Integrand size = 17, antiderivative size = 11 \[ \int \cos (3 x) \csc ^2(x) \sec ^3(x) \sin (2 x) \, dx=6 \log (\cos (x))+2 \log (\sin (x)) \]
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Time = 0.09 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {4450, 457, 78} \[ \int \cos (3 x) \csc ^2(x) \sec ^3(x) \sin (2 x) \, dx=2 \log (\sin (x))+6 \log (\cos (x)) \]
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Rule 78
Rule 457
Rule 4450
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {2-8 x^2}{x \left (1-x^2\right )} \, dx,x,\sin (x)\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {2-8 x}{(1-x) x} \, dx,x,\sin ^2(x)\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {6}{-1+x}+\frac {2}{x}\right ) \, dx,x,\sin ^2(x)\right ) \\ & = 6 \log (\cos (x))+2 \log (\sin (x)) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \cos (3 x) \csc ^2(x) \sec ^3(x) \sin (2 x) \, dx=8 \log (\cos (x))+2 \log (\tan (x)) \]
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Time = 0.18 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09
\[-6 \ln \left (\tan \left (x \right )\right )+8 \ln \left (\sin \left (x \right )\right )\]
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Time = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.55 \[ \int \cos (3 x) \csc ^2(x) \sec ^3(x) \sin (2 x) \, dx=3 \, \log \left (\cos \left (x\right )^{2}\right ) + \log \left (-\frac {1}{4} \, \cos \left (x\right )^{2} + \frac {1}{4}\right ) \]
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Time = 92.25 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.36 \[ \int \cos (3 x) \csc ^2(x) \sec ^3(x) \sin (2 x) \, dx=3 \log {\left (\sin ^{2}{\left (x \right )} - 1 \right )} + 2 \log {\left (\sin {\left (x \right )} \right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (11) = 22\).
Time = 0.28 (sec) , antiderivative size = 54, normalized size of antiderivative = 4.91 \[ \int \cos (3 x) \csc ^2(x) \sec ^3(x) \sin (2 x) \, dx=3 \, \log \left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1\right ) + \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) + \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) \]
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Time = 0.27 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.45 \[ \int \cos (3 x) \csc ^2(x) \sec ^3(x) \sin (2 x) \, dx=\log \left (-\cos \left (x\right )^{2} + 1\right ) + 6 \, \log \left ({\left | \cos \left (x\right ) \right |}\right ) \]
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Time = 16.29 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \cos (3 x) \csc ^2(x) \sec ^3(x) \sin (2 x) \, dx=6\,\ln \left (\cos \left (x\right )\right )+\ln \left ({\sin \left (x\right )}^2\right ) \]
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