Integrand size = 10, antiderivative size = 14 \[ \int \sqrt {-1+\csc ^2(x)} \, dx=\sqrt {\cot ^2(x)} \log (\sin (x)) \tan (x) \]
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Time = 0.03 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4206, 3739, 3556} \[ \int \sqrt {-1+\csc ^2(x)} \, dx=\tan (x) \sqrt {\cot ^2(x)} \log (\sin (x)) \]
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Rule 3556
Rule 3739
Rule 4206
Rubi steps \begin{align*} \text {integral}& = \int \sqrt {\cot ^2(x)} \, dx \\ & = \left (\sqrt {\cot ^2(x)} \tan (x)\right ) \int \cot (x) \, dx \\ & = \sqrt {\cot ^2(x)} \log (\sin (x)) \tan (x) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.29 \[ \int \sqrt {-1+\csc ^2(x)} \, dx=\sqrt {\cot ^2(x)} (\log (\cos (x))+\log (\tan (x))) \tan (x) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(33\) vs. \(2(12)=24\).
Time = 0.57 (sec) , antiderivative size = 34, normalized size of antiderivative = 2.43
method | result | size |
default | \(-\frac {\tan \left (x \right ) \left (\ln \left (\frac {2}{\cos \left (x \right )+1}\right )-\ln \left (\csc \left (x \right )-\cot \left (x \right )\right )\right ) \sqrt {\cot \left (x \right )^{2}}\, \sqrt {4}}{2}\) | \(34\) |
risch | \(-\frac {\left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {\left ({\mathrm e}^{2 i x}+1\right )^{2}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, x}{{\mathrm e}^{2 i x}+1}-\frac {i \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {\left ({\mathrm e}^{2 i x}+1\right )^{2}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{2 i x}-1\right )}{{\mathrm e}^{2 i x}+1}\) | \(92\) |
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Time = 0.25 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.50 \[ \int \sqrt {-1+\csc ^2(x)} \, dx=-\log \left (\frac {1}{2} \, \sin \left (x\right )\right ) \]
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\[ \int \sqrt {-1+\csc ^2(x)} \, dx=\int \sqrt {\csc ^{2}{\left (x \right )} - 1}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93 \[ \int \sqrt {-1+\csc ^2(x)} \, dx=-\frac {1}{2} \, \log \left (\tan \left (x\right )^{2} + 1\right ) + \log \left (\tan \left (x\right )\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (12) = 24\).
Time = 0.33 (sec) , antiderivative size = 44, normalized size of antiderivative = 3.14 \[ \int \sqrt {-1+\csc ^2(x)} \, dx=\frac {1}{2} \, {\left (2 \, \log \left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right ) \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )^{4} - 1\right ) - \log \left (\tan \left (\frac {1}{2} \, x\right )^{2}\right ) \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )^{4} - 1\right )\right )} \mathrm {sgn}\left (\sin \left (x\right )\right ) \]
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Timed out. \[ \int \sqrt {-1+\csc ^2(x)} \, dx=\int \sqrt {\frac {1}{{\sin \left (x\right )}^2}-1} \,d x \]
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