Integrand size = 10, antiderivative size = 5 \[ \int \sqrt {1+\cot ^2(x)} \, dx=-\text {arcsinh}(\cot (x)) \]
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Time = 0.02 (sec) , antiderivative size = 5, 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 = {3738, 4207, 221} \[ \int \sqrt {1+\cot ^2(x)} \, dx=-\text {arcsinh}(\cot (x)) \]
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Rule 221
Rule 3738
Rule 4207
Rubi steps \begin{align*} \text {integral}& = \int \sqrt {\csc ^2(x)} \, dx \\ & = -\text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,\cot (x)\right ) \\ & = -\text {arcsinh}(\cot (x)) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(28\) vs. \(2(5)=10\).
Time = 0.03 (sec) , antiderivative size = 28, normalized size of antiderivative = 5.60 \[ \int \sqrt {1+\cot ^2(x)} \, dx=\sqrt {\csc ^2(x)} \left (-\log \left (\cos \left (\frac {x}{2}\right )\right )+\log \left (\sin \left (\frac {x}{2}\right )\right )\right ) \sin (x) \]
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Time = 0.03 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.20
method | result | size |
derivativedivides | \(-\operatorname {arcsinh}\left (\cot \left (x \right )\right )\) | \(6\) |
default | \(-\operatorname {arcsinh}\left (\cot \left (x \right )\right )\) | \(6\) |
risch | \(-2 \sqrt {-\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}+1\right ) \sin \left (x \right )+2 \sqrt {-\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}-1\right ) \sin \left (x \right )\) | \(62\) |
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Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (5) = 10\).
Time = 0.29 (sec) , antiderivative size = 53, normalized size of antiderivative = 10.60 \[ \int \sqrt {1+\cot ^2(x)} \, dx=-\frac {1}{2} \, \log \left (\frac {1}{2} \, \sqrt {2} \sqrt {-\frac {1}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) + 1\right ) + \frac {1}{2} \, \log \left (-\frac {1}{2} \, \sqrt {2} \sqrt {-\frac {1}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) + 1\right ) \]
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\[ \int \sqrt {1+\cot ^2(x)} \, dx=\int \sqrt {\cot ^{2}{\left (x \right )} + 1}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (5) = 10\).
Time = 0.36 (sec) , antiderivative size = 35, normalized size of antiderivative = 7.00 \[ \int \sqrt {1+\cot ^2(x)} \, dx=-\frac {1}{2} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) + \frac {1}{2} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) \]
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none
Time = 0.27 (sec) , antiderivative size = 10, normalized size of antiderivative = 2.00 \[ \int \sqrt {1+\cot ^2(x)} \, dx=\log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right ) \mathrm {sgn}\left (\sin \left (x\right )\right ) \]
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Time = 12.97 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \sqrt {1+\cot ^2(x)} \, dx=-\mathrm {asinh}\left (\mathrm {cot}\left (x\right )\right ) \]
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