Integrand size = 21, antiderivative size = 11 \[ \int \frac {\cos (x) (-3+2 \sin (x))}{2-3 \sin (x)+\sin ^2(x)} \, dx=\log \left (2-3 \sin (x)+\sin ^2(x)\right ) \]
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Time = 0.03 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {4419, 642} \[ \int \frac {\cos (x) (-3+2 \sin (x))}{2-3 \sin (x)+\sin ^2(x)} \, dx=\log \left (\sin ^2(x)-3 \sin (x)+2\right ) \]
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Rule 642
Rule 4419
Rubi steps \begin{align*} \text {integral}& = \text {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*}
Time = 0.20 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.55 \[ \int \frac {\cos (x) (-3+2 \sin (x))}{2-3 \sin (x)+\sin ^2(x)} \, dx=2 (\text {arctanh}(3-2 \sin (x))+\log (1-\sin (x))) \]
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Time = 0.30 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09
method | result | size |
derivativedivides | \(\ln \left (2-3 \sin \left (x \right )+\sin ^{2}\left (x \right )\right )\) | \(12\) |
default | \(\ln \left (2-3 \sin \left (x \right )+\sin ^{2}\left (x \right )\right )\) | \(12\) |
risch | \(-2 i x +2 \ln \left ({\mathrm e}^{i x}-i\right )+\ln \left (-4 i {\mathrm e}^{i x}+{\mathrm e}^{2 i x}-1\right )\) | \(33\) |
norman | \(2 \ln \left (\tan \left (\frac {x}{2}\right )-1\right )-2 \ln \left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )+\ln \left (\tan ^{2}\left (\frac {x}{2}\right )-\tan \left (\frac {x}{2}\right )+1\right )\) | \(37\) |
parallelrisch | \(2 \ln \left (-\cot \left (x \right )+\csc \left (x \right )-1\right )-2 \ln \left (\frac {1}{\cos \left (x \right )+1}\right )+\ln \left (\frac {-\sin \left (x \right )+2}{4 \cos \left (x \right )+4}\right )\) | \(38\) |
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Time = 0.26 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.36 \[ \int \frac {\cos (x) (-3+2 \sin (x))}{2-3 \sin (x)+\sin ^2(x)} \, dx=\log \left (-\frac {1}{2} \, \sin \left (x\right ) + 1\right ) + \log \left (-\sin \left (x\right ) + 1\right ) \]
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Time = 0.09 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09 \[ \int \frac {\cos (x) (-3+2 \sin (x))}{2-3 \sin (x)+\sin ^2(x)} \, dx=\log {\left (\sin {\left (x \right )} - 2 \right )} + \log {\left (\sin {\left (x \right )} - 1 \right )} \]
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Time = 0.19 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {\cos (x) (-3+2 \sin (x))}{2-3 \sin (x)+\sin ^2(x)} \, dx=\log \left (\sin \left (x\right )^{2} - 3 \, \sin \left (x\right ) + 2\right ) \]
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Time = 0.32 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.36 \[ \int \frac {\cos (x) (-3+2 \sin (x))}{2-3 \sin (x)+\sin ^2(x)} \, dx=\log \left (-\sin \left (x\right ) + 2\right ) + \log \left (-\sin \left (x\right ) + 1\right ) \]
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Time = 0.09 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {\cos (x) (-3+2 \sin (x))}{2-3 \sin (x)+\sin ^2(x)} \, dx=\ln \left ({\sin \left (x\right )}^2-3\,\sin \left (x\right )+2\right ) \]
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