Integrand size = 15, antiderivative size = 24 \[ \int \cot (x) \sqrt {a+a \tan ^2(x)} \, dx=-\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a \sec ^2(x)}}{\sqrt {a}}\right ) \]
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Time = 0.09 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {3738, 4209, 65, 213} \[ \int \cot (x) \sqrt {a+a \tan ^2(x)} \, dx=-\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a \sec ^2(x)}}{\sqrt {a}}\right ) \]
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Rule 65
Rule 213
Rule 3738
Rule 4209
Rubi steps \begin{align*} \text {integral}& = \int \cot (x) \sqrt {a \sec ^2(x)} \, dx \\ & = \frac {1}{2} a \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a x}} \, dx,x,\sec ^2(x)\right ) \\ & = \text {Subst}\left (\int \frac {1}{-1+\frac {x^2}{a}} \, dx,x,\sqrt {a \sec ^2(x)}\right ) \\ & = -\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a \sec ^2(x)}}{\sqrt {a}}\right ) \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.33 \[ \int \cot (x) \sqrt {a+a \tan ^2(x)} \, dx=\arctan \left (\sqrt {-\cos ^2(x)}\right ) \sqrt {-\cos ^2(x)} \sqrt {a \sec ^2(x)} \]
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Time = 0.37 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83
method | result | size |
default | \(\ln \left (-\cot \left (x \right )+\csc \left (x \right )\right ) \cos \left (x \right ) \sqrt {a \sec \left (x \right )^{2}}\) | \(20\) |
risch | \(-2 \sqrt {\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}+1\right ) \cos \left (x \right )+2 \sqrt {\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}-1\right ) \cos \left (x \right )\) | \(62\) |
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none
Time = 0.27 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.62 \[ \int \cot (x) \sqrt {a+a \tan ^2(x)} \, dx=\left [\frac {1}{2} \, \sqrt {a} \log \left (\frac {a \tan \left (x\right )^{2} - 2 \, \sqrt {a \tan \left (x\right )^{2} + a} \sqrt {a} + 2 \, a}{\tan \left (x\right )^{2}}\right ), \sqrt {-a} \arctan \left (\frac {\sqrt {a \tan \left (x\right )^{2} + a} \sqrt {-a}}{a}\right )\right ] \]
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\[ \int \cot (x) \sqrt {a+a \tan ^2(x)} \, dx=\int \sqrt {a \left (\tan ^{2}{\left (x \right )} + 1\right )} \cot {\left (x \right )}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 38 vs. \(2 (18) = 36\).
Time = 0.54 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.58 \[ \int \cot (x) \sqrt {a+a \tan ^2(x)} \, dx=-\frac {1}{2} \, \sqrt {a} {\left (\log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) - \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right )\right )} \]
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none
Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \cot (x) \sqrt {a+a \tan ^2(x)} \, dx=\frac {a \arctan \left (\frac {\sqrt {a \tan \left (x\right )^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} \]
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Time = 0.19 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.50 \[ \int \cot (x) \sqrt {a+a \tan ^2(x)} \, dx=-\sqrt {a}\,\mathrm {atanh}\left (\sqrt {\frac {1}{{\cos \left (x\right )}^2}}\right ) \]
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