Optimal. Leaf size=85 \[ -\frac{1}{16} \sqrt{\pi } \text{FresnelC}\left (\frac{2 x+1}{\sqrt{\pi }}\right )+\frac{1}{16} \sqrt{\pi } S\left (\frac{2 x+1}{\sqrt{\pi }}\right )+\frac{x^3}{6}-\frac{1}{8} x \sin \left (2 x^2+2 x+\frac{1}{2}\right )+\frac{1}{16} \sin \left (2 x^2+2 x+\frac{1}{2}\right ) \]
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Rubi [A] time = 0.0619234, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467, Rules used = {3467, 3464, 3445, 3351, 3462, 3446, 3352} \[ -\frac{1}{16} \sqrt{\pi } \text{FresnelC}\left (\frac{2 x+1}{\sqrt{\pi }}\right )+\frac{1}{16} \sqrt{\pi } S\left (\frac{2 x+1}{\sqrt{\pi }}\right )+\frac{x^3}{6}-\frac{1}{8} x \sin \left (2 x^2+2 x+\frac{1}{2}\right )+\frac{1}{16} \sin \left (2 x^2+2 x+\frac{1}{2}\right ) \]
Antiderivative was successfully verified.
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Rule 3467
Rule 3464
Rule 3445
Rule 3351
Rule 3462
Rule 3446
Rule 3352
Rubi steps
\begin{align*} \int x^2 \sin ^2\left (\frac{1}{4}+x+x^2\right ) \, dx &=\int \left (\frac{x^2}{2}-\frac{1}{2} x^2 \cos \left (\frac{1}{2}+2 x+2 x^2\right )\right ) \, dx\\ &=\frac{x^3}{6}-\frac{1}{2} \int x^2 \cos \left (\frac{1}{2}+2 x+2 x^2\right ) \, dx\\ &=\frac{x^3}{6}-\frac{1}{8} x \sin \left (\frac{1}{2}+2 x+2 x^2\right )+\frac{1}{8} \int \sin \left (\frac{1}{2}+2 x+2 x^2\right ) \, dx+\frac{1}{4} \int x \cos \left (\frac{1}{2}+2 x+2 x^2\right ) \, dx\\ &=\frac{x^3}{6}+\frac{1}{16} \sin \left (\frac{1}{2}+2 x+2 x^2\right )-\frac{1}{8} x \sin \left (\frac{1}{2}+2 x+2 x^2\right )-\frac{1}{8} \int \cos \left (\frac{1}{2}+2 x+2 x^2\right ) \, dx+\frac{1}{8} \int \sin \left (\frac{1}{8} (2+4 x)^2\right ) \, dx\\ &=\frac{x^3}{6}+\frac{1}{16} \sqrt{\pi } S\left (\frac{1+2 x}{\sqrt{\pi }}\right )+\frac{1}{16} \sin \left (\frac{1}{2}+2 x+2 x^2\right )-\frac{1}{8} x \sin \left (\frac{1}{2}+2 x+2 x^2\right )-\frac{1}{8} \int \cos \left (\frac{1}{8} (2+4 x)^2\right ) \, dx\\ &=\frac{x^3}{6}-\frac{1}{16} \sqrt{\pi } C\left (\frac{1+2 x}{\sqrt{\pi }}\right )+\frac{1}{16} \sqrt{\pi } S\left (\frac{1+2 x}{\sqrt{\pi }}\right )+\frac{1}{16} \sin \left (\frac{1}{2}+2 x+2 x^2\right )-\frac{1}{8} x \sin \left (\frac{1}{2}+2 x+2 x^2\right )\\ \end{align*}
Mathematica [A] time = 0.144895, size = 77, normalized size = 0.91 \[ \frac{1}{48} \left (-3 \sqrt{\pi } \text{FresnelC}\left (\frac{2 x+1}{\sqrt{\pi }}\right )+3 \sqrt{\pi } S\left (\frac{2 x+1}{\sqrt{\pi }}\right )+8 x^3-6 x \sin \left (\frac{1}{2} (2 x+1)^2\right )+3 \sin \left (\frac{1}{2} (2 x+1)^2\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 64, normalized size = 0.8 \begin{align*}{\frac{{x}^{3}}{6}}+{\frac{1}{16}\sin \left ({\frac{1}{2}}+2\,x+2\,{x}^{2} \right ) }-{\frac{x}{8}\sin \left ({\frac{1}{2}}+2\,x+2\,{x}^{2} \right ) }-{\frac{\sqrt{\pi }}{16}{\it FresnelC} \left ({\frac{1+2\,x}{\sqrt{\pi }}} \right ) }+{\frac{\sqrt{\pi }}{16}{\it FresnelS} \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 = 2.32955, size = 231, normalized size = 2.72 \begin{align*} \frac{8192 \, x^{4} + 4096 \, x^{3} - x{\left (3072 i \, e^{\left (2 i \, x^{2} + 2 i \, x + \frac{1}{2} i\right )} - 3072 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 (192 i - 192\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 (192 i + 192\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 (384 i + 384\right ) \, \sqrt{2} \Gamma \left (\frac{3}{2}, 2 i \, x^{2} + 2 i \, x + \frac{1}{2} i\right ) - \left (384 i - 384\right ) \, \sqrt{2} \Gamma \left (\frac{3}{2}, -2 i \, x^{2} - 2 i \, x - \frac{1}{2} i\right )\right )} - 1536 i \, e^{\left (2 i \, x^{2} + 2 i \, x + \frac{1}{2} i\right )} + 1536 i \, e^{\left (-2 i \, x^{2} - 2 i \, x - \frac{1}{2} i\right )}}{24576 \,{\left (2 \, x + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65278, size = 215, normalized size = 2.53 \begin{align*} \frac{1}{6} \, x^{3} - \frac{1}{8} \,{\left (2 \, x - 1\right )} \cos \left (x^{2} + x + \frac{1}{4}\right ) \sin \left (x^{2} + x + \frac{1}{4}\right ) - \frac{1}{16} \, \sqrt{\pi } \operatorname{C}\left (\frac{2 \, x + 1}{\sqrt{\pi }}\right ) + \frac{1}{16} \, \sqrt{\pi } \operatorname{S}\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \sin ^{2}{\left (x^{2} + x + \frac{1}{4} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.35223, size = 86, normalized size = 1.01 \begin{align*} \frac{1}{6} \, x^{3} - \frac{1}{32} \,{\left (-2 i \, x + i\right )} e^{\left (2 i \, x^{2} + 2 i \, x + \frac{1}{2} i\right )} - \frac{1}{32} \,{\left (2 i \, x - i\right )} e^{\left (-2 i \, x^{2} - 2 i \, x - \frac{1}{2} i\right )} + \frac{1}{32} i \, \sqrt{\pi } \operatorname{erf}\left (\left (i - 1\right ) \, x + \frac{1}{2} i - \frac{1}{2}\right ) - \frac{1}{32} i \, \sqrt{\pi } \operatorname{erf}\left (-\left (i + 1\right ) \, x - \frac{1}{2} i - \frac{1}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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