Optimal. Leaf size=24 \[ \sqrt{\frac{\pi }{2}} S\left (\frac{2 x+1}{\sqrt{2 \pi }}\right ) \]
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Rubi [A] time = 0.0060308, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {3445, 3351} \[ \sqrt{\frac{\pi }{2}} S\left (\frac{2 x+1}{\sqrt{2 \pi }}\right ) \]
Antiderivative was successfully verified.
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Rule 3445
Rule 3351
Rubi steps
\begin{align*} \int \sin \left (\frac{1}{4}+x+x^2\right ) \, dx &=\int \sin \left (\frac{1}{4} (1+2 x)^2\right ) \, dx\\ &=\sqrt{\frac{\pi }{2}} S\left (\frac{1+2 x}{\sqrt{2 \pi }}\right )\\ \end{align*}
Mathematica [A] time = 0.0265427, size = 24, normalized size = 1. \[ \sqrt{\frac{\pi }{2}} S\left (\frac{2 x+1}{\sqrt{2 \pi }}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 20, normalized size = 0.8 \begin{align*}{\frac{\sqrt{2}\sqrt{\pi }}{2}{\it FresnelS} \left ({\frac{\sqrt{2}}{\sqrt{\pi }} \left ( x+{\frac{1}{2}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 3.76987, size = 95, normalized size = 3.96 \begin{align*} \frac{1}{16} \, \sqrt{\pi }{\left (\left (i + 1\right ) \, \sqrt{2} \operatorname{erf}\left (-\frac{1}{2} \, \left (-1\right )^{\frac{3}{4}}{\left (2 i \, x + i\right )}\right ) + \left (i + 1\right ) \, \sqrt{2} \operatorname{erf}\left (-\left (\frac{1}{4} i - \frac{1}{4}\right ) \, \sqrt{2}{\left (2 i \, x + i\right )}\right ) - \left (i - 1\right ) \, \sqrt{2} \operatorname{erf}\left (-\left (\frac{1}{4} i + \frac{1}{4}\right ) \, \sqrt{2}{\left (2 i \, x + i\right )}\right ) + \left (i - 1\right ) \, \sqrt{2} \operatorname{erf}\left (\frac{2 i \, x + i}{2 \, \sqrt{-i}}\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.35415, size = 89, normalized size = 3.71 \begin{align*} \frac{1}{2} \, \sqrt{2} \sqrt{\pi } \operatorname{S}\left (\frac{\sqrt{2}{\left (2 \, x + 1\right )}}{2 \, \sqrt{\pi }}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.499589, size = 29, normalized size = 1.21 \begin{align*} \frac{\sqrt{2} \sqrt{\pi } S\left (\frac{\sqrt{2} \left (2 x + 1\right )}{2 \sqrt{\pi }}\right )}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.17521, size = 53, normalized size = 2.21 \begin{align*} \left (\frac{1}{8} i - \frac{1}{8}\right ) \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (\left (\frac{1}{4} i - \frac{1}{4}\right ) \, \sqrt{2}{\left (2 \, x + 1\right )}\right ) - \left (\frac{1}{8} i + \frac{1}{8}\right ) \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\left (\frac{1}{4} i + \frac{1}{4}\right ) \, \sqrt{2}{\left (2 \, x + 1\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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