Optimal. Leaf size=43 \[ \frac{1}{3} \log \left (2 \cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )-\frac{1}{3} \log \left (\cos \left (\frac{x}{2}\right )-2 \sin \left (\frac{x}{2}\right )\right ) \]
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Rubi [A] time = 0.0176698, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {2660, 616, 31} \[ \frac{1}{3} \log \left (2 \cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )-\frac{1}{3} \log \left (\cos \left (\frac{x}{2}\right )-2 \sin \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Rule 2660
Rule 616
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{4-5 \sin (x)} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{4-10 x+4 x^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )\\ &=\frac{4}{3} \operatorname{Subst}\left (\int \frac{1}{-8+4 x} \, dx,x,\tan \left (\frac{x}{2}\right )\right )-\frac{4}{3} \operatorname{Subst}\left (\int \frac{1}{-2+4 x} \, dx,x,\tan \left (\frac{x}{2}\right )\right )\\ &=-\frac{1}{3} \log \left (1-2 \tan \left (\frac{x}{2}\right )\right )+\frac{1}{3} \log \left (2-\tan \left (\frac{x}{2}\right )\right )\\ \end{align*}
Mathematica [A] time = 0.0118386, size = 43, normalized size = 1. \[ \frac{1}{3} \log \left (2 \cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )-\frac{1}{3} \log \left (\cos \left (\frac{x}{2}\right )-2 \sin \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 22, normalized size = 0.5 \begin{align*} -{\frac{1}{3}\ln \left ( 2\,\tan \left ( x/2 \right ) -1 \right ) }+{\frac{1}{3}\ln \left ( \tan \left ({\frac{x}{2}} \right ) -2 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.936871, size = 41, normalized size = 0.95 \begin{align*} -\frac{1}{3} \, \log \left (\frac{2 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} - 1\right ) + \frac{1}{3} \, \log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} - 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.29027, size = 105, normalized size = 2.44 \begin{align*} \frac{1}{6} \, \log \left (\frac{3}{2} \, \cos \left (x\right ) - 2 \, \sin \left (x\right ) + \frac{5}{2}\right ) - \frac{1}{6} \, \log \left (-\frac{3}{2} \, \cos \left (x\right ) - 2 \, \sin \left (x\right ) + \frac{5}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.222954, size = 20, normalized size = 0.47 \begin{align*} \frac{\log{\left (\tan{\left (\frac{x}{2} \right )} - 2 \right )}}{3} - \frac{\log{\left (\tan{\left (\frac{x}{2} \right )} - \frac{1}{2} \right )}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08024, size = 31, normalized size = 0.72 \begin{align*} -\frac{1}{3} \, \log \left ({\left | 2 \, \tan \left (\frac{1}{2} \, x\right ) - 1 \right |}\right ) + \frac{1}{3} \, \log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) - 2 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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