operatorname{S}\left (\frac{{\left (2 \, c x - b\right )} \sqrt{\frac{c}{\pi }}}{c}\right ) \sin \left (\frac{b^{2} + 4 \, a c}{2 \, c}\right ) - 2 \, c x}{4 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.14479, size = 88, normalized size = 0.88 \begin{align*} \frac{x}{2} - \frac{\sqrt{\pi } \sqrt{- \frac{1}{c}} \left (- \sin{\left (2 a + \frac{b^{2}}{2 c} \right )} S\left (\frac{2 b - 4 c x}{2 \sqrt{\pi } \sqrt{- c}}\right ) + \cos{\left (2 a + \frac{b^{2}}{2 c} \right )} C\left (\frac{2 b - 4 c x}{2 \sqrt{\pi } \sqrt{- c}}\right )\right )}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.34575, size = 167, normalized size = 1.67 \begin{align*} \frac{1}{2} \, x + \frac{\sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{c}{\left (2 \, x - \frac{b}{c}\right )}{\left (-\frac{i \, c}{{\left | c \right |}} + 1\right )}\right ) e^{\left (-\frac{i \, b^{2} + 4 i \, a c}{2 \, c}\right )}}{8 \, \sqrt{c}{\left (-\frac{i \, c}{{\left | c \right |}} + 1\right )}} + \frac{\sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{c}{\left (2 \, x - \frac{b}{c}\right )}{\left (\frac{i \, c}{{\left | c \right |}} + 1\right )}\right ) e^{\left (-\frac{-i \, b^{2} - 4 i \, a c}{2 \, c}\right )}}{8 \, \sqrt{c}{\left (\frac{i \, c}{{\left | c \right |}} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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