Optimal. Leaf size=41 \[ -\frac{1}{2} \sqrt{\frac{\pi }{2}} S\left (\frac{2 x+1}{\sqrt{2 \pi }}\right )-\frac{1}{2} \cos \left (x^2+x+\frac{1}{4}\right ) \]
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Rubi [A] time = 0.0129375, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {3461, 3445, 3351} \[ -\frac{1}{2} \sqrt{\frac{\pi }{2}} S\left (\frac{2 x+1}{\sqrt{2 \pi }}\right )-\frac{1}{2} \cos \left (x^2+x+\frac{1}{4}\right ) \]
Antiderivative was successfully verified.
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Rule 3461
Rule 3445
Rule 3351
Rubi steps
\begin{align*} \int x \sin \left (\frac{1}{4}+x+x^2\right ) \, dx &=-\frac{1}{2} \cos \left (\frac{1}{4}+x+x^2\right )-\frac{1}{2} \int \sin \left (\frac{1}{4}+x+x^2\right ) \, dx\\ &=-\frac{1}{2} \cos \left (\frac{1}{4}+x+x^2\right )-\frac{1}{2} \int \sin \left (\frac{1}{4} (1+2 x)^2\right ) \, dx\\ &=-\frac{1}{2} \cos \left (\frac{1}{4}+x+x^2\right )-\frac{1}{2} \sqrt{\frac{\pi }{2}} S\left (\frac{1+2 x}{\sqrt{2 \pi }}\right )\\ \end{align*}
Mathematica [A] time = 0.0500704, size = 39, normalized size = 0.95 \[ \frac{1}{4} \left (-\sqrt{2 \pi } S\left (\frac{2 x+1}{\sqrt{2 \pi }}\right )-2 \cos \left (x^2+x+\frac{1}{4}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 30, normalized size = 0.7 \begin{align*} -{\frac{1}{2}\cos \left ({\frac{1}{4}}+x+{x}^{2} \right ) }-{\frac{\sqrt{2}\sqrt{\pi }}{4}{\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 = 4.14306, size = 166, normalized size = 4.05 \begin{align*} -\frac{2048 \, x{\left (e^{\left (i \, x^{2} + i \, x + \frac{1}{4} i\right )} + e^{\left (-i \, x^{2} - i \, x - \frac{1}{4} i\right )}\right )} - \sqrt{4 \, x^{2} + 4 \, x + 1}{\left (-\left (256 i + 256\right ) \, \sqrt{2} \sqrt{\pi }{\left (\operatorname{erf}\left (\sqrt{i \, x^{2} + i \, x + \frac{1}{4} i}\right ) - 1\right )} + \left (256 i - 256\right ) \, \sqrt{2} \sqrt{\pi }{\left (\operatorname{erf}\left (\sqrt{-i \, x^{2} - i \, x - \frac{1}{4} i}\right ) - 1\right )}\right )} + 1024 \, e^{\left (i \, x^{2} + i \, x + \frac{1}{4} i\right )} + 1024 \, e^{\left (-i \, x^{2} - i \, x - \frac{1}{4} i\right )}}{4096 \,{\left (2 \, x + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.39032, size = 124, normalized size = 3.02 \begin{align*} -\frac{1}{4} \, \sqrt{2} \sqrt{\pi } \operatorname{S}\left (\frac{\sqrt{2}{\left (2 \, x + 1\right )}}{2 \, \sqrt{\pi }}\right ) - \frac{1}{2} \, \cos \left (x^{2} + x + \frac{1}{4}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.14774, size = 160, normalized size = 3.9 \begin{align*} - \frac{3 \sqrt{2} \sqrt{\pi } x S\left (\frac{\sqrt{2} x}{\sqrt{\pi }} + \frac{\sqrt{2}}{2 \sqrt{\pi }}\right ) \Gamma \left (\frac{3}{4}\right )}{8 \Gamma \left (\frac{7}{4}\right )} + \frac{\sqrt{2} \sqrt{\pi } x S\left (\frac{\sqrt{2} x}{\sqrt{\pi }} + \frac{\sqrt{2}}{2 \sqrt{\pi }}\right )}{2} - \frac{3 \cos{\left (\left (x + \frac{1}{2}\right )^{2} \right )} \Gamma \left (\frac{3}{4}\right )}{8 \Gamma \left (\frac{7}{4}\right )} - \frac{3 \sqrt{2} \sqrt{\pi } S\left (\frac{\sqrt{2} x}{\sqrt{\pi }} + \frac{\sqrt{2}}{2 \sqrt{\pi }}\right ) \Gamma \left (\frac{3}{4}\right )}{16 \Gamma \left (\frac{7}{4}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.29688, size = 88, normalized size = 2.15 \begin{align*} -\left (\frac{1}{16} i - \frac{1}{16}\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}{16} i + \frac{1}{16}\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 ) - \frac{1}{4} \, e^{\left (i \, x^{2} + i \, x + \frac{1}{4} i\right )} - \frac{1}{4} \, e^{\left (-i \, x^{2} - i \, x - \frac{1}{4} i\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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