Integrand size = 10, antiderivative size = 15 \[ \int \frac {1}{\sqrt {-1+\csc ^2(x)}} \, dx=-\frac {\cot (x) \log (\cos (x))}{\sqrt {\cot ^2(x)}} \]
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Time = 0.02 (sec) , antiderivative size = 15, 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} \[ \int \frac {1}{\sqrt {-1+\csc ^2(x)}} \, dx=-\frac {\cot (x) \log (\cos (x))}{\sqrt {\cot ^2(x)}} \]
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Rule 3556
Rule 3739
Rule 4206
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\sqrt {\cot ^2(x)}} \, dx \\ & = \frac {\cot (x) \int \tan (x) \, dx}{\sqrt {\cot ^2(x)}} \\ & = -\frac {\cot (x) \log (\cos (x))}{\sqrt {\cot ^2(x)}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {-1+\csc ^2(x)}} \, dx=-\frac {\cot (x) \log (\cos (x))}{\sqrt {\cot ^2(x)}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.56 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.60
method | result | size |
default | \(-\frac {\operatorname {csgn}\left (\cot \left (x \right )\right ) \left (\ln \left (\csc \left (x \right )-\cot \left (x \right )+1\right )+\ln \left (-\cot \left (x \right )+\csc \left (x \right )-1\right )-\ln \left (\frac {2}{\cos \left (x \right )+1}\right )\right ) \sqrt {4}}{2}\) | \(39\) |
risch | \(-\frac {\left ({\mathrm e}^{2 i x}+1\right ) x}{\sqrt {-\frac {\left ({\mathrm e}^{2 i x}+1\right )^{2}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}-1\right )}-\frac {i \left ({\mathrm e}^{2 i x}+1\right ) \ln \left ({\mathrm e}^{2 i x}+1\right )}{\sqrt {-\frac {\left ({\mathrm e}^{2 i x}+1\right )^{2}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}-1\right )}\) | \(92\) |
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none
Time = 0.24 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.33 \[ \int \frac {1}{\sqrt {-1+\csc ^2(x)}} \, dx=\log \left (-\cos \left (x\right )\right ) \]
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\[ \int \frac {1}{\sqrt {-1+\csc ^2(x)}} \, dx=\int \frac {1}{\sqrt {\csc ^{2}{\left (x \right )} - 1}}\, dx \]
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none
Time = 0.28 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.60 \[ \int \frac {1}{\sqrt {-1+\csc ^2(x)}} \, dx=\frac {1}{2} \, \log \left (\tan \left (x\right )^{2} + 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (13) = 26\).
Time = 0.30 (sec) , antiderivative size = 35, normalized size of antiderivative = 2.33 \[ \int \frac {1}{\sqrt {-1+\csc ^2(x)}} \, dx=-\frac {1}{2} \, \log \left (\tan \left (\frac {1}{2} \, x\right )^{2} + \frac {1}{\tan \left (\frac {1}{2} \, x\right )^{2}} + 2\right ) + \frac {1}{2} \, \log \left (\tan \left (\frac {1}{2} \, x\right )^{2} + \frac {1}{\tan \left (\frac {1}{2} \, x\right )^{2}} - 2\right ) \]
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Timed out. \[ \int \frac {1}{\sqrt {-1+\csc ^2(x)}} \, dx=\int \frac {1}{\sqrt {\frac {1}{{\sin \left (x\right )}^2}-1}} \,d x \]
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