Optimal. Leaf size=24 \[ \frac{1}{8} \tan ^2\left (\frac{x}{2}\right )+\frac{1}{4} \log \left (\tan \left (\frac{x}{2}\right )\right ) \]
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Rubi [A] time = 0.0268354, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {12, 14} \[ \frac{1}{8} \tan ^2\left (\frac{x}{2}\right )+\frac{1}{4} \log \left (\tan \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rubi steps
\begin{align*} \int \frac{1}{2 \sin (x)+\sin (2 x)} \, dx &=2 \operatorname{Subst}\left (\int \frac{1+x^2}{8 x} \, dx,x,\tan \left (\frac{x}{2}\right )\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{1+x^2}{x} \, dx,x,\tan \left (\frac{x}{2}\right )\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (\frac{1}{x}+x\right ) \, dx,x,\tan \left (\frac{x}{2}\right )\right )\\ &=\frac{1}{4} \log \left (\tan \left (\frac{x}{2}\right )\right )+\frac{1}{8} \tan ^2\left (\frac{x}{2}\right )\\ \end{align*}
Mathematica [A] time = 0.0286321, size = 39, normalized size = 1.62 \[ \frac{1-2 \cos ^2\left (\frac{x}{2}\right ) \left (\log \left (\cos \left (\frac{x}{2}\right )\right )-\log \left (\sin \left (\frac{x}{2}\right )\right )\right )}{4 (\cos (x)+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 24, normalized size = 1. \begin{align*}{\frac{1}{4\,\cos \left ( x \right ) +4}}-{\frac{\ln \left ( \cos \left ( x \right ) +1 \right ) }{8}}+{\frac{\ln \left ( \cos \left ( x \right ) -1 \right ) }{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.963876, size = 297, normalized size = 12.38 \begin{align*} \frac{4 \, \cos \left (2 \, x\right ) \cos \left (x\right ) + 8 \, \cos \left (x\right )^{2} -{\left (2 \,{\left (2 \, \cos \left (x\right ) + 1\right )} \cos \left (2 \, x\right ) + \cos \left (2 \, x\right )^{2} + 4 \, \cos \left (x\right )^{2} + \sin \left (2 \, x\right )^{2} + 4 \, \sin \left (2 \, x\right ) \sin \left (x\right ) + 4 \, \sin \left (x\right )^{2} + 4 \, \cos \left (x\right ) + 1\right )} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) +{\left (2 \,{\left (2 \, \cos \left (x\right ) + 1\right )} \cos \left (2 \, x\right ) + \cos \left (2 \, x\right )^{2} + 4 \, \cos \left (x\right )^{2} + \sin \left (2 \, x\right )^{2} + 4 \, \sin \left (2 \, x\right ) \sin \left (x\right ) + 4 \, \sin \left (x\right )^{2} + 4 \, \cos \left (x\right ) + 1\right )} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) + 4 \, \sin \left (2 \, x\right ) \sin \left (x\right ) + 8 \, \sin \left (x\right )^{2} + 4 \, \cos \left (x\right )}{8 \,{\left (2 \,{\left (2 \, \cos \left (x\right ) + 1\right )} \cos \left (2 \, x\right ) + \cos \left (2 \, x\right )^{2} + 4 \, \cos \left (x\right )^{2} + \sin \left (2 \, x\right )^{2} + 4 \, \sin \left (2 \, x\right ) \sin \left (x\right ) + 4 \, \sin \left (x\right )^{2} + 4 \, \cos \left (x\right ) + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.34319, size = 132, normalized size = 5.5 \begin{align*} -\frac{{\left (\cos \left (x\right ) + 1\right )} \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) -{\left (\cos \left (x\right ) + 1\right )} \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) - 2}{8 \,{\left (\cos \left (x\right ) + 1\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 \frac{1}{2 \sin{\left (x \right )} + \sin{\left (2 x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06128, size = 38, normalized size = 1.58 \begin{align*} -\frac{\cos \left (x\right ) - 1}{8 \,{\left (\cos \left (x\right ) + 1\right )}} + \frac{1}{8} \, \log \left (-\frac{\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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