Integrand size = 9, antiderivative size = 4 \[ \int \frac {1}{-\cos (x)+\sec (x)} \, dx=-\csc (x) \]
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Time = 0.02 (sec) , antiderivative size = 4, 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} \[ \int \frac {1}{-\cos (x)+\sec (x)} \, dx=-\csc (x) \]
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Rule 8
Rule 2686
Rule 4482
Rubi steps \begin{align*} \text {integral}& = \int \cot (x) \csc (x) \, dx \\ & = -\text {Subst}(\int 1 \, dx,x,\csc (x)) \\ & = -\csc (x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 4, normalized size of antiderivative = 1.00 \[ \int \frac {1}{-\cos (x)+\sec (x)} \, dx=-\csc (x) \]
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Time = 0.40 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.25
method | result | size |
parallelrisch | \(-\csc \left (x \right )\) | \(5\) |
default | \(-\frac {1}{\sin \left (x \right )}\) | \(7\) |
norman | \(\frac {-\frac {1}{2}-\frac {\tan \left (\frac {x}{2}\right )^{2}}{2}}{\tan \left (\frac {x}{2}\right )}\) | \(18\) |
risch | \(-\frac {2 i {\mathrm e}^{i x}}{{\mathrm e}^{2 i x}-1}\) | \(18\) |
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none
Time = 0.25 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.50 \[ \int \frac {1}{-\cos (x)+\sec (x)} \, dx=-\frac {1}{\sin \left (x\right )} \]
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\[ \int \frac {1}{-\cos (x)+\sec (x)} \, dx=- \int \frac {1}{\cos {\left (x \right )} - \sec {\left (x \right )}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 21 vs. \(2 (4) = 8\).
Time = 0.21 (sec) , antiderivative size = 21, normalized size of antiderivative = 5.25 \[ \int \frac {1}{-\cos (x)+\sec (x)} \, dx=-\frac {\cos \left (x\right ) + 1}{2 \, \sin \left (x\right )} - \frac {\sin \left (x\right )}{2 \, {\left (\cos \left (x\right ) + 1\right )}} \]
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none
Time = 0.26 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.50 \[ \int \frac {1}{-\cos (x)+\sec (x)} \, dx=-\frac {1}{\sin \left (x\right )} \]
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Time = 25.78 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.50 \[ \int \frac {1}{-\cos (x)+\sec (x)} \, dx=-\frac {1}{\sin \left (x\right )} \]
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