Optimal. Leaf size=17 \[ \log (2-\sin (x))-\log (1-\sin (x)) \]
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Rubi [A] time = 0.0270485, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3258, 616, 31} \[ \log (2-\sin (x))-\log (1-\sin (x)) \]
Antiderivative was successfully verified.
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Rule 3258
Rule 616
Rule 31
Rubi steps
\begin{align*} \int \frac{\cos (x)}{2-3 \sin (x)+\sin ^2(x)} \, dx &=\operatorname{Subst}\left (\int \frac{1}{2-3 x+x^2} \, dx,x,\sin (x)\right )\\ &=\operatorname{Subst}\left (\int \frac{1}{-2+x} \, dx,x,\sin (x)\right )-\operatorname{Subst}\left (\int \frac{1}{-1+x} \, dx,x,\sin (x)\right )\\ &=-\log (1-\sin (x))+\log (2-\sin (x))\\ \end{align*}
Mathematica [A] time = 0.0560068, size = 26, normalized size = 1.53 \[ \log (2-\sin (x))-2 \log \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 14, normalized size = 0.8 \begin{align*} -\ln \left ( -1+\sin \left ( x \right ) \right ) +\ln \left ( \sin \left ( x \right ) -2 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.959675, size = 18, normalized size = 1.06 \begin{align*} -\log \left (\sin \left (x\right ) - 1\right ) + \log \left (\sin \left (x\right ) - 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.35779, size = 55, normalized size = 3.24 \begin{align*} \log \left (-\frac{1}{2} \, \sin \left (x\right ) + 1\right ) - \log \left (-\sin \left (x\right ) + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.209983, size = 12, normalized size = 0.71 \begin{align*} \log{\left (\sin{\left (x \right )} - 2 \right )} - \log{\left (\sin{\left (x \right )} - 1 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15078, size = 23, normalized size = 1.35 \begin{align*} \log \left (-\sin \left (x\right ) + 2\right ) - \log \left (-\sin \left (x\right ) + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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