Optimal. Leaf size=27 \[ \frac{x}{2}-\frac{1}{4} \sqrt{\pi } \text{FresnelC}\left (\frac{2 x+1}{\sqrt{\pi }}\right ) \]
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Rubi [A] time = 0.0146485, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {3449, 3446, 3352} \[ \frac{x}{2}-\frac{1}{4} \sqrt{\pi } \text{FresnelC}\left (\frac{2 x+1}{\sqrt{\pi }}\right ) \]
Antiderivative was successfully verified.
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Rule 3449
Rule 3446
Rule 3352
Rubi steps
\begin{align*} \int \sin ^2\left (\frac{1}{4}+x+x^2\right ) \, dx &=\int \left (\frac{1}{2}-\frac{1}{2} \cos \left (\frac{1}{2}+2 x+2 x^2\right )\right ) \, dx\\ &=\frac{x}{2}-\frac{1}{2} \int \cos \left (\frac{1}{2}+2 x+2 x^2\right ) \, dx\\ &=\frac{x}{2}-\frac{1}{2} \int \cos \left (\frac{1}{8} (2+4 x)^2\right ) \, dx\\ &=\frac{x}{2}-\frac{1}{4} \sqrt{\pi } C\left (\frac{1+2 x}{\sqrt{\pi }}\right )\\ \end{align*}
Mathematica [A] time = 0.0125249, size = 27, normalized size = 1. \[ \frac{1}{4} \left (2 x-\sqrt{\pi } \text{FresnelC}\left (\frac{2 x+1}{\sqrt{\pi }}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 20, normalized size = 0.7 \begin{align*}{\frac{x}{2}}-{\frac{\sqrt{\pi }}{4}{\it FresnelC} \left ({\frac{1+2\,x}{\sqrt{\pi }}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.73747, size = 46, normalized size = 1.7 \begin{align*} \frac{1}{16} \, \sqrt{\pi }{\left (\left (i - 1\right ) \, \operatorname{erf}\left (\frac{2 i \, x + i}{\sqrt{2 i}}\right ) + \left (i + 1\right ) \, \operatorname{erf}\left (\frac{2 i \, x + i}{\sqrt{-2 i}}\right )\right )} + \frac{1}{2} \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.68478, size = 74, normalized size = 2.74 \begin{align*} -\frac{1}{4} \, \sqrt{\pi } \operatorname{C}\left (\frac{2 \, x + 1}{\sqrt{\pi }}\right ) + \frac{1}{2} \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.887441, size = 22, normalized size = 0.81 \begin{align*} \frac{x}{2} - \frac{\sqrt{\pi } C\left (\frac{4 x + 2}{2 \sqrt{\pi }}\right )}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.28109, size = 35, normalized size = 1.3 \begin{align*} \left (\frac{1}{16} i + \frac{1}{16}\right ) \, \sqrt{\pi } \operatorname{erf}\left (\left (i - 1\right ) \, x + \frac{1}{2} i - \frac{1}{2}\right ) - \left (\frac{1}{16} i - \frac{1}{16}\right ) \, \sqrt{\pi } \operatorname{erf}\left (-\left (i + 1\right ) \, x - \frac{1}{2} i - \frac{1}{2}\right ) + \frac{1}{2} \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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