Optimal. Leaf size=21 \[ \frac{1}{3} \log (\cos (x)+2)-\frac{1}{3} \log (1-\cos (x)) \]
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Rubi [A] time = 0.0244769, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {3259, 616, 31} \[ \frac{1}{3} \log (\cos (x)+2)-\frac{1}{3} \log (1-\cos (x)) \]
Antiderivative was successfully verified.
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Rule 3259
Rule 616
Rule 31
Rubi steps
\begin{align*} \int \frac{\sin (x)}{-2+\cos (x)+\cos ^2(x)} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{-2+x+x^2} \, dx,x,\cos (x)\right )\\ &=-\left (\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{-1+x} \, dx,x,\cos (x)\right )\right )+\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{2+x} \, dx,x,\cos (x)\right )\\ &=-\frac{1}{3} \log (1-\cos (x))+\frac{1}{3} \log (2+\cos (x))\\ \end{align*}
Mathematica [A] time = 0.0251045, size = 19, normalized size = 0.9 \[ \frac{1}{3} \left (\log (\cos (x)+2)-2 \log \left (\sin \left (\frac{x}{2}\right )\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.025, size = 16, normalized size = 0.8 \begin{align*}{\frac{\ln \left ( 2+\cos \left ( x \right ) \right ) }{3}}-{\frac{\ln \left ( \cos \left ( x \right ) -1 \right ) }{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.948131, size = 20, normalized size = 0.95 \begin{align*} \frac{1}{3} \, \log \left (\cos \left (x\right ) + 2\right ) - \frac{1}{3} \, \log \left (\cos \left (x\right ) - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.31343, size = 68, normalized size = 3.24 \begin{align*} \frac{1}{3} \, \log \left (\cos \left (x\right ) + 2\right ) - \frac{1}{3} \, \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.242954, size = 15, normalized size = 0.71 \begin{align*} - \frac{\log{\left (\cos{\left (x \right )} - 1 \right )}}{3} + \frac{\log{\left (\cos{\left (x \right )} + 2 \right )}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.78329, size = 23, normalized size = 1.1 \begin{align*} \frac{1}{3} \, \log \left (\cos \left (x\right ) + 2\right ) - \frac{1}{3} \, \log \left (-\cos \left (x\right ) + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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