Integrand size = 9, antiderivative size = 26 \[ \int \cot ^2(x) \csc ^3(x) \, dx=\frac {1}{8} \text {arctanh}(\cos (x))+\frac {1}{8} \cot (x) \csc (x)-\frac {1}{4} \cot (x) \csc ^3(x) \]
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Time = 0.03 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2691, 3853, 3855} \[ \int \cot ^2(x) \csc ^3(x) \, dx=\frac {1}{8} \text {arctanh}(\cos (x))-\frac {1}{4} \cot (x) \csc ^3(x)+\frac {1}{8} \cot (x) \csc (x) \]
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Rule 2691
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{4} \cot (x) \csc ^3(x)-\frac {1}{4} \int \csc ^3(x) \, dx \\ & = \frac {1}{8} \cot (x) \csc (x)-\frac {1}{4} \cot (x) \csc ^3(x)-\frac {1}{8} \int \csc (x) \, dx \\ & = \frac {1}{8} \text {arctanh}(\cos (x))+\frac {1}{8} \cot (x) \csc (x)-\frac {1}{4} \cot (x) \csc ^3(x) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(71\) vs. \(2(26)=52\).
Time = 0.03 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.73 \[ \int \cot ^2(x) \csc ^3(x) \, dx=\frac {1}{32} \csc ^2\left (\frac {x}{2}\right )-\frac {1}{64} \csc ^4\left (\frac {x}{2}\right )+\frac {1}{8} \log \left (\cos \left (\frac {x}{2}\right )\right )-\frac {1}{8} \log \left (\sin \left (\frac {x}{2}\right )\right )-\frac {1}{32} \sec ^2\left (\frac {x}{2}\right )+\frac {1}{64} \sec ^4\left (\frac {x}{2}\right ) \]
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Time = 0.13 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.38
method | result | size |
default | \(-\frac {\cos ^{3}\left (x \right )}{4 \sin \left (x \right )^{4}}-\frac {\cos ^{3}\left (x \right )}{8 \sin \left (x \right )^{2}}-\frac {\cos \left (x \right )}{8}-\frac {\ln \left (\csc \left (x \right )-\cot \left (x \right )\right )}{8}\) | \(36\) |
risch | \(-\frac {{\mathrm e}^{7 i x}+7 \,{\mathrm e}^{5 i x}+7 \,{\mathrm e}^{3 i x}+{\mathrm e}^{i x}}{4 \left ({\mathrm e}^{2 i x}-1\right )^{4}}-\frac {\ln \left ({\mathrm e}^{i x}-1\right )}{8}+\frac {\ln \left ({\mathrm e}^{i x}+1\right )}{8}\) | \(58\) |
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Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (20) = 40\).
Time = 0.25 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.62 \[ \int \cot ^2(x) \csc ^3(x) \, dx=-\frac {2 \, \cos \left (x\right )^{3} - {\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + {\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + 2 \, \cos \left (x\right )}{16 \, {\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )}} \]
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Time = 0.07 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.58 \[ \int \cot ^2(x) \csc ^3(x) \, dx=\frac {- \cos ^{3}{\left (x \right )} - \cos {\left (x \right )}}{8 \cos ^{4}{\left (x \right )} - 16 \cos ^{2}{\left (x \right )} + 8} - \frac {\log {\left (\cos {\left (x \right )} - 1 \right )}}{16} + \frac {\log {\left (\cos {\left (x \right )} + 1 \right )}}{16} \]
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none
Time = 0.19 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.46 \[ \int \cot ^2(x) \csc ^3(x) \, dx=-\frac {\cos \left (x\right )^{3} + \cos \left (x\right )}{8 \, {\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )}} + \frac {1}{16} \, \log \left (\cos \left (x\right ) + 1\right ) - \frac {1}{16} \, \log \left (\cos \left (x\right ) - 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (20) = 40\).
Time = 0.26 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.81 \[ \int \cot ^2(x) \csc ^3(x) \, dx=-\frac {\frac {1}{\cos \left (x\right )} + \cos \left (x\right )}{8 \, {\left ({\left (\frac {1}{\cos \left (x\right )} + \cos \left (x\right )\right )}^{2} - 4\right )}} + \frac {1}{32} \, \log \left ({\left | \frac {1}{\cos \left (x\right )} + \cos \left (x\right ) + 2 \right |}\right ) - \frac {1}{32} \, \log \left ({\left | \frac {1}{\cos \left (x\right )} + \cos \left (x\right ) - 2 \right |}\right ) \]
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Time = 0.34 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \cot ^2(x) \csc ^3(x) \, dx=\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^4}{64}-\frac {1}{64\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4}-\frac {\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{8} \]
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