Optimal. Leaf size=46 \[ \frac{1}{8} \sqrt{\pi } \text{FresnelC}\left (\frac{2 x+1}{\sqrt{\pi }}\right )+\frac{x^2}{4}-\frac{1}{8} \sin \left (2 x^2+2 x+\frac{1}{2}\right ) \]
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Rubi [A] time = 0.0304473, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {3467, 3462, 3446, 3352} \[ \frac{1}{8} \sqrt{\pi } \text{FresnelC}\left (\frac{2 x+1}{\sqrt{\pi }}\right )+\frac{x^2}{4}-\frac{1}{8} \sin \left (2 x^2+2 x+\frac{1}{2}\right ) \]
Antiderivative was successfully verified.
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Rule 3467
Rule 3462
Rule 3446
Rule 3352
Rubi steps
\begin{align*} \int x \sin ^2\left (\frac{1}{4}+x+x^2\right ) \, dx &=\int \left (\frac{x}{2}-\frac{1}{2} x \cos \left (\frac{1}{2}+2 x+2 x^2\right )\right ) \, dx\\ &=\frac{x^2}{4}-\frac{1}{2} \int x \cos \left (\frac{1}{2}+2 x+2 x^2\right ) \, dx\\ &=\frac{x^2}{4}-\frac{1}{8} \sin \left (\frac{1}{2}+2 x+2 x^2\right )+\frac{1}{4} \int \cos \left (\frac{1}{2}+2 x+2 x^2\right ) \, dx\\ &=\frac{x^2}{4}-\frac{1}{8} \sin \left (\frac{1}{2}+2 x+2 x^2\right )+\frac{1}{4} \int \cos \left (\frac{1}{8} (2+4 x)^2\right ) \, dx\\ &=\frac{x^2}{4}+\frac{1}{8} \sqrt{\pi } C\left (\frac{1+2 x}{\sqrt{\pi }}\right )-\frac{1}{8} \sin \left (\frac{1}{2}+2 x+2 x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0607905, size = 42, normalized size = 0.91 \[ \frac{1}{8} \left (\sqrt{\pi } \text{FresnelC}\left (\frac{2 x+1}{\sqrt{\pi }}\right )+2 x^2-\sin \left (\frac{1}{2} (2 x+1)^2\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 35, normalized size = 0.8 \begin{align*}{\frac{{x}^{2}}{4}}-{\frac{1}{8}\sin \left ({\frac{1}{2}}+2\,x+2\,{x}^{2} \right ) }+{\frac{\sqrt{\pi }}{8}{\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 = 2.04188, size = 182, normalized size = 3.96 \begin{align*} \frac{65536 \, x^{3} + 32768 \, x^{2} + x{\left (16384 i \, e^{\left (2 i \, x^{2} + 2 i \, x + \frac{1}{2} i\right )} - 16384 i \, e^{\left (-2 i \, x^{2} - 2 i \, x - \frac{1}{2} i\right )}\right )} + \sqrt{8 \, x^{2} + 8 \, x + 2}{\left (-\left (2048 i - 2048\right ) \, \sqrt{2} \sqrt{\pi }{\left (\operatorname{erf}\left (\sqrt{2 i \, x^{2} + 2 i \, x + \frac{1}{2} i}\right ) - 1\right )} + \left (2048 i + 2048\right ) \, \sqrt{2} \sqrt{\pi }{\left (\operatorname{erf}\left (\sqrt{-2 i \, x^{2} - 2 i \, x - \frac{1}{2} i}\right ) - 1\right )}\right )} + 8192 i \, e^{\left (2 i \, x^{2} + 2 i \, x + \frac{1}{2} i\right )} - 8192 i \, e^{\left (-2 i \, x^{2} - 2 i \, x - \frac{1}{2} i\right )}}{131072 \,{\left (2 \, x + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65995, size = 135, normalized size = 2.93 \begin{align*} \frac{1}{4} \, x^{2} - \frac{1}{4} \, \cos \left (x^{2} + x + \frac{1}{4}\right ) \sin \left (x^{2} + x + \frac{1}{4}\right ) + \frac{1}{8} \, \sqrt{\pi } \operatorname{C}\left (\frac{2 \, x + 1}{\sqrt{\pi }}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.6479, size = 121, normalized size = 2.63 \begin{align*} \frac{x^{2}}{4} - \frac{\sqrt{\pi } x C\left (\frac{2 x}{\sqrt{\pi }} + \frac{1}{\sqrt{\pi }}\right )}{4} + \frac{\sqrt{\pi } x C\left (\frac{2 x}{\sqrt{\pi }} + \frac{1}{\sqrt{\pi }}\right ) \Gamma \left (\frac{1}{4}\right )}{16 \Gamma \left (\frac{5}{4}\right )} - \frac{\sin{\left (2 \left (x + \frac{1}{2}\right )^{2} \right )} \Gamma \left (\frac{1}{4}\right )}{32 \Gamma \left (\frac{5}{4}\right )} + \frac{\sqrt{\pi } C\left (\frac{2 x}{\sqrt{\pi }} + \frac{1}{\sqrt{\pi }}\right ) \Gamma \left (\frac{1}{4}\right )}{32 \Gamma \left (\frac{5}{4}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.28864, size = 73, normalized size = 1.59 \begin{align*} \frac{1}{4} \, x^{2} - \left (\frac{1}{32} i + \frac{1}{32}\right ) \, \sqrt{\pi } \operatorname{erf}\left (\left (i - 1\right ) \, x + \frac{1}{2} i - \frac{1}{2}\right ) + \left (\frac{1}{32} i - \frac{1}{32}\right ) \, \sqrt{\pi } \operatorname{erf}\left (-\left (i + 1\right ) \, x - \frac{1}{2} i - \frac{1}{2}\right ) + \frac{1}{16} i \, e^{\left (2 i \, x^{2} + 2 i \, x + \frac{1}{2} i\right )} - \frac{1}{16} i \, e^{\left (-2 i \, x^{2} - 2 i \, x - \frac{1}{2} i\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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