Integrand size = 7, antiderivative size = 7 \[ \int \cos (x) \csc (2 x) \, dx=-\frac {1}{2} \text {arctanh}(\cos (x)) \]
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Time = 0.01 (sec) , antiderivative size = 7, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4372, 3855} \[ \int \cos (x) \csc (2 x) \, dx=-\frac {1}{2} \text {arctanh}(\cos (x)) \]
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Rule 3855
Rule 4372
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \int \csc (x) \, dx \\ & = -\frac {1}{2} \text {arctanh}(\cos (x)) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(21\) vs. \(2(7)=14\).
Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 3.00 \[ \int \cos (x) \csc (2 x) \, dx=\frac {1}{2} \left (-\log \left (\cos \left (\frac {x}{2}\right )\right )+\log \left (\sin \left (\frac {x}{2}\right )\right )\right ) \]
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Time = 1.41 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.29
method | result | size |
default | \(-\frac {\ln \left (\cot \left (x \right )+\csc \left (x \right )\right )}{2}\) | \(9\) |
risch | \(-\frac {\ln \left ({\mathrm e}^{i x}+1\right )}{2}+\frac {\ln \left ({\mathrm e}^{i x}-1\right )}{2}\) | \(22\) |
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Leaf count of result is larger than twice the leaf count of optimal. 19 vs. \(2 (5) = 10\).
Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 2.71 \[ \int \cos (x) \csc (2 x) \, dx=-\frac {1}{4} \, \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + \frac {1}{4} \, \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 15 vs. \(2 (7) = 14\).
Time = 0.43 (sec) , antiderivative size = 15, normalized size of antiderivative = 2.14 \[ \int \cos (x) \csc (2 x) \, dx=\frac {\log {\left (\cos {\left (x \right )} - 1 \right )}}{4} - \frac {\log {\left (\cos {\left (x \right )} + 1 \right )}}{4} \]
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Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (5) = 10\).
Time = 0.22 (sec) , antiderivative size = 35, normalized size of antiderivative = 5.00 \[ \int \cos (x) \csc (2 x) \, dx=-\frac {1}{4} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) + \frac {1}{4} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 17 vs. \(2 (5) = 10\).
Time = 0.26 (sec) , antiderivative size = 17, normalized size of antiderivative = 2.43 \[ \int \cos (x) \csc (2 x) \, dx=-\frac {1}{4} \, \log \left (\cos \left (x\right ) + 1\right ) + \frac {1}{4} \, \log \left (-\cos \left (x\right ) + 1\right ) \]
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Time = 0.04 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.71 \[ \int \cos (x) \csc (2 x) \, dx=-\frac {\mathrm {atanh}\left (\cos \left (x\right )\right )}{2} \]
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