Integrand size = 12, antiderivative size = 13 \[ \int \frac {\csc (x)}{\sec (x)-\tan (x)} \, dx=-\log (1-\sin (x))+\log (\sin (x)) \]
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Time = 0.07 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {4476, 2786, 36, 29, 31} \[ \int \frac {\csc (x)}{\sec (x)-\tan (x)} \, dx=\log (\sin (x))-\log (1-\sin (x)) \]
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Rule 29
Rule 31
Rule 36
Rule 2786
Rule 4476
Rubi steps \begin{align*} \text {integral}& = \int \frac {\cot (x)}{1-\sin (x)} \, dx \\ & = \text {Subst}\left (\int \frac {1}{x (1+x)} \, dx,x,-\sin (x)\right ) \\ & = \text {Subst}\left (\int \frac {1}{x} \, dx,x,-\sin (x)\right )-\text {Subst}\left (\int \frac {1}{1+x} \, dx,x,-\sin (x)\right ) \\ & = -\log (1-\sin (x))+\log (\sin (x)) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.69 \[ \int \frac {\csc (x)}{\sec (x)-\tan (x)} \, dx=-2 \log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )+\log (\sin (x)) \]
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Time = 0.32 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.62
method | result | size |
derivativedivides | \(-\ln \left (\csc \left (x \right )-1\right )\) | \(8\) |
default | \(-\ln \left (\csc \left (x \right )-1\right )\) | \(8\) |
risch | \(-2 \ln \left ({\mathrm e}^{i x}-i\right )+\ln \left ({\mathrm e}^{2 i x}-1\right )\) | \(21\) |
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none
Time = 0.24 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.15 \[ \int \frac {\csc (x)}{\sec (x)-\tan (x)} \, dx=\log \left (\frac {1}{2} \, \sin \left (x\right )\right ) - \log \left (-\sin \left (x\right ) + 1\right ) \]
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\[ \int \frac {\csc (x)}{\sec (x)-\tan (x)} \, dx=\int \frac {\csc {\left (x \right )}}{- \tan {\left (x \right )} + \sec {\left (x \right )}}\, dx \]
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none
Time = 0.20 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.92 \[ \int \frac {\csc (x)}{\sec (x)-\tan (x)} \, dx=-2 \, \log \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1} - 1\right ) + \log \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right ) \]
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none
Time = 0.28 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08 \[ \int \frac {\csc (x)}{\sec (x)-\tan (x)} \, dx=-\log \left (-\sin \left (x\right ) + 1\right ) + \log \left ({\left | \sin \left (x\right ) \right |}\right ) \]
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Time = 22.12 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.15 \[ \int \frac {\csc (x)}{\sec (x)-\tan (x)} \, dx=\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )-2\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )-1\right ) \]
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