Integrand size = 9, antiderivative size = 7 \[ \int \csc ^2(x) \sec ^2(x) \, dx=-\cot (x)+\tan (x) \]
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Time = 0.02 (sec) , antiderivative size = 7, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2700, 14} \[ \int \csc ^2(x) \sec ^2(x) \, dx=\tan (x)-\cot (x) \]
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Rule 14
Rule 2700
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1+x^2}{x^2} \, dx,x,\tan (x)\right ) \\ & = \text {Subst}\left (\int \left (1+\frac {1}{x^2}\right ) \, dx,x,\tan (x)\right ) \\ & = -\cot (x)+\tan (x) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.86 \[ \int \csc ^2(x) \sec ^2(x) \, dx=-2 \cot (2 x) \]
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Time = 0.21 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.57
method | result | size |
parallelrisch | \(-2 \cot \left (x \right )+\sec \left (x \right ) \csc \left (x \right )\) | \(11\) |
default | \(\frac {1}{\cos \left (x \right ) \sin \left (x \right )}-2 \cot \left (x \right )\) | \(15\) |
risch | \(-\frac {4 i}{\left ({\mathrm e}^{2 i x}+1\right ) \left ({\mathrm e}^{2 i x}-1\right )}\) | \(22\) |
norman | \(\frac {\frac {1}{2}-3 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+\frac {\left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{2}}{\left (\tan ^{2}\left (\frac {x}{2}\right )-1\right ) \tan \left (\frac {x}{2}\right )}\) | \(36\) |
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Leaf count of result is larger than twice the leaf count of optimal. 18 vs. \(2 (7) = 14\).
Time = 0.23 (sec) , antiderivative size = 18, normalized size of antiderivative = 2.57 \[ \int \csc ^2(x) \sec ^2(x) \, dx=-\frac {2 \, \cos \left (x\right )^{2} - 1}{\cos \left (x\right ) \sin \left (x\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 12 vs. \(2 (5) = 10\).
Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.71 \[ \int \csc ^2(x) \sec ^2(x) \, dx=- \frac {2 \cos {\left (2 x \right )}}{\sin {\left (2 x \right )}} \]
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none
Time = 0.20 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.29 \[ \int \csc ^2(x) \sec ^2(x) \, dx=-\frac {1}{\tan \left (x\right )} + \tan \left (x\right ) \]
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none
Time = 0.29 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.29 \[ \int \csc ^2(x) \sec ^2(x) \, dx=-\frac {1}{\tan \left (x\right )} + \tan \left (x\right ) \]
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Time = 0.06 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.86 \[ \int \csc ^2(x) \sec ^2(x) \, dx=-2\,\mathrm {cot}\left (2\,x\right ) \]
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