Optimal. Leaf size=11 \[ \log \left (\sin ^2(x)-3 \sin (x)+2\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 11, normalized size of antiderivative = 1.00, 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 628
Rule 4334
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*}
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Mathematica [B] time = 0.09, 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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fricas [A] time = 0.70, size = 15, normalized size = 1.36 \[ \log \left (-\frac {1}{2} \, \sin \relax (x) + 1\right ) + \log \left (-\sin \relax (x) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 15, normalized size = 1.36 \[ \log \left (-\sin \relax (x) + 2\right ) + \log \left (-\sin \relax (x) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 12, normalized size = 1.09 \[ \ln \left (2-3 \sin \relax (x )+\sin ^{2}\relax (x )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 11, normalized size = 1.00 \[ \log \left (\sin \relax (x)^{2} - 3 \, \sin \relax (x) + 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 11, normalized size = 1.00 \[ \ln \left ({\sin \relax (x)}^2-3\,\sin \relax (x)+2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.19, size = 12, normalized size = 1.09 \[ \log {\left (\sin {\relax (x )} - 2 \right )} + \log {\left (\sin {\relax (x )} - 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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