Integrand size = 7, antiderivative size = 7 \[ \int \csc (2 x) \sin (x) \, dx=\frac {1}{2} \text {arctanh}(\sin (x)) \]
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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 = {4373, 3855} \[ \int \csc (2 x) \sin (x) \, dx=\frac {1}{2} \text {arctanh}(\sin (x)) \]
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Rule 3855
Rule 4373
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \int \sec (x) \, dx \\ & = \frac {1}{2} \text {arctanh}(\sin (x)) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 7, normalized size of antiderivative = 1.00 \[ \int \csc (2 x) \sin (x) \, dx=\frac {1}{2} \text {arctanh}(\sin (x)) \]
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Time = 0.29 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.29
method | result | size |
default | \(\frac {\ln \left (\sec \left (x \right )+\tan \left (x \right )\right )}{2}\) | \(9\) |
risch | \(\frac {\ln \left (i+{\mathrm e}^{i x}\right )}{2}-\frac {\ln \left ({\mathrm e}^{i x}-i\right )}{2}\) | \(24\) |
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Leaf count of result is larger than twice the leaf count of optimal. 17 vs. \(2 (5) = 10\).
Time = 0.24 (sec) , antiderivative size = 17, normalized size of antiderivative = 2.43 \[ \int \csc (2 x) \sin (x) \, dx=\frac {1}{4} \, \log \left (\sin \left (x\right ) + 1\right ) - \frac {1}{4} \, \log \left (-\sin \left (x\right ) + 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 15 vs. \(2 (5) = 10\).
Time = 0.40 (sec) , antiderivative size = 15, normalized size of antiderivative = 2.14 \[ \int \csc (2 x) \sin (x) \, dx=- \frac {\log {\left (\sin {\left (x \right )} - 1 \right )}}{4} + \frac {\log {\left (\sin {\left (x \right )} + 1 \right )}}{4} \]
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Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (5) = 10\).
Time = 0.30 (sec) , antiderivative size = 35, normalized size of antiderivative = 5.00 \[ \int \csc (2 x) \sin (x) \, dx=\frac {1}{4} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \sin \left (x\right ) + 1\right ) - \frac {1}{4} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \sin \left (x\right ) + 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 17 vs. \(2 (5) = 10\).
Time = 0.26 (sec) , antiderivative size = 17, normalized size of antiderivative = 2.43 \[ \int \csc (2 x) \sin (x) \, dx=\frac {1}{4} \, \log \left (\sin \left (x\right ) + 1\right ) - \frac {1}{4} \, \log \left (-\sin \left (x\right ) + 1\right ) \]
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Time = 26.99 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.71 \[ \int \csc (2 x) \sin (x) \, dx=\frac {\mathrm {atanh}\left (\sin \left (x\right )\right )}{2} \]
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