Integrand size = 10, antiderivative size = 15 \[ \int \csc (2 x) (\cos (x)+\sin (x)) \, dx=-\frac {1}{2} \text {arctanh}(\cos (x))+\frac {1}{2} \text {arctanh}(\sin (x)) \]
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Time = 0.05 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4486, 4372, 3855, 4373} \[ \int \csc (2 x) (\cos (x)+\sin (x)) \, dx=\frac {1}{2} \text {arctanh}(\sin (x))-\frac {1}{2} \text {arctanh}(\cos (x)) \]
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Rule 3855
Rule 4372
Rule 4373
Rule 4486
Rubi steps \begin{align*} \text {integral}& = \int (\cos (x) \csc (2 x)+\csc (2 x) \sin (x)) \, dx \\ & = \int \cos (x) \csc (2 x) \, dx+\int \csc (2 x) \sin (x) \, dx \\ & = \frac {1}{2} \int \csc (x) \, dx+\frac {1}{2} \int \sec (x) \, dx \\ & = -\frac {1}{2} \text {arctanh}(\cos (x))+\frac {1}{2} \text {arctanh}(\sin (x)) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 2.00 \[ \int \csc (2 x) (\cos (x)+\sin (x)) \, dx=\frac {1}{2} \text {arctanh}(\sin (x))-\frac {1}{2} \log \left (\cos \left (\frac {x}{2}\right )\right )+\frac {1}{2} \log \left (\sin \left (\frac {x}{2}\right )\right ) \]
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Time = 1.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.20
method | result | size |
parts | \(-\frac {\ln \left (\cot \left (x \right )+\csc \left (x \right )\right )}{2}+\frac {\ln \left (\sec \left (x \right )+\tan \left (x \right )\right )}{2}\) | \(18\) |
default | \(\frac {\ln \left (\sec \left (x \right )+\tan \left (x \right )\right )}{2}+\frac {\ln \left (\csc \left (x \right )-\cot \left (x \right )\right )}{2}\) | \(20\) |
risch | \(-\frac {\ln \left ({\mathrm e}^{2 i x}+\left (1-i\right ) {\mathrm e}^{i x}-i\right )}{2}+\frac {\ln \left ({\mathrm e}^{2 i x}+\left (-1+i\right ) {\mathrm e}^{i x}-i\right )}{2}\) | \(42\) |
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Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (11) = 22\).
Time = 0.25 (sec) , antiderivative size = 35, normalized size of antiderivative = 2.33 \[ \int \csc (2 x) (\cos (x)+\sin (x)) \, dx=-\frac {1}{4} \, \log \left (-\frac {1}{2} \, {\left (\cos \left (x\right ) + 1\right )} \sin \left (x\right ) + \frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + \frac {1}{4} \, \log \left (-\frac {1}{2} \, {\left (\cos \left (x\right ) - 1\right )} \sin \left (x\right ) - \frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 32 vs. \(2 (12) = 24\).
Time = 0.73 (sec) , antiderivative size = 32, normalized size of antiderivative = 2.13 \[ \int \csc (2 x) (\cos (x)+\sin (x)) \, dx=- \frac {\log {\left (\sin {\left (x \right )} - 1 \right )}}{4} + \frac {\log {\left (\sin {\left (x \right )} + 1 \right )}}{4} + \frac {\log {\left (\cos {\left (x \right )} - 1 \right )}}{4} - \frac {\log {\left (\cos {\left (x \right )} + 1 \right )}}{4} \]
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Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (11) = 22\).
Time = 0.28 (sec) , antiderivative size = 69, normalized size of antiderivative = 4.60 \[ \int \csc (2 x) (\cos (x)+\sin (x)) \, dx=-\frac {1}{4} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) + \frac {1}{4} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) + \frac {1}{4} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \sin \left (x\right ) + 1\right ) - \frac {1}{4} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \sin \left (x\right ) + 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 29 vs. \(2 (11) = 22\).
Time = 0.29 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.93 \[ \int \csc (2 x) (\cos (x)+\sin (x)) \, dx=\frac {1}{2} \, \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) + 1 \right |}\right ) - \frac {1}{2} \, \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) - 1 \right |}\right ) + \frac {1}{2} \, \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right ) \]
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Time = 27.14 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.60 \[ \int \csc (2 x) (\cos (x)+\sin (x)) \, dx=\frac {\ln \left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+\mathrm {tan}\left (\frac {x}{2}\right )\right )}{2}-\frac {\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )-1\right )}{2} \]
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