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 = 0.03, antiderivative size = 24, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.03, 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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fricas [B] time = 0.60, size = 35, normalized size = 1.46 \[ -\frac {{\left (\cos \relax (x) + 1\right )} \log \left (\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) - {\left (\cos \relax (x) + 1\right )} \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) - 2}{8 \, {\left (\cos \relax (x) + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 28, normalized size = 1.17 \[ -\frac {\cos \relax (x) - 1}{8 \, {\left (\cos \relax (x) + 1\right )}} + \frac {1}{8} \, \log \left (-\frac {\cos \relax (x) - 1}{\cos \relax (x) + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 24, normalized size = 1.00 \[ \frac {\ln \left (-1+\cos \relax (x )\right )}{8}+\frac {1}{4+4 \cos \relax (x )}-\frac {\ln \left (1+\cos \relax (x )\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 220, normalized size = 9.17 \[ \frac {4 \, \cos \left (2 \, x\right ) \cos \relax (x) + 8 \, \cos \relax (x)^{2} - {\left (2 \, {\left (2 \, \cos \relax (x) + 1\right )} \cos \left (2 \, x\right ) + \cos \left (2 \, x\right )^{2} + 4 \, \cos \relax (x)^{2} + \sin \left (2 \, x\right )^{2} + 4 \, \sin \left (2 \, x\right ) \sin \relax (x) + 4 \, \sin \relax (x)^{2} + 4 \, \cos \relax (x) + 1\right )} \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \cos \relax (x) + 1\right ) + {\left (2 \, {\left (2 \, \cos \relax (x) + 1\right )} \cos \left (2 \, x\right ) + \cos \left (2 \, x\right )^{2} + 4 \, \cos \relax (x)^{2} + \sin \left (2 \, x\right )^{2} + 4 \, \sin \left (2 \, x\right ) \sin \relax (x) + 4 \, \sin \relax (x)^{2} + 4 \, \cos \relax (x) + 1\right )} \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \cos \relax (x) + 1\right ) + 4 \, \sin \left (2 \, x\right ) \sin \relax (x) + 8 \, \sin \relax (x)^{2} + 4 \, \cos \relax (x)}{8 \, {\left (2 \, {\left (2 \, \cos \relax (x) + 1\right )} \cos \left (2 \, x\right ) + \cos \left (2 \, x\right )^{2} + 4 \, \cos \relax (x)^{2} + \sin \left (2 \, x\right )^{2} + 4 \, \sin \left (2 \, x\right ) \sin \relax (x) + 4 \, \sin \relax (x)^{2} + 4 \, \cos \relax (x) + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.07, size = 16, normalized size = 0.67 \[ \frac {\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{4}+\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{2 \sin {\relax (x )} + \sin {\left (2 x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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