Integrand size = 9, antiderivative size = 2 \[ \int \frac {1}{\csc (x)-\sin (x)} \, dx=\sec (x) \]
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Time = 0.02 (sec) , antiderivative size = 2, 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 = {4482, 2686, 8} \[ \int \frac {1}{\csc (x)-\sin (x)} \, dx=\sec (x) \]
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Rule 8
Rule 2686
Rule 4482
Rubi steps \begin{align*} \text {integral}& = \int \sec (x) \tan (x) \, dx \\ & = \text {Subst}(\int 1 \, dx,x,\sec (x)) \\ & = \sec (x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 2, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\csc (x)-\sin (x)} \, dx=\sec (x) \]
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Time = 0.40 (sec) , antiderivative size = 5, normalized size of antiderivative = 2.50
method | result | size |
default | \(\frac {1}{\cos \left (x \right )}\) | \(5\) |
parallelrisch | \(1+\sec \left (x \right )\) | \(5\) |
norman | \(-\frac {2}{\tan \left (\frac {x}{2}\right )^{2}-1}\) | \(13\) |
risch | \(\frac {2 \,{\mathrm e}^{i x}}{{\mathrm e}^{2 i x}+1}\) | \(17\) |
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none
Time = 0.23 (sec) , antiderivative size = 4, normalized size of antiderivative = 2.00 \[ \int \frac {1}{\csc (x)-\sin (x)} \, dx=\frac {1}{\cos \left (x\right )} \]
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\[ \int \frac {1}{\csc (x)-\sin (x)} \, dx=\int \frac {1}{- \sin {\left (x \right )} + \csc {\left (x \right )}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 17 vs. \(2 (2) = 4\).
Time = 0.23 (sec) , antiderivative size = 17, normalized size of antiderivative = 8.50 \[ \int \frac {1}{\csc (x)-\sin (x)} \, dx=-\frac {2}{\frac {\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - 1} \]
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Leaf count of result is larger than twice the leaf count of optimal. 17 vs. \(2 (2) = 4\).
Time = 0.27 (sec) , antiderivative size = 17, normalized size of antiderivative = 8.50 \[ \int \frac {1}{\csc (x)-\sin (x)} \, dx=\frac {2}{\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1} + 1} \]
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Time = 29.31 (sec) , antiderivative size = 12, normalized size of antiderivative = 6.00 \[ \int \frac {1}{\csc (x)-\sin (x)} \, dx=-\frac {2}{{\mathrm {tan}\left (\frac {x}{2}\right )}^2-1} \]
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