Optimal. Leaf size=11 \[ \log \left (\sin ^2(x)-3 \sin (x)+2\right ) \]
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Rubi [A] time = 0.047502, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {4334, 628} \[ \log \left (\sin ^2(x)-3 \sin (x)+2\right ) \]
Antiderivative was successfully verified.
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Rule 4334
Rule 628
Rubi steps
\begin{align*} \int \frac{\cos (x) (-3+2 \sin (x))}{2-3 \sin (x)+\sin ^2(x)} \, dx &=\operatorname{Subst}\left (\int \frac{-3+2 x}{2-3 x+x^2} \, dx,x,\sin (x)\right )\\ &=\log \left (2-3 \sin (x)+\sin ^2(x)\right )\\ \end{align*}
Mathematica [B] time = 0.08709, size = 26, normalized size = 2.36 \[ \log (2-\sin (x))+2 \log \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 12, normalized size = 1.1 \begin{align*} \ln \left ( 2-3\,\sin \left ( x \right ) + \left ( \sin \left ( x \right ) \right ) ^{2} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.93726, size = 15, normalized size = 1.36 \begin{align*} \log \left (\sin \left (x\right )^{2} - 3 \, \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 = 2.11957, size = 55, normalized size = 5. \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 = 0.215164, size = 12, normalized size = 1.09 \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 = 1.06162, size = 20, normalized size = 1.82 \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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